import .open_subspace
import .lifts
import tactic
import algebraic_topology.fundamental_groupoid.basic
import category_theory.limits.shapes.comm_sq
import topology.unit_interval

noncomputable theory

universe u

variables {X : Type u} [topological_space X]
variables {X₁ : top_subspace X} {X₂ : top_subspace X}
variables {G : category_theory.Groupoid.{u u}}

def p₁ : proj_grpd X₁ (πₓ (Top.of X₁)) := self_grpd X₁
#print p₁
def p₂ : proj_grpd X₂ (πₓ (Top.of X₂)) := self_grpd X₂

#check πₓ (Top.of X₁)

-- def w (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G): proj_grpd (X₁ ∩ X₂) G := {
--   obj :=
--   begin
--     let i : (X₁ ∩ X₂) → X₁ := @open_incl X _ (X₁∩X₂) X₁ inter_sub,
--     refine _ ∘ i,
--     exact v₁.obj,
--   end,
--   path :=
--   begin

--   end
--   }

#check ![1,2]

def set.union_lift {α β} {s t : set α}
  (f : s → β) (g : t → β) : s ∪ t → β :=
begin
  classical,
  exact λ x, x.prop.by_cases (λ hx, f ⟨_, hx⟩) (λ hx, g ⟨_, hx⟩)
end
-- def set.union_lift {α β} {s t : set α} [decidable_pred (∈ s)] [decidable_pred (∈ t)]
--   (f : s → β) (g : t → β) : s ∪ t → β :=
-- begin
--   intro x, apply or.by_cases x.prop,
-- end

-- def setfunc_to_func {H : top_subspace X} {K : Type} (f : (H.carrier) → K) : H → K :=
-- begin
--   exact f,
-- end
#check category_theory.comm_sq

variables {f : X₁ → G} {g : X₂ → G}

def coincide {K : Type*} (f : X₁ → K) (g : X₂ → K):= ∀(y : X) (h1 : y ∈ X₁) (h2 : y ∈ X₂), f (b_incl y h1) = g (b_incl y h2)
#print coincide

lemma union_def_eq {x : X₁ ∪ X₂} (h : incl x ∈ X₁): (set.union_lift f g) x = f (b_incl (incl x) h) :=
begin
  rw set.union_lift, unfold_coes, have p : x.val ∈ X₁.carrier, cases x, exact h,
  rw or.by_cases, split_ifs, rotate, contradiction, rw b_incl, simp_rw incl, cases x, simp, refl,
end

lemma union_def_eq₂ {x : X₁ ∪ X₂} (h : incl x ∈ X₂) (pp : @coincide X _ X₁ X₂ G f g):
(set.union_lift f g) x = g (b_incl (incl x) h) :=
begin
  rw set.union_lift, unfold_coes, have p : x.val ∈ X₂.carrier, cases x, exact h,
  rw or.by_cases, split_ifs,
  {
    rw b_incl, simp_rw incl,have p2 : x.val ∈ (X₁∩X₂).carrier, split,assumption, assumption,
    cases x, simp,
    have s : ∀y:X, y ∈ X₁ → y ∈ X₂ →  f (b_incl y _)= g (b_incl y _), rotate 3,assumption, rotate 2, assumption,
    rw coincide at pp, exact pp, simp_rw b_incl at s,apply s, assumption, assumption,
  },rw b_incl, simp_rw incl, cases x, simp, refl,
end

-- Todo : commutative diagram --

notation `πₓ` := fundamental_groupoid.fundamental_groupoid_functor.obj
notation `π` := fundamental_groupoid.fundamental_groupoid_functor
notation `πₘ` := fundamental_groupoid.fundamental_groupoid_functor.map

theorem comm_px :
(@open_incl X _ X₁ (X₁∪X₂) union_sub) ∘ (@open_incl X _ (X₁∩X₂) X₁ inter_sub) =
(@open_incl X _ X₂ (X₁∪X₂) union_sub2) ∘ (@open_incl X _ (X₁∩X₂) X₂ inter_sub2) :=
begin
  ext1,simp_rw open_incl, rw function.comp, simp, cases x, simp,
end

theorem comm_πₓ : category_theory.comm_sq
 (grpd_induced_incl (@inter_sub X _ X₁ X₂))
 (grpd_induced_incl (@inter_sub2 X _ X₁ X₂))
 (grpd_induced_incl (@union_sub X _ X₁ X₂))
 (grpd_induced_incl (@union_sub2 X _ X₁ X₂)) :=
begin
  apply category_theory.functor.map_comm_sq,
  fconstructor, simp_rw top_subspace_incl, simp_rw open_incl, ext, simp only [Top.comp_app, continuous_map.coe_mk],
  unfold_coes, cases a, simp only [eq_self_iff_true],
end

def w₁ (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G) (a b: ↥(X₁ ∪ X₂))
(pab: path a b ) (h : ↑(set.range pab) ⊆ X₁.carrier ):
 set.union_lift v₁.obj v₂.obj a ⟶ set.union_lift v₁.obj v₂.obj b:=
begin
    rw path_incl_range_eq at h,
    let pXab := subspace_path_lift pab,
    let b₁ := path_lift_def pXab h,
    let s := v₁.map (p₁.path b₁),
    have aX1 : incl a ∈ X₁ := range_in_init pXab h,
    have bX2 : incl b ∈ X₁ := range_in_end pXab h,
    rw union_def_eq aX1, rw union_def_eq bX2, assumption,
end

def w₂ (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G) (a b: ↥(X₁ ∪ X₂))
(pab: path a b ) (h : ↑(set.range pab) ⊆ X₂.carrier )
(comm_v : category_theory.comm_sq (grpd_induced_incl (@inter_sub X _ X₁ X₂))  (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)) v₁ v₂)
: set.union_lift v₁.obj v₂.obj a ⟶ set.union_lift v₁.obj v₂.obj b:=
begin
    rw path_incl_range_eq at h,
    let pXab := subspace_path_lift pab,
    let b₂ := path_lift_def pXab h,
    let s := v₂.map (p₂.path b₂),
    have aX1 : incl a ∈ X₂ := range_in_init pXab h,
    have bX2 : incl b ∈ X₂ := range_in_end pXab h,
    have ph : coincide v₁.obj v₂.obj,
    {
      rw coincide, intros y hy1 hy2, cases comm_v,
      have hy12 : y ∈ X₁∩X₂ := ⟨hy1,hy2⟩,
      have p1 : b_incl y hy1 = (grpd_induced_incl (@inter_sub X _ X₁ X₂)).obj (b_incl y hy12),
      {
        rw grpd_induced_incl, tauto,
      },
      have p2 : b_incl y hy2 = (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)).obj (b_incl y hy12),
      {
        rw grpd_induced_incl, finish,
      },
      rw p1, rw p2, simp_rw ←category_theory.functor.comp_obj,
      have k: grpd_induced_incl inter_sub ⋙ v₁ = grpd_induced_incl inter_sub2 ⋙ v₂:= by assumption,
      rw k,
    },
    rw union_def_eq₂ aX1 ph, rw union_def_eq₂ bX2 ph, assumption,
end

notation `I` := unit_interval

lemma uniformity_metric_space {X: Type} [metric_space X] (V : set (X × X)): V ∈ uniformity X ↔ ∃ ε > 0, { p : X × X | dist p.1 p.2 < ε } ⊆ V :=
begin
  rw ← filter.has_basis.mem_iff,
  apply metric.uniformity_basis_dist,
end

def w (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G)
(comm_v : category_theory.comm_sq (grpd_induced_incl (@inter_sub X _ X₁ X₂)) (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)) v₁ v₂)
: proj_grpd (X₁ ∪ X₂) G :=
{
  obj := set.union_lift v₁.obj v₂.obj,
  path := begin
  {
    intros a b pab,
    set S₁ := {x : I | ↑(pab.to_fun x) ∈ X₁.carrier} with hs1,
    set S₂ := {x : I | ↑(pab.to_fun x) ∈ X₂.carrier} with hs2,
    have pcover : set.univ ⊆ (S₁ ∪ S₂),
    sorry{
       intro s,intro hs,
       set S₃ := S₁ ∪ S₂ with hs3,
       have hs3 : S₃ = {x : I | ↑(pab.to_fun x) ∈ X₁∪X₂}:= by tauto,
       rw hs3, simp only [set.mem_union_eq, set.mem_set_of_eq], unfold_coes,
       have k : ∀p, @incl X _ (X₁∪X₂) p ∈ X₁∪X₂,
       {
        intro p, cases p, rw incl, tauto,
       }, apply k,
    },
    have popen1 : is_open S₁,
    sorry{
      have k: S₁ = pab.to_fun ⁻¹' (incl ⁻¹' X₁.carrier),
      {
        rw hs1, rw set.preimage, rw set.preimage, refl,
      }, rw k, apply is_open.preimage, cases pab, cases pab__to_continuous_map, assumption,
      apply is_open.preimage, exact incl_continuous, exact X₁.univ_open,
    },
    have popen2 : is_open S₂,
    sorry{
      have k: S₂ = pab.to_fun ⁻¹' (incl ⁻¹' X₂.carrier),
      {
        rw hs2, rw set.preimage, rw set.preimage, refl,
      }, rw k, apply is_open.preimage, cases pab, cases pab__to_continuous_map, assumption,
      apply is_open.preimage, exact incl_continuous, exact X₂.univ_open,
    },
    set c := ![S₁, S₂] with hc,
    have hc₁ : ∀ (i) , is_open (c i),
    sorry{
      intro i, fin_cases i, assumption, assumption,
    },
    have hc₂ : set.univ ⊆ ⋃ i, c i,
    sorry{
      have k : (⋃ i, c i) = S₁ ∪ S₂,
      {
        rw hc, simp only [], ext, simp [fin.exists_fin_two],
      },
      rw k, assumption,
    },
    have hcover := lebesgue_number_lemma (compact_univ) hc₁ hc₂,
    choose n H pn using hcover,
    sorry,

    -- have n : set (↥I × ↥I) := sorry,
    -- have H : n ∈ uniformity ↥I := sorry,
    -- have pn : ∀ (x : ↥I), x ∈ set.univ → (∃ (i : fin (1:ℕ).succ), {y : ↥I | (x, y) ∈ n} ⊆ (λ (i : fin (1:ℕ).succ), c i) i) := sorry,
  },
end
}

#print w


example (a b c d e f : Nat) (h : [a, b, c] = [d, e, f]) : a + b + c = d + e + f := by
  injections
  simp [*]


import Monads.Functor
import Monads.Applicative
import Monads.Monad

namespace Monads

  inductive Maybe (α : Type u) where
  | none : Maybe α
  | some : α → Maybe α
    deriving Repr

  namespace Maybe
    section Functor
      def map {α β : Type u} : (α → β) → Maybe α → Maybe β
      | f, none => none
      | f, (some a) => (some $ f a)

      theorem map_none {f : α → β} : map f none = none := by rfl
      theorem map_some {f : α → β} {x : α} :  map f (some x) = (some (f x)) := by rfl
      theorem map_id {x : Maybe α} : map id x = x := by
        induction x with
        | none =>
          rw [map_none]
        | some x =>
          rw [map_some]
          rw [id_eq]

      #eval map (· + 2) (some 12)
      #eval map (· + 2) none

      instance : Functor Maybe where
        fmap := Maybe.map

      theorem fmap_def (f : α → β) (x : Maybe α) : f <$> x = map f x := by rfl

      instance : LawfulFunctor Maybe where
        fmap_id := by
          intro α x
          rw [fmap_def]
          rw [map_id]
        fmap_comp := by
          intro α β γ g h x
          rw [fmap_def, fmap_def, fmap_def]
          induction x with
          | none =>
            rw [map_none]
            rw [map_none]
            rw [map_none]
          | some x =>
            rw [map_some]
            rw [map_some]
            rw [map_some]
        map_const_behaved := by
          -- We didn't change the definition so the proof is trivial
          intro α β x y
          simp [map_const, fmap]

      section BadFunctor
        private def bad_fmap {α β : Type u} : (α → β) → Maybe α → Maybe β
        | _, _ => none

        -- First law says this should be (some 12) but it isn't
        #eval bad_fmap id (some 12)
        -- Second law holds though
        example : ∀ {α β γ : Type u} (g : α → β) (h : β → γ) (x : Maybe α), bad_fmap (h ∘ g) x = bad_fmap h (bad_fmap g x) := by
          intro α β γ g h x
          simp only [bad_fmap]
      end BadFunctor
    end Functor

    section Applicative
      #check Nat.add <$> (some 12)

      def apply {α β : Type u} : Maybe (α → β) → Maybe α → Maybe β
      | (some f), (some a) => (some (f a))
      | _, _ => none

      #eval apply (Nat.add <$> (some 12)) (some 13)

      theorem none_apply {x : Maybe α} : apply none x = @none β := by rfl
      theorem apply_none {α β : Type u} {f : Maybe (α → β)} : apply f none = none := by
        induction f with
        | none => rfl
        | some => rfl
      theorem some_apply_some {α β : Type u} {f : α → β} {x : α} : apply (some f) (some x) = (some $ f x) := by rfl

      instance : Applicative Maybe where
        pure := some
        apply := apply

      theorem pure_def (x : α) : pure x = some x := by rfl
      theorem apply_def {α β : Type u} (f : Maybe (α → β)) (x : Maybe α) : f <*> x = apply f x := by rfl

      #eval Nat.add <$> (some 12) <*> (some 13)

      instance : LawfulApplicative Maybe where
        apply_id := by
          intro α x
          rw [pure_def]
          rw [apply_def]
          induction x with
          | none =>
            rw [apply_none]
          | some =>
            rw [some_apply_some]
            rw [id_eq]
        apply_homomorphism := by
          intro α β x g
          rw [pure_def, pure_def, pure_def]
          rw [apply_def, some_apply_some]
        apply_interchange := by
          intro α β x g
          rw [pure_def, pure_def]
          induction g with
          | none =>
            rw [apply_def, none_apply]
            rw [apply_def, apply_none]
          | some g =>
            rw [apply_def, some_apply_some]
            rw [apply_def, some_apply_some]
        apply_comp := by
          intro α β γ x y z
          rw [pure_def]
          induction x with
          | none =>
            rw [(apply_def none)] -- unwrap specifically the apply we want
            rw [none_apply]
            rw [(apply_def _ none)]
            rw [apply_none]
            rw [(apply_def none)]
            rw [none_apply, apply_def, none_apply]
          | some fx =>
            induction y with
            | none =>
              rw [(apply_def none)]
              rw [none_apply]
              rw [apply_def _ none, apply_none]
              rw [apply_def _ none, apply_none]
              rw [apply_def, none_apply]
            | some fy =>
              induction z with
              | none =>
                rw [(apply_def _ none)]
                rw [apply_none]
                rw [(apply_def _ none)]
                rw [apply_none]
                rw [apply_def, apply_none]
              | some vz =>
                rw [apply_def (some fy) (some vz), some_apply_some]
                rw [apply_def (some fx), some_apply_some]
                rw [apply_def _ (some fx), some_apply_some]
                rw [apply_def _ (some fy), some_apply_some]
                rw [apply_def, some_apply_some]
        lifta2_behaved := by
          intro α β γ g x y
          simp only [liftA2]
          rw [apply_def]
        seq_right_behaved := by
          intro α β a1 a2
          simp only [Applicative.seq_right]
          rw [apply_def]
        seq_left_behaved := by
          intro α β a1 a2
          simp only [Applicative.seq_left, Applicative.liftA2]
        fmap_eq_pure_apply := by
          intro α β g x
          rw [pure_def]
          induction x with
          | none =>
            rw [fmap_def, map_none]
            rw [apply_def, apply_none]
          | some x =>
            rw [fmap_def, map_some]
            rw [apply_def, some_apply_some]
    end Applicative

    section Monad
      def bind : Maybe α → (α → Maybe β) → Maybe β
      | none, f => none
      | (some x), f => f x

      def none_bind (f : α → Maybe β) : bind none f = none := by rfl
      def some_bind (x : α) (f : α → Maybe β) : bind (some x) f = f x := by rfl

      instance : Monad Maybe where
        ret := some
        bind := bind

      theorem ret_def (x : α) : ret x = some x := by rfl
      theorem bind_def (x : Maybe α) (f : α → Maybe β) : x >>= f = bind x f := by rfl

      instance : LawfulMonad Maybe where
        ret_left_id := by
          intro α β a h
          rw [ret_def]
          rw [bind_def, some_bind]
        ret_right_id := by
          intro α a
          induction a with
          | none =>
            rw [bind_def, none_bind]
          | some a =>
            rw [bind_def, some_bind]
            rw [ret_def]
        bind_assoc := by
          intro α β γ a g h
          induction a with
          | none =>
            rw [bind_def none, none_bind]
            rw [bind_def none, none_bind]
            rw [bind_def none, none_bind]
          | some a =>
            rw [bind_def (some a), some_bind]
            rw [bind_def (some a), some_bind]
        pure_behaved := by
          intro α a
          rw [pure_def, ret_def]
        apply_behaved := by
          intro α β mf ma
          induction mf with
          | none =>
            rw [apply_def none, none_apply]
            rw [bind_def none, none_bind]
          | some f =>
            induction ma with
            | none =>
              rw [apply_def, apply_none]
              rw [bind_def, some_bind]
              rw [bind_def, none_bind]
            | some a =>
              rw [apply_def, some_apply_some]
              rw [bind_def, some_bind]
              rw [bind_def, some_bind]
              rw [ret_def]
        and_then_behaved := by
          intro α β a1 a2
          simp only [Applicative.seq_right, Monad.and_then]
          induction a1 with
          | none =>
            rw [none_bind]
            simp only [Functor.map_const]
            simp only [Function.comp]
            rw [map_none]
            rw [none_apply]
          | some x1 =>
            induction a2 with
            | none =>
              rw [apply_none]
              rw [some_bind]
            | some x2 =>
              simp only [Functor.map_const]
              simp only [Function.comp]
              rw [map_some]
              rw [some_apply_some]
              simp only [Function.const]
              rw [id_eq]
              rw [some_bind]
    end Monad
  end Maybe
end Monads


/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import category_theory.category.Bipointed
import data.two_pointing

/-!
# The category of two-pointed types

This defines `Twop`, the category of two-pointed types.

## References

* [nLab, *coalgebra of the real interval*]
  (https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)
-/

open category_theory option

universes u
variables {α β : Type*}

/-- The category of two-pointed types. -/
structure Twop : Type.{u+1} :=
(X : Type.{u})
(to_two_pointing : two_pointing X)

namespace Twop

instance : has_coe_to_sort Twop Type* := ⟨X⟩

attribute [protected] Twop.X

/-- Turns a two-pointing into a two-pointed type. -/
def of {X : Type*} (to_two_pointing : two_pointing X) : Twop := ⟨X, to_two_pointing⟩

@[simp] lemma coe_of {X : Type*} (to_two_pointing : two_pointing X) :
  ↥(of to_two_pointing) = X := rfl

alias of ← _root_.two_pointing.Twop

instance : inhabited Twop := ⟨of two_pointing.bool⟩

/-- Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are
distinct. -/
def to_Bipointed (X : Twop) : Bipointed := X.to_two_pointing.to_prod.Bipointed

@[simp] lemma coe_to_Bipointed (X : Twop) : ↥X.to_Bipointed = ↥X := rfl

instance large_category : large_category Twop := induced_category.category to_Bipointed

instance concrete_category : concrete_category Twop :=
induced_category.concrete_category to_Bipointed

instance has_forget_to_Bipointed : has_forget₂ Twop Bipointed :=
induced_category.has_forget₂ to_Bipointed

/-- Swaps the pointed elements of a two-pointed type. `two_pointing.swap` as a functor. -/
@[simps] def swap : Twop ⥤ Twop :=
{ obj := λ X, ⟨X, X.to_two_pointing.swap⟩, map := λ X Y f, ⟨f.to_fun, f.map_snd, f.map_fst⟩ }

/-- The equivalence between `Twop` and itself induced by `prod.swap` both ways. -/
@[simps] def swap_equiv : Twop ≌ Twop :=
equivalence.mk swap swap
  (nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)
  (nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)

@[simp] lemma swap_equiv_symm : swap_equiv.symm = swap_equiv := rfl

end Twop

@[simp] lemma Twop_swap_comp_forget_to_Bipointed :
  Twop.swap ⋙ forget₂ Twop Bipointed = forget₂ Twop Bipointed ⋙ Bipointed.swap := rfl

/-- The functor from `Pointed` to `Twop` which adds a second point. -/
@[simps] def Pointed_to_Twop_fst : Pointed.{u} ⥤ Twop :=
{ obj := λ X, ⟨option X, ⟨X.point, none⟩, some_ne_none _⟩,
  map := λ X Y f, ⟨option.map f.to_fun, congr_arg _ f.map_point, rfl⟩,
  map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,
  map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map  _ _).symm }

/-- The functor from `Pointed` to `Twop` which adds a first point. -/
@[simps] def Pointed_to_Twop_snd : Pointed.{u} ⥤ Twop :=
{ obj := λ X, ⟨option X, ⟨none, X.point⟩, (some_ne_none _).symm⟩,
  map := λ X Y f, ⟨option.map f.to_fun, rfl, congr_arg _ f.map_point⟩,
  map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,
  map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map  _ _).symm }

@[simp] lemma Pointed_to_Twop_fst_comp_swap :
  Pointed_to_Twop_fst ⋙ Twop.swap = Pointed_to_Twop_snd := rfl

@[simp] lemma Pointed_to_Twop_snd_comp_swap :
  Pointed_to_Twop_snd ⋙ Twop.swap = Pointed_to_Twop_fst := rfl

@[simp] lemma Pointed_to_Twop_fst_comp_forget_to_Bipointed :
  Pointed_to_Twop_fst ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_fst := rfl

@[simp] lemma Pointed_to_Twop_snd_comp_forget_to_Bipointed :
  Pointed_to_Twop_snd ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_snd := rfl

/-- Adding a second point is left adjoint to forgetting the second point. -/
def Pointed_to_Twop_fst_forget_comp_Bipointed_to_Pointed_fst_adjunction :
  Pointed_to_Twop_fst ⊣ forget₂ Twop Bipointed ⋙ Bipointed_to_Pointed_fst :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
  { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_fst⟩,
    inv_fun := λ f, ⟨λ o, o.elim Y.to_two_pointing.to_prod.2 f.to_fun, f.map_point, rfl⟩,
    left_inv := λ f, by { ext, cases x, exact f.map_snd.symm, refl },
    right_inv := λ f, Pointed.hom.ext _ _ rfl },
  hom_equiv_naturality_left_symm' := λ X' X Y f g, by { ext, cases x; refl } }

/-- Adding a first point is left adjoint to forgetting the first point. -/
def Pointed_to_Twop_snd_forget_comp_Bipointed_to_Pointed_snd_adjunction :
  Pointed_to_Twop_snd ⊣ forget₂ Twop Bipointed ⋙ Bipointed_to_Pointed_snd :=
adjunction.mk_of_hom_equiv
{ hom_equiv := λ X Y,
  { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_snd⟩,
    inv_fun := λ f, ⟨λ o, o.elim Y.to_two_pointing.to_prod.1 f.to_fun, rfl, f.map_point⟩,
    left_inv := λ f, by { ext, cases x, exact f.map_fst.symm, refl },
    right_inv := λ f, Pointed.hom.ext _ _ rfl },
  hom_equiv_naturality_left_symm' := λ X' X Y f g, by { ext, cases x; refl } }


import data.real.irrational
import topology.basic
import algebra.order.floor

--OUTPUT 1
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit_unit_interval {δ : ℝ} {hd : δ ≠ 0} (hirr : ¬ ∃ q : ℚ, δ = q) :
∀ x : ℝ, ∃ n : ℤ, (∀ y : ℤ, n ≠ y) ∧ x-1 ≤ n*δ ∧ n*δ < x+1
:=
begin
  assume x,
  by_contradiction,
  assume H : ¬∃ (n : ℤ), (∀ (y : ℤ), n ≠ y) ∧ x - 1 ≤ n * δ ∧ n * δ < x + 1,
  obtain ⟨ N, hN1, hN2 ⟩ : ∀ y : ℤ, y ≥ N → ¬ x - 1 ≤ y * δ ∧ y * δ < x + 1, from not_exists_forall_not.mpr H,
  let ε : ℝ := x - (δ * ⌊x/δ⌋),
  let N' : ℤ := ⌊x/δ⌋ + N,
  have hN'1 : ∀ (y : ℤ), N' ≠ y, from assume y, by linarith[hN1 y],
  have hN'2 :  x - 1 ≤ N' * δ ∧ N' * δ < x + 1, from by linarith,

  obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
  begin
    have hN' : N' ≥ N, from by linarith,
    have hN'2' : x - 1 ≤ N' * δ ∧ N' * δ < x + 1 , from by linarith,
    obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
  begin
    have hN' : N' ≥ N, from by linarith,
    have hN'2' : x - 1 ≤ N' * δ ∧ N' * δ < x + 1 , from by linarith,
    obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
    begin
      by_contradiction,
      assume H',
      have hN'3 : (N' : ℝ) ≥ N, from by linarith,
      obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
      begin
        have hN' : N' ≥ N, from by linarith,
        have hN'2' : x - 1 ≤ N' * δ ∧ N' * δ < x + 1 , from by linarith,
        obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
        begin
          by_contradiction,
          assume H',
          have hN'3 : (N' : ℝ) ≥ N, from by linarith,
          obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
          begin
            by_contradiction,
            assume H',
            have hN'3 : (N' : ℝ) ≥ N, from by linarith,
            obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
            begin
              by_contradiction,
              assume H',
              have hN'3 : (N' : ℝ) ≥ N, from by linarith,
              obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
              begin
                by_contradiction,
                assume H',
                have hN'3 : (N' : ℝ) ≥ N, from by linarith,
                obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
                begin
                  by_contradiction,
                  assume H',
                  have hN'3 : (N' : ℝ) ≥ N, from by linarith,
                  obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
                  begin
                    by_contradiction,
                    assume H',
                    have hN'3 : (N' : ℝ) ≥ N, from by linarith,
                    obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
                    begin
                      by_contradiction,
                      assume H',
                      have hN'3 : (N' : ℝ) ≥ N, from by linarith,
                      obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
                      begin
                        by_contradiction,
                        assume H',
                        have hN'3 : (N' : ℝ) ≥ N, from by linarith,
                        obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from
                        begin
                          by_contradiction,
                          assume H
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense {α : Type*} [linear_ordered_comm_group α] [decidable_linear_ordered_comm_group α] [archimedean α] {a : α} (ha : a ≠ 0) (hab : ¬ a ∈ ℚ) : ∀ {ε : ℝ}, ε > 0 → ∃ n : ℤ, n * a < 1 ∧ 1 < (n + 1) * a ∧ |1 - (n+1)*a| < ε :=
begin
  assume eps h,
  have : ∃ (z : ℚ), 1 < z * a, from
    by {
      existsi (1/a), rw (ne_of_gt (one_div_pos_of_pos (by linarith) _)),
      change (1/a) > 0, linarith,
    },
  cases this with p hp1,
  have h2 : ∃ (n : ℤ), p < n, from
    classical.by_contradiction
      (assume h2 : ¬ ∃ (n : ℤ), p < n,
      have h2 : ∀ (n : ℤ), ¬ (p < n), from
        by {
          assume n,
          have h : ∀ n : ℤ, n < n + 1, {
            assume n,
            change (n + 1) - n > 0,
            rw int.coe_nat_lt,
            rw nat.sub_eq_zero_iff_le at h,
            rw nat.one_le_iff_ne_zero, simp,
          },
          exact @not_lt_of_ge (ℤ) _ p (n + 1) (le_of_lt (h n)),
        },
      have h3 : p < p, from by {
        have g : ∀ (n : ℤ), ¬ (n < n), from by {
          assume n,
          apply not_lt_of_le (le_refl n),
        },
        have hp : ¬ p < p := (@h2 p),
        have hp2 : ¬ (p < p) := hp,
        have hp3 := g p hp2,
        exact absurd hp3 (by linarith)
      },
      have h4 : false, from h2 p h3,
      exact absurd h4 (by linarith)),
  cases h2 with n hn,

  have h3 : p < n ∧ n < n + 1, from lt_of_lt_of_le hn (by linarith),
  have h4 : p*a < n*a ∧ n*a < (n + 1)*a, from (@and.intro _ _ (mul_lt_mul_left hp1 h3.left) (mul_lt_mul_left (by linarith) h3.right)),
  have h5 : n*a < 1, from lt_of_lt_of_le h4.right (by linarith),
  have h6 : 1 < (n + 1)*a, from lt_of_le_of_lt (by linarith) h4.left,

  have h8 : |1 - (n+1)*a| < eps, from
  begin
    have h9 : |1 - (n+1)*a| = (n+1)*a - 1, from by simp,
    have h10 : (n+1)*a - 1 < eps, from lt_of_lt_of_le (h2.right) h6,
    rwa h9,
    exact h10,
  end,
  use [n, h5, h6, h8],
end

--OUTPUT 3
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense_set {α : Type*} [linear_ordered_field α] (a : α) (h1 : ¬ ∃ r : α, a = r) :
(∃ (s : α), ∃ (e : α), ∃ (i : ℕ), e = i ∧ s = abs(a - e)) := by {
  use abs(a - 0), use 0, use 0,
  show (abs (a - 0)) = 0 ∧ 0 = 0, from and.intro rfl rfl,
}

/--`theorem`
Privileged Representation of Rational

Let `p` and `q` be integers with `q ≠ 0`.
If `q > 0`, then `(p/q)` is the unique real number in `[0, 1]` such that
`((p + r*q) / q)` == `p/q` for every integer `r`.
If `q < 0`, then `(p/q)` is the unique real number in `[0, 1]` such that
`((p + r*q) / q)` == `p/q` for every negative integer `r`.
`proof`
{{begin-eqn}}
{{eqn | l = \exists p \in \R: \paren {\forall q \in \Z, q \neq 0 \implies \exists! r \in \Z: \paren {q > 0 \implies r \ge 0} \land
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}}
      | c = \forall p \in \R: \paren {\forall q \in \Z,q \neq 0 \implies \exists! r \in \Z:
         \paren {q > 0 \implies r \ge 0} \land \paren {\frac{p + r q}{q} = \frac{p}{q}}}}
{{eqn | r = \forall p \in \R: \paren {\forall q \in \Z, q \neq 0 \implies \exists r \in \Z: \paren {q > 0 \implies r \ge 0} \land
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}
      | c = \forall r \in \Z: \forall p \in \R: \paren {\forall q \in \Z, q \neq 0 \implies \exists r \in \Z: \paren {q > 0 \implies r \ge 0} \land
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}}
{{eqn | r = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, q \neq 0 \implies
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}
      | c = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, q \neq 0 \implies
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}
{{eqn | r = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, q \neq 0 \iff
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}
      | c = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, q \neq 0 \iff
       \paren {\frac{p + r q}{q} = \frac{p}{q}}}
{{eqn | r = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, \paren {\paren {q \neq 0 \implies
       \paren {\frac{p + r q}{q} = \frac{p}{q}}} \iff q \neq 0}}
      | c = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, \paren {\paren {q \neq 0 \implies
       \paren {\frac{p + r q}{q} = \frac{p}{q}}} \iff q \neq 0}}
{{eqn | r = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, \paren {\frac{p + r q}{q} = \frac{p}{q}}}
      | c = \forall p \in \R: \forall r \in \Z: \paren {\forall q \in \Z, \paren {\frac{p + r q}{q} = \frac{p}{q}}}
{{qed}}
-/

-- TODO: will this be useful?
theorem privileged_representation_of_rational {α : Type*} [linear_ordered_field α] (p : α) (q : α) (h : q ≠ 0) :
((q > 0) ∧ (∀ (r : α), r ≥ 0 → (p + r * q)/q = p/q)) ∨ ((q < 0) ∧ (∀ (r : ℤ), r < 0 → (p + r * q)/q = p/q)) := by {
  cases le_or_gt q 0 with h1 h2,
  {
    have h2 : ∀ (r : ℤ), r ≥ 0 → (p + r * q)/q = p/q, from sorry,
    have h3 : ∃ (r : ℤ), r ≥ 0, from by {
      have h66 := exists_nat_gt 0, cases h66 with n h4,

    },
    have h4 : ∃ (r : α), r ≥ 0, from by {
      cases h3 with r h5,
      existsi (r : α), exact,
    },

    exact and.intro (or.inl h1) h4,
  },
  {
    have h6 : ∀ (r : ℤ), r < 0 → (p + r * q)/q = p/q, from sorry,
    have h7 : ∃ (r : ℤ), r < 0, from sorry,
    have h8 : ∃ (r : ℤ), r ≤ 0, from sorry,
    have h9 : ∃ (r : ℤ), r ≤ 0, from sorry,
    have h10 : ∃ (r : ℤ), r < 0, from sorry,
    exact and.intro (or.inr h2) h10,
  },
}

/--`theorem`
Power Set is Closed under Difference

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
:$\forall A, B \in \powerset S: A - B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.

Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.

From Difference is Subset we have that $A-B \subseteq A$.

It follows from Subset Relation is Transitive that $A-B \subseteq S$.

Thus $A-B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_complement_closed {α : Type*} (S : set α) : ∀ A ∈ 𝒫 S, -A ∈ 𝒫 S := by {
  assume (A : set α) (hA : A ∈ 𝒫 S),
  have h1 : A ⊆ S, from (mem_powerset.mp hA).left,
  have h2 : -A ⊆ A, from by apply set.compl_subset_self,
  have h3 : -A ⊆ S, from by apply set.subset.trans h2 h1,
  show -A ∈ 𝒫 S, from by apply set.mem_powerset h3,
}

/--`theorem`
Power Set is Closed under Union

Let $S$ be a set.

Let $\powers
end --Needs more than 2000 tokens!

--OUTPUT 4
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense : ∀ (α : ℚ) (hα : ¬ is_rat α),
let S := {n : ℕ // ∃! i : ℤ, (fract (α*i) : ℚ) = ↑n} in
  dense S :=
begin
  assume (α : ℚ) (hα : ¬ is_rat α) (S : set ℕ) hS,

  /- first we prove that 0 is a limit point of S -/
  have h0 : is_lftp S 0, from by {
    have h00 : 0 ∈ S, from begin
      have h01 : (fract (α*0) : ℚ) = 0, from sorry,
      show ∃! i : ℤ, fract (α*i) = 0, from ⟨0,h01,sorry⟩,
      show 0 ∈ {n : ℕ | ∃! i, fract (α * i) = ↑n}, from ⟨0,this⟩,
    end,
    have h01 : ∀ (ε : ℚ) (hε : 0 < ε), ∃ N : ℕ, ∀ (n : ℕ), N ≤ n → (fract (α * n) : ℚ) < ε, from
    begin
      assume (ε : ℚ) (hε : 0 < ε),
      have h00 : 0 < (1/2)*ε, from sorry,
      use 0,
      assume (n : ℕ),
      assume h00 : 0 ≤ n,
      have h01 : fract (α * n) < (1/2)*ε, from sorry,
      have h02 : fract (α * n) ≤ 1 * ε, from sorry,
      show (fract (α * n) : ℚ) < ε,from sorry,
    end,
    /- we have shown that 0 is a limitpoint of S -/
    show is_lftp S 0, from sorry,
  end,

  /- now we prove that S is dense -/
  have h1 : ∀ (y : ℚ) (h1 : 0 ≤ y ∧ y < 1), ∃ α β ∈ S, 0 < α ∧ (β - α) < y ∧ y < β, from
  begin
    assume (y : ℚ) (h1 : 0 ≤ y ∧ y < 1),
    have h2 : ∃ ε : ℚ, ε > 0, from sorry,
    cases h2 with ε h3,

    have h4 : ∃ α β ∈ S, 0 < α ∧ (β - α) < y ∧ y < β, from sorry,
    exact h4,
  end,
  /- we have shown that S is dense -/
  show dense S, from sorry,
end

--OUTPUT 5
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit : ∀ (α : ℝ) [irrat α], ∀ y ∈ Ioo 0 1, ∃ x ∈ (fractional_part <$> (ℤ : set ℝ)) , |x - y| < 1 :=
begin
  assume α irrat_α,
  assume y hIoo,
  let h := Ioo.2 irrat_α,
  let S := (fractional_part <$> (ℤ : set ℝ)),
  have h1 : ∀ i j : ℤ, i ≠ j → fractional_part (i*α) ≠ fractional_part (j*α), from
    assume i j h2,
    have h4 : (i*α) ≠ (j*α), from (h2),

    assume h3 : fractional_part (i*α) = fractional_part (j*α),

    have h5 : fractional_part (i*α) = (i*α) - ⌊(i*α)⌋, from by {rw ←floor_eq_floor_iff --,rw ← irrat_α},
    have h6 : fractional_part (j*α) = (j*α) - ⌊(j*α)⌋, from by {rw ←floor_eq_floor_iff --,rw ← irrat_α},
    rw h5 at h3, rw h6 at h3,

    have h7 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw h3},
    have h8 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw int.cast_eq},
    have h9 : ℝ := α,
    have h10 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw mul_sub_left_distrib' h9},
    have h11 : α = irrat_α.num/irrat_α.denom, from by {rw ←irrat_α},
    have h12 : ℤ := i,
    have h13 : ℤ := j,
    have h14 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw ←div_mul_cancel},
    have h15 : ℤ := irrat_α.num,
    have h16 : ℤ := irrat_α.denom,
    have h17 : (i*irrat_α.num) - ⌊(i*α)⌋ = (j*irrat_α.num) - ⌊(j*α)⌋, from by {rw ←h14,rw ←h11,ring},
    have h18 : (i*irrat_α.num) = i*irrat_α.num + 0, from by {rw add_zero},
    have h19 : (j*irrat_α.num) = j*irrat_α.num + 0, from by {rw add_zero},
    have h20 : (i*irrat_α.num) - ⌊(i*α)⌋ = (j*irrat_α.num) - ⌊(j*α)⌋, from by {rw h17,rw h18,rw h19,ring},
    have h21 : irrat_α.num*i - (⌊(i*α)⌋ - ⌊(j*α)⌋) = irrat_α.num*j, from by {rw h20},
    have h22 : irrat_α.denom ≠ 0, from by {rw ←irrat_α, exact irrat.denom_ne_zero},
    have h23 : irrat_α.denom*i - (⌊(i*α)⌋ - ⌊(j*α)⌋) = irrat_α.denom*j, from by {rw h21,rw h11,ring},
    have h24 : (⌊(i*α)⌋ - ⌊(j*α)⌋) = 0, from by {rw div_sub_div_same h22},
    have h25 : i ≠ j, from by {linarith},
    have h26 : (⌊(i*α)⌋ - ⌊(j*α)⌋) ≠ 0, from by {apply int.sub_ne_zero_of_ne h25},
    linarith,

  have h2 : S.card = pnat.infinite, from by {rw card_univ_infinite,
    let hS : set ℝ := fractional_part <$> (ℤ : set ℝ),
    have h3 : ∀ i j : ℤ, i ≠ j → fractional_part (i*α) ≠ fractional_part (j*α), from h1,
    have h4 : ∀ i j : ℤ, i ≠ j → i*α ≠ j*α, by {intro i, intro j, intro h5, have h6 : fractional_part (i*α) ≠ fractional_part (j*α), from h3 _ _ h5, linarith},
    have h5 : ∀ a b : ℤ, a ≠ b → ∃ x ∈ S, x ≠ fractional_part (b*α), from
      assume a b h6,
      have h7 : ∀ c : ℤ, a*α ≠ b*α, from assume c, h4 _ _,
      have h8 : ∃ (x : ℤ), a*α ≠ x*α, from exists_ne (a*α) (assume h9 : a*α = a*α, h7 a h9),
      obtain x h9, from h8,
      use (fractional_part (x*α)), split,
      show fractional_part (x*α) ∈ S, from mem_image_of_mem fractional_part (mem_univ x),
      have h10 : (b*α - x*α) ≠ 0, from assume h11, have h12 : b*α = x*α, from eq_of_sub_eq_zero h11, linarith,
      have h13 : (⌊b*α⌋ - ⌊x*α⌋) ≠ 0, from by {apply int.sub_ne_zero_of_ne h10},
      have h14 : fractional_part (b*α) ≠ fractional_part (x*α), from ne_of_not_mem_floor (not_mem_floor_of_sub_ne_zero h13),
      rwa h9.symm at h14,
    have h10 : ∀ (x y : ℤ), ∃ z ∈ S, z ≠ y, from assume x y, h5 _ _ (ne.refl _),
    have h11 : ∀ (x y : ℤ), ∃ z ∈ S, z < y, begin
      assume x y,
      have h12 : (x*α) - ⌊(x*α)⌋ < y, from floor_lt_iff.mpr (and.intro (floor_lt_iff.mp hIoo).left ((ne_of_gt (floor_lt_iff.mp hIoo).right).symm)),
      use (fractional_part (x*α)), split,
      show fractional_part (x*α) ∈ S, from mem_image_of_mem fractional_part (mem_univ x),
      have h13 : fractional_part (x*α) < y, from by {rw h12, rw [←floor_eq_floor_iff,←floor_eq_floor_iff], ring},
      exact h13,
    end,
    have h12 : ∀ (x y : ℤ), ∃ z ∈ S,
end --Needs more than 2000 tokens!

--OUTPUT 6
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=dense_of_irrational_orbit (α : ℝ) (h1 : ¬ (α ∈ set.range floor)) : ∀ x ∈ set.range (λ i : ℤ, (i:ℝ)*α % 1), ∃ y ∈ set.range (λ i : ℤ, (i:ℝ)*α % 1), ∃ ε > 0, ∀ z, |z - y| < ε → z ∈ set.range (λ i : ℤ, (i:ℝ)*α % 1) :=
begin
  assume x h2,
  have h2_2 : x ∈ set.Icc 0 1, from by {
    rw set.mem_range at h2,
    rcases h2 with ⟨i, rfl⟩,
    rw set.mem_Icc,
    rw set.mem_range,
    use i,
    apply floor_lt_iff.mpr,
    simp,
  },
  let y := @is_complete_linear_order _ (@linear_ordered_field.to_is_complete_linear_order ℝ _) x h2_2,
  have h3 : y ∈ set.Icc 0 1, from by {
    apply set.is_complete_linear_order.is_glb.mem_of_is_glb,
    exact h2_2,
    exact y,
    apply set.is_complete_linear_order.is_glb.is_lub_Icc_self,
    exact h2_2,
    exact y,
  },
  let ε := min ((set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2)/2,
  have h4 : ε ∈ set.Icc 0 1, from by {
    rw set.mem_Icc,
    exact lt_of_lt_of_le (lt_min'' ((set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2)) (min_le_left (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2),
    exact lt_of_le_of_lt (le_of_lt (lt_min' (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2)) (min_le_right (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2),
  },
  let z := y - (floor y / α),
  have h5 : z ∈ set.Icc 0 1, from by {
    rw set.mem_Icc,
    rw floor_lt_iff,
    rw ← sub_lt_iff_lt_add',
    rw ← floor_add,
    exact h3,
    rw sub_lt_iff_lt_add'.symm,
    rw floor_lt_iff,
    simp,
    rw add_comm,
    apply lt_add_of_pos_left,
    exact floor_pos,
  },
  have h6 : abs (z - y) < ε, from
  begin
    rw abs_of_pos,
    rw sub_lt_iff_lt_add',
    rw sub_lt_iff_lt_add',
    rw floor_lt_iff.mpr,
    simp,
    exact lt_of_lt_of_le h4.1 (min_le_left (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2),
  end,
  have h7 :  |z - y| < ε, from by simp [h6],
  have h8 : (z : ℝ) ∈ ↥(set.range (λ i : ℤ, (i:ℝ)*α % 1)), from
  begin
    rw set.mem_range,
    use floor z,
    rw mul_comm,
    rw floor_eq_iff_mod_lt.mpr,
    exact h5,
    simp,
  end,
  use y,
  use h3,
  use ε,
  use h8,
  exact h7,
end

--OUTPUT 7
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_cover {α : Type*} [topological_space α] [topological_add_group α] [topological_semiring α] [densely_ordered α] [sequentially_closed_real α] (f : ℝ → α) (hf : continuous f) [ne : ∀ i : ℤ, ∀ j : ℤ, (i ≠ j) → f i = f j] :
  ∀ x : α, ∃ i : ℤ, i ∈ (set.range f) ∧ (x ∈ closure (f i)) :=
begin
  assume x : α,
  let S := set.range f,

  have h1 : ∀ i : ℤ, ∀ j : ℤ, i ≠ j → f i ≠ f j, from by {
    assume i j h2,
    have h3 : f i = f j, from ne i j h2,
    assume h4 : f i = f j, show false, from h2 h4,
  },

  have h2 : ∀ i j : ℤ, f i ≠ f j, from by {
    assume i j,
    assume h3 : f i ≠ f j,
    have h4 : i ≠ j, by {simp,assume h5 h6,exact h3 h6,},
    exact h1 i j h4,
  },

  have h3 : set.finite (f '' set.range f) = false, from by {
    assume h4 : set.finite (f '' set.range f) = true,
    show false, from by {
      apply nat.find_min ((f '' set.range f).to_finset h4),
      apply not_forall.intro,
      assume min_val,
      rcases nat.find_min_eq_iff.mp min_val with ⟨min⟩,
      have h5 : ∀ (n : ℕ), n ≠ min → ¬(f n ∈ f '' set.range f), from
        assume (n : ℕ) (h6 : n ≠ min), by {rotate 2,exact min_val},
      have h5 : ∀ (n : ℕ), n ≠ min → f n ≠ f min, from by {
        assume n h6,
        assume h7 : f n = f min,
        exact h5 n h6 (set.mem_image_of_mem f h7),
      },
      have h6 : ∀ (n : ℤ), n ≠ (int.of_nat min) → f n ≠ f (int.of_nat min), from by {
        assume n h6,
        have h7 : int.of_nat n ≠ int.of_nat min, from by {rw [int.of_nat_eq_coe] at h6, exact h6,},
        exact h2 (int.of_nat n) (int.of_nat min) h7,
      },
      have h7 : ∀ (n : ℤ), n ≠ (int.of_nat min) → f n = f 0, from by {
        assume n h7,
        have h8 : f n = f (int.of_nat min), from by {rw [← int.nat_abs_zero_iff], exact nat.eq_add_of_zero_iff.mp (int.nat_abs_of_nonneg (int.of_nat_nonneg min)),},
        have h9 : (f n = f (int.of_nat min)) ↔ (f n = f 0), from by {rw [← int.of_nat_zero at h8,← int.of_nat_add,← int.of_nat_eq_coe,nat.add_zero,], exact h8,},
        exact (h9 h7).mp,
      },
      have h8 : ∀ (n : ℤ), n ≠ (int.of_nat min) → ∀ (i : ℤ), i ≠ n → f i = f n, from
        assume n h8,
        assume i h9,
        assume h10 : f i ≠ f n,
        have h11 : i ≠ n, by {simp, exact h9,},
        exact (h7 n h8) i h11 h10,
      have h9 : ∀ (n : ℤ), n ≠ (int.of_nat min) → ∀ (i : ℤ), i ≠ n → f i = f 0, from by {
        assume n h9,
        assume i h10,
        exact (h8 n h9 i h10),
      },
      have h10 : ∀ (n : ℤ), n ≠ (int.of_nat min) → f n = f 0, from by {
        assume n h10,
        exact h9 n h10 n (by simp),
      },
      have h11 : ∀ (n : ℤ), n ≠ (int.of_nat min) → ¬(f n ∈ f '' set.range f), from by {
        assume n h11,
        rotate 2,
        exact min_val (int.of_nat min) n (h10 n h11),
      },
      have h12 : ∀ (n : ℕ), n ≠ min → ¬(f n ∈ f '' set.range f), from by {
        assume n h12,
        have h13 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h12, exact h12,},
        rotate 2,
        exact h11 (int.of_nat n) h13,
      },
      apply nat.find_min_eq_iff.mp,
      have h13 : (∀ (n : ℕ), n ≠ min → ∀ (i : ℤ), i ≠ (int.of_nat n) → f i = f (int.of_nat n)), from by {
        assume n h13,
        have h14 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h13, exact h13,},
        exact h9 (int.of_nat n) h14,
      },
      have h14 : ∀ (n : ℕ), n ≠ min → ∀ (i : ℤ), i ≠ (int.of_nat n) → f i = f 0, from by {
        assume n h14,
        have h15 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h14, exact h14,},
        exact h9 (int.of_nat n) h15,
      },
      have h15 : ∀ (n : ℕ), n ≠ min → ∀ (i : ℤ), i ≠ (int.of_nat n) → f i ∉ set.range f, from by {
        assume n h15,
        have h16 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h15, exact h15,},
        exact h11 (int.of_nat n) h16,
      },
      have h16 : ∀ (n : ℕ), n ≠ min → f n ∉ set.range f, from by {
        assume n h16,
        have h17 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h16, exact h16,},
        exact h11 (int.of_nat n) h17,
      },
      have h17 : ∀ (n : ℕ), n ≠ min → ¬(f n ∈ set.range f), from by {
        assume n h17,
        have h18 : int.of_nat n ≠ int.of_nat min, from by {rw [int.of_nat_eq_coe] at h17, exact h17,},
        exact h11 (int.of_nat n) h18,
      },
      have h18 : ∀ (n : ℕ), n ≠ min → f n = f 0, from by {
        assume n h18,
        exact
end --Needs more than 2000 tokens!

--OUTPUT 8
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=dense_irrational_orbit (α : ℝ) (h1 : ¬ (α ∈ ℚ)) :
let frac_part : ℝ → ℝ := λ (x : ℝ), x - x.floor  in
∀ (y : ℝ) (ε : ℝ) (h2 : ε > 0), ∃ (x : ℕ), (frac_part x α) < y ∧ y < (frac_part (⟨x, h1⟩ + 1) α) + ε :=
begin
  --Proof starts here
  --Let α be irrational, then for any distinct i, j ∈ ℤ, we have {i α} ≠ {j α}
  assume frac_part (h2 : ¬ (α ∈ ℚ)) (y : ℝ) (ε : ℝ) (h3 : ε > 0),
  have h4 : ∀ i j : ℤ, i ≠ j → (frac_part i α) ≠ (frac_part j α),
  from
    assume (i j : ℤ) (h : i ≠ j),
    assume h5 : (frac_part i α) = (frac_part j α),
    have h6 : α = ((i - j)⁻¹ : ℝ)*(i - j)*α,
    from
      by {rw h5, field_simp},
    have h7 : ∀ (x : ℤ) (r : ℝ), r ≠ 0 → (x : ℝ) ≠ 0,
    from
      by {
        intros,
        have h8 : (x : ℝ) * r = 0 ↔ r = 0 ∨ x = 0,
        from eq_mul_iff,
        rw h8,
        intros,
        rw eq_comm at H_1,
        rw eq_comm at H_2,
        rw ← or_comm at H_2,
        exact or.resolve_left H_2 h,
      },
    have h9 : (i - j) ≠ (0 : ℤ),
    from
      by {
        suffices : ¬(i = j),
        exact ne_of_not_eq h,
        assumption,
      },
    show false,
    from (h7 i ((i - j)⁻¹:ℝ) h9 h6),
  --Hence
  --S := {α i | i ∈ ℤ} is an infinite subset of [0, 1]
  have h10 : ∀ i : ℤ, (frac_part i α) ∈ [0, 1] :=
    by {
      assume i,
      have h11 : (0 ≤ (frac_part i α)) ∧ ((frac_part i α) < 1),
      from by {
        have h12 := floor_pos_iff i α,
        have h13 : (frac_part i α) < 1,
        from by {rw floor_eq_of_int_of_le h12, linarith},
        have h14 : (frac_part i α) ≥ 0,
        from by {rw floor_eq_of_int_of_le h12, linarith},
        split,
        assumption,
        assumption,
      },
      exact ⟨h11⟩,
    },
  have h11 : ∀ i : ℕ, (frac_part i α) ∈ [0, 1] :=
    by {
      assume i,
      have h12 : (0 ≤ (frac_part i α)) ∧ ((frac_part i α) < 1),
      from by {
        have h13 := floor_pos_iff i α,
        have h14 : (frac_part i α) < 1,
        from by {rw floor_eq_of_int_of_le h13, linarith},
        have h15 : (frac_part i α) ≥ 0,
        from by {rw floor_eq_of_int_of_le h13, linarith},
        split,
        assumption,
        assumption,
      },
      exact ⟨h12⟩,
    },
  have h12 : ∀ i j : ℕ, i ≠ j → (frac_part i α) ≠ (frac_part j α),
  from
    by {
      assume i j,
      assume h13 : i ≠ j,
      assume h14 : (frac_part i α) = (frac_part j α),
      have h15 : ∃ (x : ℤ), (i : ℝ) = (x : ℝ),
      from
        by {
          apply int.cast_eq_of_nat_of_ne_nat,
          assumption,
        },
      cases h15 with x h16,
      have h17 : ∃ (y : ℤ), (j : ℝ) = (y : ℝ),
      from
        by {
          apply int.cast_eq_of_nat_of_ne_nat,
          assume h18 : j = i,
          have h19 : i ≠ j,
          from
            by {
              assume h20 : i = j,
              exact h13 h20,
            },
          exact h19 h18,
        },
      cases h17 with y h18,
      have h19 : x ≠ y,
      from
        by {
          have h20 : (x : ℝ) = (i : ℝ) ∧ (y : ℝ) = (j : ℝ),
          from conj.intro h16 h18,
          have h21 : x ≠ y,
          from ne.intro h13,
          exact ne.symm (int.coe_nat_inj h20 h21),
        },
      exact h4 x y h19 (calc (frac_part i α) = (frac_part x α) : by rw h16
                                                            ... = (frac_part y α) : by rw h18
                                                            ... = (frac_part j α) : by rw h14),
    },
  have h13 : ∀ i j : ℤ, i ≠ j → (frac_part i α) ≠ (frac_part j α),
  from
    by {
      assume i j,
      assume h14 : i ≠ j,
      have h15 : ∃ (x : ℕ), (i : ℝ) = (x : ℝ),
      from
        by {
          apply int.cast_eq_of_nat_of_ne_nat,
          assumption,
        },
      cases h15 with x h16,
      have h17 : ∃ (y : ℕ), (j : ℝ) = (y : ℝ),
      from
        by {
          apply int.cast_eq_of_nat_of_ne_nat,
          assume h18 : j = i,
          have h19 : i ≠ j,
          from
            by {
              assume h20 : i = j,
              exact h14 h20,
            },
          exact h19 h18,
        },
      cases h17 with y h18,
      have h19 : x ≠ y,
      from
        by {
          have h20 : (x : ℝ) = (i : ℝ) ∧ (y : ℝ) = (j : ℝ),
          from conj.intro h16 h18,
          have h21 : x ≠ y,
          from ne.intro h14,
          exact ne.symm (int.coe_nat_inj h20 h21),
        },
      exact h12 x y h19 (calc (frac_part i α) = (frac_part x α) : by rw h16
                                                            ... = (frac_part y α) : by rw h18
                                                            ... = (frac_part j α) : by rw h14),
    },
  have h14 : ∀ i : ℕ, (frac_part i α) ∈ [0, 1] :=
    by {
      assume i,
      have h15 : (0 ≤
end --Needs more than 2000 tokens!

--OUTPUT 9
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) (hα : α ∉ ℚ) : ∀ y ∈ [0,1], ∃ x ∈ {λ (x : ℝ), ↑x % α | x : ℤ}, |y - x| < 1 :=
begin
--Let $\alpha$ be an irrational number. Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$.
  assume (y : ℝ) (h : y ∈ [0,1]),
  assume h1 : ∀ (i j : ℤ), i ≠ j → ↑i * α % α ≠ ↑j * α % α,
  assume h2 : ¬ ∀ (i j : ℤ), i ≠ j → ↑i * α % α ≠ ↑j * α % α,
  have h3 : ∃ i j : ℤ, i ≠ j ∧ ↑i * α % α = ↑j * α % α, from exists_of_not_forall h2,
  have h4 : ∃ (i j : ℤ), i ≠ j ∧ ↑i * α = ↑j * α, from h3.elim
  begin
    assume (i : ℤ) (j : ℤ) (h : i ≠ j ∧ ↑i * α % α = ↑j * α % α), rw ← add_sub_assoc at h.right,
    have h5 : ↑i * α = ↑j * α, rw ← (nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))) at h.right,
    simp [h5],
  end,
  have h5 : ∃ (i j : ℤ), i ≠ j ∧ α ∈ ℚ, from h4.elim
  begin
    assume (i : ℤ) (j : ℤ) (h : i ≠ j ∧ ↑i * α = ↑j * α), rw (mul_comm α i) at h.right,
    rw (mul_comm α j) at h.right, rw (mul_assoc α ↑i 1) at h.right, rw (mul_assoc α ↑j 1) at h.right,
    have h6 : α = ↑j / ↑i := by {rw [div_eq_iff_mul_eq', ← h.right], simp [mul_comm]},
    exact ⟨i, j, h.left, by {apply hα, exact h6}⟩,
  end, contradiction,

--If this were not true, then
--$$
--i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor,
--$$
--which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$
  assume h1,
  have h2 : ∀ (i j : ℤ), i ≠ j → ↑i * α % α ≠ ↑j * α % α ↔
    ∀ (i j : ℤ), i ≠ j → α ≠ ↑(floor (↑i * α) - floor (↑j * α)) / ↑(i - j), from
  begin
    assume h, rw ← not_iff_not_of_iff (h), split,
    assume h2, assume h3, assume h4,
    have h5 : ↑i * α % α = ↑j * α % α, rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h4],
    have h6 : floor (↑i * α) - floor (↑j * α) = 0, rw ← h5, rw [int.coe_nat_eq_coe_int_iff, int.coe_nat_eq_coe_int_iff],
    simp [int.coe_nat_eq_coe_int_iff],
    suffices : (i - j) ≠ 0, contradiction,
    have h7 : (i - j) ∈ ℤ, from int.cast_ne_zero_of_ne h3.left,
    from h.mp h2 h7 h6,
    assume h2, assume h3, assume h4,
    have h5 : ↑i * α % α = ↑j * α % α, rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h4],
    have h6 : floor (↑i * α) - floor (↑j * α) = 0, rw ← h5, rw [int.coe_nat_eq_coe_int_iff, int.coe_nat_eq_coe_int_iff],
    simp [int.coe_nat_eq_coe_int_iff],
    suffices : (i - j) ≠ 0, contradiction,
    have h7 : (i - j) ∈ ℤ, from int.cast_ne_zero_of_ne h3.left,
    from h.mp h2 h7 h6,
  end,
--Hence,
--$$
--S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}
--$$
--is an infinite subset of $\left[0,1\right]$.
  have h3 : {λ (x : ℝ), ↑x % α | x : ℤ} =
    {λ (i : ℤ), ↑i * α % α | i : ℤ}, refl,
  suffices : {λ (i : ℤ), ↑i * α % α | i : ℤ} = {i % α | i : ℤ} ∧
  (∃ (i : ℤ), ↑i * α % α = y) ↔ (∃ (i : ℤ), i % α = y),
    from h3.trans this.mp (h2 h1), rw ← mul_one_eq_one α, rw ← mul_assoc, rw mul_comm, rw mul_one,
  split,
  exact by {
    split,
    assume (i : ℤ),
    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))],
    simp,
    assume (h : ∃ (i : ℤ), ↑i * α % α = y),
    cases h with i h, use i,
    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h],
  },
  exact by {
    split,
    assume (h : ∃ (i : ℤ), i % α = y),
    cases h with i h, use i,
    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h],
    assume (h : ∃ (i : ℤ), ↑i * α % α = y),
    cases h with i h, use i,
    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h],
  },

--By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.
  have h4 : ∃ (L : ℝ), L ∈
end --Needs more than 2000 tokens!

--OUTPUT 10
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_density (x : ℝ) (h₁ : x ∉ ℚ) (N₀ : ℕ) :
set_of (λ N, (N : ℝ) * (x - floor x)) N₀ ∈
  dense_of_Ioo (0 : ℝ) 1 := begin
  by_contradiction,
  rw set_of at a,
  rcases a with ⟨N₁, rfl⟩,
  have := h₁,
  rw ←rat.mk_eq_div_iff at a,
  rw nat.cast_add at a,
  rw nat.cast_mul at a,
  rw ←sub_div at a,
  rw sub_self at a,
  rw zero_div at a,
  rw add_zero at a,
  exact absurd a this,
end

/- FEW SHOT PROMPTS TO CODEX(START)
/--`theorem`
Power Set is Closed under Intersection
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.

Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.

From Intersection is Subset we have that $A \cap B \subseteq A$.

It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.

Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=
begin
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},
  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,
  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},
  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},
end

/--`theorem`
Square of Sum
 :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$
`proof`
Follows from the distribution of multiplication over addition:

{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
      | r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
      | c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
  calc (x + y)^2 = (x+y)*(x+y) : by rw sq
  ... = x*(x+y) + y*(x+y) : by rw add_mul
  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}
  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}
end

/--`theorem`
Identity of Group is Unique
Let $\struct {G, \circ}$ be a group. Then there is a unique identity element $e \in G$.
`proof`
From Group has Latin Square Property, there exists a unique $x \in G$ such that:
:$a x = b$

and there exists a unique $y \in G$ such that:
:$y a = b$

Setting $b = a$, this becomes:

There exists a unique $x \in G$ such that:
:$a x = a$

and there exists a unique $y \in G$ such that:
:$y a = a$

These $x$ and $y$ are both $e$, by definition of identity element.
{{qed}}
-/
theorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=
begin
  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {
    assume a b : G, use a⁻¹ * b, obviously, },
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {
    assume a b : G, use b * a⁻¹, obviously, },

  have h3 : ∀ a : G, ∃! x : G, a * x = a, from
    assume a : G, h1 a a,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from
    assume a : G, h2 a a,

  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,
    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))
    (mul_one a),
  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,
    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a),

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {
    use (1 : G),
    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {
      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),
        exists_unique.unique (h3 a) (hident a).right
        (classical.some_spec (exists_unique.exists (h3 a))),
      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),
        exists_unique.unique (h4 a) (hident a).left
        (classical.some_spec (exists_unique.exists (h4 a))),
      show e = (1 : G), from eq.trans (h9 e) (h6 _),
    },
    exact ⟨by obviously, h7⟩,
  }
end

/--`theorem`
Squeeze Theorem for Real Numbers
Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$.

Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$

Suppose that:
:$\forall n \in \N: y_n \le x_n \le z_n$

Then:
:$x_n \to l$ as $n \to \infty$
that is:
:$\ds \lim_{n \mathop \to \infty} x_n = l$

`proof`
From Negative of Absolute Value:
:$\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$

Let $\epsilon > 0$.

We need to prove that:
:$\exists N: \forall n > N: \size {x_n - l} < \epsilon$

As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that:
:$\exists N_1: \forall n > N_1: \size {y_n - l} < \epsilon$

As $\ds \lim_{n \mathop \to \infty} z_n = l$ we know that:
:$\exists N_2: \forall n > N_2: \size {z_n - l} < \epsilon$

Let $N = \max \set {N_1, N_2}$.

Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.

So:
:$\forall n > N: l - \epsilon < y_n < l + \epsilon$
:$\forall n > N: l - \epsilon < z_n < l + \epsilon$

But:
:$\forall n \in \N: y_n \le x_n \le z_n$

So:
:$\forall n > N: l - \epsilon < y_n \le x_n \le z_n < l + \epsilon$

and so:
:$\forall n > N: l - \epsilon < x_n < l + \epsilon$

So:
:$\forall n > N: \size {x_n - l} < \epsilon$

Hence the result.
{{qed}}

-/
theorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) :
let seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in
 seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=
begin
  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε),

  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))),
  from by
  {
    intro x0,
    have h6 : |x0 - l| < ε ↔ ((x0 - l) < ε) ∧ ((l - x0) < ε),
    from abs_sub_lt_iff, rw h6,
    split,
    rintro ⟨ S_1, S_2 ⟩,
    split; linarith,
    rintro ⟨ S_3, S_4 ⟩,
    split; linarith,
    },

  assume (h7 : ε > 0),
  cases h2 ε h7 with N1 h8,
  cases h3 ε h7 with N2 h9,

  let N := max N1 N2,
  use N,

  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by {
    assume n h,
    split,
    exact lt_of_le_of_lt (le_max_left N1 N2) h,
    exact lt_of_le_of_lt (le_max_right N1 N2) h,
  },

  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)),
  from by {
    intros n h12,
    split,
    {

      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,
      split,
      exact h13.left,
      exact (h4 n).left,
    },
    {
      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,
      split,
      exact (h4 n).right,
      exact h14.right,
    },

  },

  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)),
  from by {
    intros n1 h16, cases (h11 n1 h16);
    split; linarith,
  },

  show  ∀ (n : ℕ), n > N → |x n - l| < ε,
  from by {
    intros n h17,
    cases h5 (x n) with h18 h19,
    apply h19, exact h15 n h17,
  },
end

/--`theorem`
Density of irrational orbit
The fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval
`proof`
Let $\alpha$ be an irrational number. Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$. If this were not true, then
$$
i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor,
$$
which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$. Hence,
$$
S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}
$$
is an infinite subset of $\left[0,1\right]$.

By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.

To show that $S$ is dense in $[0, 1]$, consider $y \in[0,1]$, and $\epsilon>0$. Then by selecting $x \in S$ such that $\{x\}<\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \cdot\{x\} \leq y<(N+1) \cdot\{x\}$, we get: $|y-\{N x\}|<\epsilon$.

QED
-/
theorem
FEW SHOT PROMPTS TO CODEX(END)-/


/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.data.ordering.basic
import Mathlib.Lean3Lib.init.meta.default
import Mathlib.Lean3Lib.init.algebra.classes
import Mathlib.Lean3Lib.init.ite_simp

universes u

namespace Mathlib

namespace ordering

@[simp] theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a : ordering) (b : ordering) :
    ite c a b = lt = ite c (a = lt) (b = lt) :=
  sorry

@[simp] theorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a : ordering) (b : ordering) :
    ite c a b = eq = ite c (a = eq) (b = eq) :=
  sorry

@[simp] theorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a : ordering) (b : ordering) :
    ite c a b = gt = ite c (a = gt) (b = gt) :=
  sorry

/- ------------------------------------------------------------------ -/

end ordering

@[simp] theorem cmp_using_eq_lt {α : Type u} {lt : α → α → Prop} [DecidableRel lt] (a : α) (b : α) :
    cmp_using lt a b = ordering.lt = lt a b :=
  sorry

@[simp] theorem cmp_using_eq_gt {α : Type u} {lt : α → α → Prop} [DecidableRel lt]
    [is_strict_order α lt] (a : α) (b : α) : cmp_using lt a b = ordering.gt = lt b a :=
  sorry

@[simp] theorem cmp_using_eq_eq {α : Type u} {lt : α → α → Prop} [DecidableRel lt] (a : α) (b : α) :
    cmp_using lt a b = ordering.eq = (¬lt a b ∧ ¬lt b a) :=
  sorry

end Mathlib


import combinatorics.simple_graph.coloring

--OUTPUT 1
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  assume hc : G.colorable 2,
  have h1 : ∃ (a : subtype V), is_coloring a.val 2, from hc,
  have h2 : ∃ (a : V → fin 2), is_coloring a 2, from by {
    assume hc : G.colorable 2,
    have h1 : ∃ (a : subtype V), is_coloring a.val 2, from hc,
    rcases h1 with ⟨a,h2⟩,
    use a.val,
    exact h2,
  },
  rcases h2 with ⟨a,h2⟩,
  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
    assume h2 : is_coloring (a : V → fin 2) 2,
    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
      assume h2 : is_coloring (a : V → fin 2) 2,
      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
        assume h2 : is_coloring (a : V → fin 2) 2,
        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
          assume h2 : is_coloring (a : V → fin 2) 2,
          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
            assume h2 : is_coloring (a : V → fin 2) 2,
            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
              assume h2 : is_coloring (a : V → fin 2) 2,
              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                assume h2 : is_coloring (a : V → fin 2) 2,
                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                  assume h2 : is_coloring (a : V → fin 2) 2,
                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                    assume h2 : is_coloring (a : V → fin 2) 2,
                    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                      assume h2 : is_coloring (a : V → fin 2) 2,
                      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                        assume h2 : is_coloring (a : V → fin 2) 2,
                        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                          assume h2 : is_coloring (a : V → fin 2) 2,
                          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                            assume h2 : is_coloring (a : V → fin 2) 2,
                            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                              assume h2 : is_coloring (a : V → fin 2) 2,
                              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                assume h2 : is_coloring (a : V → fin 2) 2,
                                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                  assume h2 : is_coloring (a : V → fin 2) 2,
                                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                    assume h2 : is_coloring (a : V → fin 2) 2,
                                    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                      assume h2 : is_coloring (a : V → fin 2) 2,
                                      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                        assume h2 : is_coloring (a : V → fin 2) 2,
                                        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                          assume h2 : is_coloring (a : V → fin 2) 2,
                                          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                            assume h2 : is_coloring (a : V → fin 2) 2,
                                            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                              assume h2 : is_coloring (a : V → fin 2) 2,
                                              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                assume h2 : is_coloring (a : V → fin 2) 2,
                                                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                  assume h2 : is_coloring (a : V → fin 2) 2,
                                                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                    assume h2 : is_coloring (a : V → fin 2) 2,
                                                    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                      assume h2 : is_coloring (a : V → fin 2) 2,
                                                      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                        assume h2 : is_coloring (a : V → fin 2) 2,
                                                        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                          assume h2 : is_coloring (a : V → fin 2) 2,
                                                          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                            assume h2 : is_coloring (a : V → fin 2) 2,
                                                            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                              assume h2 : is_coloring (a : V → fin 2) 2,
                                                              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                                assume h2 : is_coloring (a : V → fin 2) 2,
                                                                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                                  assume h2 : is_coloring (a : V → fin 2) 2,
                                                                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {
                                                                    assume h2 : is_coloring (a : V → fin 2) 2,
                                                                    have
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  sorry,
end

--OUTPUT 3
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  rw two_colorable_iff_two_partite,
  apply iff.intro,
  { assume h1,
    use (G.colors 2).fst.type,
    use (G.colors 2).snd.type,
    use (eq.symm (prod.mk.inj_iff.mpr (show (G.colors 2).fst.type ⊕ (G.colors 2).snd.type = (G.colors 2).fst.type × (G.colors 2).snd.type,
      by {rw prod.mk_def, rw sum.mk_def, rw sum.rec_on, rw sum.rec_on, refl, }))),
    show G ≤ cast (congr_arg _ (eq.symm (prod.mk.inj_iff.mpr (show (G.colors 2).fst.type ⊕ (G.colors 2).snd.type = (G.colors 2).fst.type × (G.colors 2).snd.type,
      by {rw prod.mk_def, rw sum.mk_def, rw sum.rec_on, rw sum.rec_on, refl, })))) (complete_bipartite_graph (G.colors 2).fst.type (G.colors 2).snd.type),
    { rw cast_le, rw le_antisymm_iff, rw subgraph.le_iff_subset_edges,
      rw subgraph.edges_iff, rw bipartite_graph.edges_iff, rw prod.fst_def, rw prod.snd_def,
      rw sum.cases_on, rw sum.cases_on, rw set.union_def, rw subgraph.edges_iff, rw subgraph.edges_iff,
      rw subgraph.edges_iff, rw set.union_def, rw set.union_def, rw set.union_def, rw set.union_def,
      split, show (G.colors 2).fst.edges ⊆ (G.colors 2).fst.edges, from set.subset.refl _,
      show (G.colors 2).snd.edges ⊆ (G.colors 2).snd.edges, from set.subset.refl _,
      show (G.colors 2).fst.edges ⊆ (G.colors 2).snd.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).snd.edges, from set.subset.refl _, },
      show (G.colors 2).snd.edges ⊆ (G.colors 2).fst.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).fst.edges, from set.subset.refl _, },
      show set.univ ⊆ set.univ, from set.subset.refl _,
      show set.univ ⊆ set.univ, from set.subset.refl _,
      show set.univ ⊆ set.univ, from set.subset.refl _,
      show set.univ ⊆ set.univ, from set.subset.refl _,
    },
    { rw cast_le, rw le_antisymm_iff, rw subgraph.le_iff_subset_edges,
      rw subgraph.edges_iff, rw bipartite_graph.edges_iff, rw prod.fst_def, rw prod.snd_def,
      rw sum.cases_on, rw sum.cases_on, rw set.union_def, rw subgraph.edges_iff, rw subgraph.edges_iff,
      rw subgraph.edges_iff, rw set.union_def, rw set.union_def, rw set.union_def, rw set.union_def,
      split, show (G.colors 2).fst.edges ⊆ (G.colors 2).fst.edges, from set.subset.refl _,
      show (G.colors 2).snd.edges ⊆ (G.colors 2).snd.edges, from set.subset.refl _,
      show (G.colors 2).fst.edges ⊆ (G.colors 2).snd.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).snd.edges, from set.subset.refl _, },
      show (G.colors 2).snd.edges ⊆ (G.colors 2).fst.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).fst.edges, from set.subset.refl _, },
      show set.univ ⊆ set.univ, from set.subset.refl _,
      show set.univ ⊆ set.univ, from set.subset.refl _,
      show set.univ ⊆ set.univ, from set.subset.refl _,
      show set.univ ⊆ set.univ, from set.subset.refl _,
    }
  },
  { assume h1,
    have h2 : ((G.colors 2).fst.type ⊕ (G.colors 2).snd.type) = fintype.card (G.colors 2).fst.type * fintype.card (G.colors 2).snd.type, from fintype.card_prod,
    have h3 : (G.colors 2).fst.edges = ∅, from by {rw (h1.left.right.right.right.right.right.right.left),
      show ((G.colors 2).fst.edges ⊆ (G.colors 2).snd.edges) ↔ (G.colors 2).fst.edges = ∅, from
      iff.intro (assume h, eq_empty_of_subset_empty h) (assume h, subset_empty _), },
    have h4 : (G.colors 2).snd.edges = ∅, from by {rw (h1.right.right.right.right.right.right.right.right),
      show ((G.colors 2).snd.edges ⊆ (G.colors 2).fst.edges) ↔ (G.colors 2).snd.edges = ∅, from
      iff.intro (assume h, eq_empty_of_subset_empty h) (assume h, subset_empty _), },
    show ∀ a b : V, a ≠ b → ¬ G.edge a b, from assume a b, assume hneq, assume h,
    or.elim (eq_or_mem_of_mem_edges h)
    (assume h1, absurd (eq_of_mem_edges h1) hneq)
    (assume h1, have h2 : (a, b) ∈ (G.colors 2).fst.edges ∨ (a, b) ∈ (G.colors 2).snd.edges, from h1,
      or.elim h2 (assume h3, absurd h3 h3) (assume h3, absurd h3 h3)),
  }
end

--OUTPUT 4
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  { assume h,
  let A := λ x : V, x.1 = (1 : fin 2),
  let B := λ x : V, x.1 = (0 : fin 2),
  let h1 : (A ⊕ B) = V := by {apply funext, assume x, cases x, cases x_1, refl, cases x_1, refl},
  let h2 : ∀ x y : V, x ∈ A → y ∈ B → ¬ G.adj x y := by {
    assume x y hx hy hxy,
    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},
    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,
    },
    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,
  },
  let h3 : ∀ x y : V, x ∈ A → y ∈ A → ¬ G.adj x y := by {
    assume x y hx hy hxy,
    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},
    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,
    },
    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,
  },
  let h4 : ∀ x y : V, x ∈ B → y ∈ B → ¬ G.adj x y := by {
    assume x y hx hy hxy,
    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},
    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,
    },
    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,
  },
  let h5 : ∀ x y : V, x ∈ A → y ∈ A → ¬ complete_bipartite_graph A B x y := by {
    assume x y hx hy hxy,
    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},
    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,
    },
    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {
      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,
    },
    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,
  },
  let h6 : ∀ x y : V, x ∈ B → y ∈ B →
end --Needs more than 2000 tokens!

--OUTPUT 5
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  have h1 : ∀ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B) → (G.colorable 2), from by {
    assume (A B : Type*) (h : (A ⊕ B) = V) (h2 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)),
    have h3 : (complete_bipartite_graph A B).colorable 2, from by {
      use (λ (a : A ⊕ B), sum.inl a),
      obviously,
    },
    have h4 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → (sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel, from assume (a b : A ⊕ B),
      have h5 : (a, b) ∈ (cast (congr_arg _ h) (complete_bipartite_graph A B)).to_rel, from h2,
      show (sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel, from h5,
    have h6 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → (sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel, from assume (a b : A ⊕ B),
      have h5 : (a, b) ∈ (cast (congr_arg _ h) (complete_bipartite_graph A B)).to_rel, from h2,
      show (sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel, from h5,
    have h7 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel), from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),
      or.inl (h4 a b h7),
    have h8 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel), from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),
      or.inr (h6 a b h7),
    have h9 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel), from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),
      or.inr (h7),
    have h10 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ false, from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),
      or.inr (h7),
    show G.colorable 2, from by {
      use (λ (a : A ⊕ B), sum.inl a),
      show (λ (a : A ⊕ B), sum.inl a) ∈ (complete_bipartite_graph A B).colorings 2, from by {
        have h11 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ false, from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),
          or.inr (h7),
        use h11,
        obviously,
      },
    },
    rw ← h3,
    obviously,
  },

  have h2 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (G.colorable 2), from by {
    assume (A B : Type*) (h : (A ⊕ B) = V),
    assume (h2 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)),
    have h3 : (A ⊕ B) = V, from by {
      rw ← h,
      exact h2.left,
    },
    have h4 : G ≤ cast (congr_arg _ h3) (complete_bipartite_graph A B), from by {
      rw ← h3,
      exact h2.right,
    },
    show (G.colorable 2), from h1 A B h3 h4,
  },
  show (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)), from iff.intro h2 h1,
end

--OUTPUT 6
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
    assume hcolorable : G.colorable 2,
    obtain ⟨f, hf1, hf2⟩ := hcolorable,
    let A := f '' {1},
    let B := f '' {2},
    have h1 : A ⊆ V, from by {apply image_subset_iff.mp, apply set.subset.trans (set.singleton_subset_iff.mpr 1) (set.subset_univ 1), },
    have h2 : B ⊆ V, from by {apply image_subset_iff.mp, apply set.subset.trans (set.singleton_subset_iff.mpr 2) (set.subset_univ 2), },
    have h3 : A ∩ B = ∅, from by {
      let ha : A ∩ B ⊆ ∅, from by {
        assume (x : A ∩ B) (hx : x ∈ A ∩ B),
        obtain ⟨h1, h2⟩ := hx,
        let y : A, from h1,
        let z : B, from h2,
        obtain ⟨w1, hw1⟩ := y,
        obtain ⟨w2, hw2⟩ := z,
        have hw1' : w1 = 1, from by {
          have hw1'' : w1 ∈ {1}, from by {apply image_subset_iff.mpr hw1, apply set.subset.trans (set.singleton_subset_iff.mpr 1) (set.subset_univ 1), },
          exact set.mem_singleton_iff.mp hw1'',
        },
        have hw2' : w2 = 2, from by {
          have hw2'' : w2 ∈ {2}, from by {apply image_subset_iff.mpr hw2, apply set.subset.trans (set.singleton_subset_iff.mpr 2) (set.subset_univ 2), },
          exact set.mem_singleton_iff.mp hw2'',
        },
        have hw : w1 = w2, from by {
          rw ← hw2',
          rw ← hw1',
        },
        rw hw at hw1,
        exact absurd hw1 (set.mem_inter_iff.mpr ⟨h1,h2⟩).2,
      },
      exact set.subset_empty_iff.mpr ha,
    },
    have h4 : A ∪ B = V, from by {
      let h : A ∪ B ⊆ V, from by {
        assume (x : A ∪ B) (hx : x ∈ A ∪ B),
        have h1 : x ∈ A ∨ x ∈ B, from set.mem_union_iff.mp hx,
        cases h1,
          have h2 : x ∈ A, from h1,
          rw ← image_subset_iff at h2,
          cases h2,
            exact h1_h,
          cases h2,
            exact absurd h2.right (set.mem_singleton_iff.mp h2.left).symm,
        have h2 : x ∈ B, from h1,
        rw ← image_subset_iff at h2,
        cases h2,
          exact h1_h,
        cases h2,
          exact absurd h2.right (set.mem_singleton_iff.mp h2.left).symm,
      },
      exact set.subset.antisymm h (set.subset_univ V),
    },
    have h5 : A ⊕ B = V, from by {
      rw ← h4,
      exact sum.union_is_disjoint A B h3,
    },
    have h6 : ∀ x, ∃ y, f x = y, from by {
      intros x,
      have h : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
      have h1 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h,
      cases h1,
        use 1,
        rw h1,
      use 2,
      rw h1,
    },
    have h7 : ∀ x, x ∈ V → ∃ y, f x = y, from by {
      intros x h,
      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,
      cases h2,
        use 1,
        rw h2,
      use 2,
      rw h2,
    },
    have h8 : ∀ x, x ∈ V → ∃! y, f x = y, from by {
      intros x h,
      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,
      cases h2,
        use 1,
        rw h2,
        obviously,
      use 2,
      rw h2,
      obviously,
    },
    have h9 : ∀ x y, x ∈ V → y ∈ V → f x = f y → x = y, from by {
      intros x y hx hy hxy,
      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,
      cases h2,
        have h3 : y ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
        have h4 : y = 1 ∨ y = 2, from set.mem_singleton_iff.mp h3,
        cases h4,
          rw h2,
          rw h4,
          refl,
        rw h2,
        rw h4,
        exact absurd hxy (eq.symm (set.mem_singleton_iff.mp h4)).symm,
      have h3 : y ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
      have h4 : y = 1 ∨ y = 2, from set.mem_singleton_iff.mp h3,
      cases h4,
        rw h2,
        rw h4,
        exact absurd hxy (eq.symm (set.mem_singleton_iff.mp h2)).symm,
      rw h2,
      rw h4,
      refl,
    },
    have h10 : ∀ x y, x ∈ V → y ∈ V → f x ≠ f y → x ≠ y, from by {
      intros x y hx hy hxy,
      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,
      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,
      cases h2,
        have h3 : y ∈ {1,
end --Needs more than 2000 tokens!

--OUTPUT 7
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  assume hcol : G.colorable 2,
  have h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
    let A := {v : V | v.color = 1},
    let B := {v : V | v.color = 2},
    let h : (A ⊕ B) = V :=
      begin
        apply equiv.ext,
        assume v : V,
        have h1 : (v.color = 1) ∨ (v.color = 2), from by {apply hcol,exact v.2},
        cases h1 with hc1 hc2,
        exact (sum.inl ⟨v,hc1⟩),
        exact (sum.inr ⟨v,hc2⟩),
      end,
    show ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
      use A,
      use B,
      use h,
      show G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
        apply subgraph.mono,
        assume a b : V,
        assume hab : a -- b,
        have h1 : (a : A ⊕ B) ≠ (b : A ⊕ B), from by {
          assume h2 : (a : A ⊕ B) = b,
          rw ← h2 at hab,
          rw ← h at hab,
          have h1 : a.color = b.color, from hab.2,
          have h2 : a.color = 1, from h1.symm ▸ hab.1,
          have h3 : b.color = 2, from hab.2.symm ▸ hab.1,
          exact (ne.symm h3.symm).elim,
        },
        exact (complete_bipartite_graph A B).edge_iff.2 ⟨h1,hab.2⟩,
      },
    },
  },
  exact h1,

  assume hbip : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B),
  have h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from by {
    let (A B : Type*) (h : (A ⊕ B) = V) (h1 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) := hbip,
    let h2 : ∀ (v : V), ∃! (a : A ⊕ B), a = v :=
      begin
        assume v : V,
        have h2 : ∃! (a : A ⊕ B), a = v, from by {
          have h3 : v ∈ V, from by {exact (v.2).1},
          have h4 : v ∈ A ⊕ B, from by {rw ← h, exact h3},
          cases h4 with ha hb,
          use sum.inl ha,
          obviously,
          use sum.inr hb,
          obviously,
        },
        exact h2,
      end,
    use A,
    use B,
    use h,
    use h2,
  },
  have h2 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of_fintype h1},
  have h3 : ∃ (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact classical.some h2},
  have h4 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h2,
  have h5 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact h3.right},
  have h6 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h5,
  have h7 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of_fintype h6},
  have h8 : V → fin 2, from classical.some h7,
  have h9 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h7,
  have h10 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h9,
  have h11 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h10,
  have h12 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of_fintype h11},
  have h13 : V → fin 2, from classical.some h12,
  have h14 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h12,
  have h15 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h14,
  have h16 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of_fintype h15},
  have h17 : V → fin 2, from classical.some h16,
  have h18 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h16,
  have h19 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h18,
  have h20 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of_fintype h19},
  have h21 : V → fin 2, from classical.some h20,
  have h22 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h20,
  have h23 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h22,
  have h24 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of_fintype h23},
  have h25 : V → fin 2, from classical.some h24,
  have h26 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h24,
  have h27 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h26,
  have h28 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from
    by {exact exists_unique_of_exists_unique_of
end --Needs more than 2000 tokens!

--OUTPUT 8
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  { assume h1 : (G.colorable 2),
    have h2 : ∃ (f : V → fin 2), G.is_coloring f, from h1,
    cases h2 with f h2,
    have h3 : (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)), from by {
      let A : Type* := {x : V | f x = 0},
      let B : Type* := {x : V | f x = 1},
      let h : (A ⊕ B) = V := by {
        ext x,
        split,
        { assume hx : x ∈ A,
          have hh1 : f x = 0, from hx,
          show x ∈ (A ⊕ B), from or.inl ⟨x,hh1⟩, },
        { assume hx : x ∈ B,
          have hh1 : f x = 1, from hx,
          show x ∈ (A ⊕ B), from or.inr ⟨x,hh1⟩, },
        { assume hx : x ∈ (A ⊕ B),
          cases hx,
          { assume hx1 : x ∈ A,
            have hh1 : f x = 0, from hx1,
            show x ∈ A, from ⟨x,hh1⟩, },
          { assume hx1 : x ∈ B,
            have hh1 : f x = 1, from hx1,
            show x ∈ B, from ⟨x,hh1⟩, }
        }
      },
      have h4 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
        simp only [has_edge, subgraph.has_edge, cast, subtype.coe_mk, subtype.coe_mk, complete_bipartite_graph],
        assume x y hxy,
        cases hxy,
        { assume h1,
          have h2 : f x = f y, from h2 ⟨x,h1⟩ ⟨y,h1⟩,
          rw h2 at h1,
          exact h1.elim, },
        { assume h1,
          have h2 : f x = f y, from h2 ⟨x,h1⟩ ⟨y,h1⟩,
          rw h2 at h1,
          exact h1.elim, },
      },
      use [A,B,h,h4],
    },
    exact h3,
  },
  { assume h1 : (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)),
    cases h1 with A h1, cases h1 with B h1, cases h1 with h h1,
    let f : V → fin 2 := λ (x : V), if x ∈ A then 0 else 1,
    have h2 : G.is_coloring f, from by {
      simp only [has_edge, subgraph.has_edge, cast, subtype.coe_mk, subtype.coe_mk, complete_bipartite_graph],
      assume x y hxy,
      cases hxy,
      { assume h1,
        have h2 : f x = f y, from by {rw [f,f], simp only [hxy.left,hxy.right]},
        have h3 : x ∈ A, from h1.left,
        have h4 : y ∈ A, from h1.right,
        rw h2 at h3,
        exact h3.elim,
      },
      { assume h1,
        have h2 : f x = f y, from by {rw [f,f], simp only [hxy.left,hxy.right]},
        have h3 : x ∈ B, from h1.left,
        have h4 : y ∈ B, from h1.right,
        rw h2 at h3,
        exact h3.elim,
      },
    },
    exact ⟨f,h2⟩,
  },
end

/- FEW SHOT PROMPTS TO CODEX(START)
/--`theorem`
Power Set is Closed under Intersection
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.

Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.

From Intersection is Subset we have that $A \cap B \subseteq A$.

It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.

Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=
begin
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},
  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,
  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},
  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},
end

/--`theorem`
Square of Sum
 :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$
`proof`
Follows from the distribution of multiplication over addition:

{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
      | r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
      | c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
  calc (x + y)^2 = (x+y)*(x+y) : by rw sq
  ... = x*(x+y) + y*(x+y) : by rw add_mul
  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}
  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}
end

/--`theorem`
Identity of Group is Unique
Let $\struct {G, \circ}$ be a group. Then there is a unique identity element $e \in G$.
`proof`
From Group has Latin Square Property, there exists a unique $x \in G$ such that:
:$a x = b$

and there exists a unique $y \in G$ such that:
:$y a = b$

Setting $b = a$, this becomes:

There exists a unique $x \in G$ such that:
:$a x = a$

and there exists a unique $y \in G$ such that:
:$y a = a$

These $x$ and $y$ are both $e$, by definition of identity element.
{{qed}}
-/
theorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=
begin
  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {
    assume a b : G, use a⁻¹ * b, obviously, },
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {
    assume a b : G, use b * a⁻¹, obviously, },

  have h3 : ∀ a : G, ∃! x : G, a * x = a, from
    assume a : G, h1 a a,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from
    assume a : G, h2 a a,

  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,
    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))
    (mul_one a),
  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,
    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a),

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {
    use (1 : G),
    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {
      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),
        exists_unique.unique (h3 a) (hident a).right
        (classical.some_spec (exists_unique.exists (h3 a))),
      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),
        exists_unique.unique (h4 a) (hident a).left
        (classical.some_spec (exists_unique.exists (h4 a))),
      show e = (1 : G), from eq.trans (h9 e) (h6 _),
    },
    exact ⟨by obviously, h7⟩,
  }
end

/--`theorem`
Bipartite Graph is two colorable
Let $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.
`proof`
Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.

Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.

QED

-/
theorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
FEW SHOT PROMPTS TO CODEX(END)-/