/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad, Simon Hudon
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.pfunctor.multivariate.basic
import Mathlib.data.qpf.multivariate.basic
import Mathlib.PostPort

universes u u_1

namespace Mathlib

/-!
# The composition of QPFs is itself a QPF

We define composition between one `n`-ary functor and `n` `m`-ary functors
and show that it preserves the QPF structure
-/

namespace mvqpf

/-- Composition of an `n`-ary functor with `n` `m`-ary
functors gives us one `m`-ary functor -/
def comp {n : ℕ} {m : ℕ} (F : typevec n → Type u_1) (G : fin2 n → typevec m → Type u) (v : typevec m) :=
  F fun (i : fin2 n) => G i v

namespace comp

protected instance inhabited {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} {G : fin2 n → typevec m → Type u} {α : typevec m} [I : Inhabited (F fun (i : fin2 n) => G i α)] : Inhabited (comp F G α) :=
  I

/-- Constructor for functor composition -/
protected def mk {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} {G : fin2 n → typevec m → Type u} {α : typevec m} (x : F fun (i : fin2 n) => G i α) : comp F G α :=
  x

/-- Destructor for functor composition -/
protected def get {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} {G : fin2 n → typevec m → Type u} {α : typevec m} (x : comp F G α) : F fun (i : fin2 n) => G i α :=
  x

@[simp] protected theorem mk_get {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} {G : fin2 n → typevec m → Type u} {α : typevec m} (x : comp F G α) : comp.mk (comp.get x) = x :=
  rfl

@[simp] protected theorem get_mk {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} {G : fin2 n → typevec m → Type u} {α : typevec m} (x : F fun (i : fin2 n) => G i α) : comp.get (comp.mk x) = x :=
  rfl

/-- map operation defined on a vector of functors -/
protected def map' {n : ℕ} {m : ℕ} {G : fin2 n → typevec m → Type u} [fG : (i : fin2 n) → mvfunctor (G i)] {α : typevec m} {β : typevec m} (f : typevec.arrow α β) : typevec.arrow (fun (i : fin2 n) => G i α) fun (i : fin2 n) => G i β :=
  fun (i : fin2 n) => mvfunctor.map f

/-- The composition of functors is itself functorial -/
protected def map {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} [fF : mvfunctor F] {G : fin2 n → typevec m → Type u} [fG : (i : fin2 n) → mvfunctor (G i)] {α : typevec m} {β : typevec m} (f : typevec.arrow α β) : comp F G α → comp F G β :=
  mvfunctor.map fun (i : fin2 n) => mvfunctor.map f

protected instance mvfunctor {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} [fF : mvfunctor F] {G : fin2 n → typevec m → Type u} [fG : (i : fin2 n) → mvfunctor (G i)] : mvfunctor (comp F G) :=
  mvfunctor.mk fun (α β : typevec m) => comp.map

theorem map_mk {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} [fF : mvfunctor F] {G : fin2 n → typevec m → Type u} [fG : (i : fin2 n) → mvfunctor (G i)] {α : typevec m} {β : typevec m} (f : typevec.arrow α β) (x : F fun (i : fin2 n) => G i α) : mvfunctor.map f (comp.mk x) = comp.mk (mvfunctor.map (fun (i : fin2 n) (x : G i α) => mvfunctor.map f x) x) :=
  rfl

theorem get_map {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} [fF : mvfunctor F] {G : fin2 n → typevec m → Type u} [fG : (i : fin2 n) → mvfunctor (G i)] {α : typevec m} {β : typevec m} (f : typevec.arrow α β) (x : comp F G α) : comp.get (mvfunctor.map f x) = mvfunctor.map (fun (i : fin2 n) (x : G i α) => mvfunctor.map f x) (comp.get x) :=
  rfl

protected instance mvqpf {n : ℕ} {m : ℕ} {F : typevec n → Type u_1} [fF : mvfunctor F] [q : mvqpf F] {G : fin2 n → typevec m → Type u} [fG : (i : fin2 n) → mvfunctor (G i)] [q' : (i : fin2 n) → mvqpf (G i)] : mvqpf (comp F G) :=
  mk (mvpfunctor.comp (P F) fun (i : fin2 n) => P (G i))
    (fun (α : typevec m) => comp.mk ∘ (mvfunctor.map fun (i : fin2 n) => abs) ∘ abs ∘ mvpfunctor.comp.get)
    (fun (α : typevec m) => mvpfunctor.comp.mk ∘ repr ∘ (mvfunctor.map fun (i : fin2 n) => repr) ∘ comp.get) sorry sorry


/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import category_theory.functor.basic

/-!
# Unbundled functors, as a typeclass decorating the object-level function.

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
-/

namespace category_theory

-- declare the `v`'s first; see `category_theory.category` for an explanation
universes v v₁ v₂ v₃ u u₁ u₂ u₃

variables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]

/-- A unbundled functor. -/
-- Perhaps in the future we could redefine `functor` in terms of this, but that isn't the
-- immediate plan.
class functorial (F : C → D) : Type (max v₁ v₂ u₁ u₂) :=
(map       : Π {X Y : C}, (X ⟶ Y) → ((F X) ⟶ (F Y)))
(map_id'   : ∀ (X : C), map (𝟙 X) = 𝟙 (F X) . obviously)
(map_comp' : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = (map f) ≫ (map g) . obviously)

/--
If `F : C → D` (just a function) has `[functorial F]`,
we can write `map F f : F X ⟶ F Y` for the action of `F` on a morphism `f : X ⟶ Y`.
-/
def map (F : C → D) [functorial.{v₁ v₂} F] {X Y : C} (f : X ⟶ Y) : F X ⟶ F Y :=
functorial.map.{v₁ v₂} f

@[simp] lemma map_as_map {F : C → D} [functorial.{v₁ v₂} F] {X Y : C} {f : X ⟶ Y} :
  functorial.map.{v₁ v₂} f = map F f := rfl

@[simp] lemma functorial.map_id {F : C → D} [functorial.{v₁ v₂} F] {X : C} :
  map F (𝟙 X) = 𝟙 (F X) := functorial.map_id' X

@[simp] lemma functorial.map_comp
  {F : C → D} [functorial.{v₁ v₂} F] {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} :
  map F (f ≫ g) = map F f ≫ map F g := functorial.map_comp' f g

namespace functor

/--
Bundle a functorial function as a functor.
-/
def of (F : C → D) [I : functorial.{v₁ v₂} F] : C ⥤ D :=
{ obj := F,
  ..I }

end functor

instance (F : C ⥤ D) : functorial.{v₁ v₂} (F.obj) := { .. F }

@[simp]
lemma map_functorial_obj (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) : map F.obj f = F.map f := rfl

instance functorial_id : functorial.{v₁ v₁} (id : C → C) :=
{ map := λ X Y f, f }

section
variables {E : Type u₃} [category.{v₃} E]

/--
`G ∘ F` is a functorial if both `F` and `G` are.
-/
-- This is no longer viable as an instance in Lean 3.7,
-- #lint reports an instance loop
-- Will this be a problem?
def functorial_comp (F : C → D) [functorial.{v₁ v₂} F] (G : D → E) [functorial.{v₂ v₃} G] :
  functorial.{v₁ v₃} (G ∘ F) :=
{ ..(functor.of F ⋙ functor.of G) }

end

end category_theory


/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import category_theory.limits.shapes.biproducts
import category_theory.limits.preserves.shapes.zero

/-!
# Preservation of biproducts

We define the image of a (binary) bicone under a functor that preserves zero morphisms and define
classes `preserves_biproduct` and `preserves_binary_biproduct`. We then

* show that a functor that preserves biproducts of a two-element type preserves binary biproducts,
* give the canonical isomorphism between the image of a biproduct and the biproduct of the images,
* show that in a preadditive category, a functor preserves a biproduct if and only if it preserves
  the corresponding product if and only if it preserves the corresponding coproduct.

-/

universes v u u₂

noncomputable theory

open category_theory
open category_theory.limits

namespace category_theory

variables {C : Type u} [category.{v} C] {D : Type u₂} [category.{v} D]

section has_zero_morphisms
variables [has_zero_morphisms C] [has_zero_morphisms D]

namespace functor

section map
variables (F : C ⥤ D) [preserves_zero_morphisms F]

section bicone
variables {J : Type v}

/-- The image of a bicone under a functor. -/
@[simps]
def map_bicone {f : J → C} (b : bicone f) : bicone (F.obj ∘ f) :=
{ X := F.obj b.X,
  π := λ j, F.map (b.π j),
  ι := λ j, F.map (b.ι j),
  ι_π := λ j j',
  begin
    rw ← F.map_comp,
    split_ifs,
    { subst h,
      simp only [bicone_ι_π_self, category_theory.functor.map_id, eq_to_hom_refl] },
    { rw [bicone_ι_π_ne _ h, F.map_zero] }
  end }

end bicone

/-- The image of a binary bicone under a functor. -/
@[simps]
def map_binary_bicone {X Y : C} (b : binary_bicone X Y) : binary_bicone (F.obj X) (F.obj Y) :=
{ X := F.obj b.X,
  fst := F.map b.fst,
  snd := F.map b.snd,
  inl := F.map b.inl,
  inr := F.map b.inr,
  inl_fst' := by rw [← F.map_comp, b.inl_fst, F.map_id],
  inl_snd' := by rw [← F.map_comp, b.inl_snd, F.map_zero],
  inr_fst' := by rw [← F.map_comp, b.inr_fst, F.map_zero],
  inr_snd' := by rw [← F.map_comp, b.inr_snd, F.map_id] }

end map

end functor

open category_theory.functor

namespace limits

section bicone
variables {J : Type v}

/-- A functor `F` preserves biproducts of `f` if `F` maps every bilimit bicone over `f` to a
    bilimit bicone over `F.obj ∘ f`. -/
class preserves_biproduct (f : J → C) (F : C ⥤ D) [preserves_zero_morphisms F] :=
(preserves : Π {b : bicone f}, b.is_bilimit → (F.map_bicone b).is_bilimit)

/-- A functor `F` preserves biproducts of `f` if `F` maps every bilimit bicone over `f` to a
    bilimit bicone over `F.obj ∘ f`. -/
def is_bilimit_of_preserves {f : J → C} (F : C ⥤ D) [preserves_zero_morphisms F]
  [preserves_biproduct f F] {b : bicone f} (hb : b.is_bilimit) : (F.map_bicone b).is_bilimit :=
preserves_biproduct.preserves hb

variables (J)

/-- A functor `F` preserves biproducts of shape `J` if it preserves biproducts of `f` for every
    `f : J → C`. -/
class preserves_biproducts_of_shape (F : C ⥤ D) [preserves_zero_morphisms F] :=
(preserves : Π {f : J → C}, preserves_biproduct f F)

attribute [instance, priority 100] preserves_biproducts_of_shape.preserves

end bicone

/-- A functor `F` preserves finite biproducts if it preserves biproducts of shape `J` whenever
    `J` is a fintype. -/
class preserves_finite_biproducts (F : C ⥤ D) [preserves_zero_morphisms F] :=
(preserves : Π {J : Type v} [fintype J], preserves_biproducts_of_shape J F)

attribute [instance, priority 100] preserves_finite_biproducts.preserves

/-- A functor `F` preserves biproducts if it preserves biproducts of any (small) shape `J`. -/
class preserves_biproducts (F : C ⥤ D) [preserves_zero_morphisms F] :=
(preserves : Π {J : Type v}, preserves_biproducts_of_shape J F)

attribute [instance, priority 100] preserves_biproducts.preserves

@[priority 100]
instance preserves_finite_biproducts_of_preserves_biproducts (F : C ⥤ D)
  [preserves_zero_morphisms F] [preserves_biproducts F] : preserves_finite_biproducts F :=
{ preserves := λ J _, infer_instance }

/-- A functor `F` preserves binary biproducts of `X` and `Y` if `F` maps every bilimit bicone over
    `X` and `Y` to a bilimit bicone over `F.obj X` and `F.obj Y`. -/
class preserves_binary_biproduct (X Y : C) (F : C ⥤ D) [preserves_zero_morphisms F] :=
(preserves : Π {b : binary_bicone X Y}, b.is_bilimit → (F.map_binary_bicone b).is_bilimit)

/-- A functor `F` preserves binary biproducts of `X` and `Y` if `F` maps every bilimit bicone over
    `X` and `Y` to a bilimit bicone over `F.obj X` and `F.obj Y`. -/
def is_binary_bilimit_of_preserves {X Y : C} (F : C ⥤ D) [preserves_zero_morphisms F]
  [preserves_binary_biproduct X Y F] {b : binary_bicone X Y} (hb : b.is_bilimit) :
  (F.map_binary_bicone b).is_bilimit :=
preserves_binary_biproduct.preserves hb

/-- A functor `F` preserves binary biproducts if it preserves the binary biproduct of `X` and `Y`
    for all `X` and `Y`. -/
class preserves_binary_biproducts (F : C ⥤ D) [preserves_zero_morphisms F] :=
(preserves : Π {X Y : C}, preserves_binary_biproduct X Y F . tactic.apply_instance)

/-- A functor that preserves biproducts of a pair preserves binary biproducts. -/
def preserves_binary_biproduct_of_preserves_biproduct (F : C ⥤ D) [preserves_zero_morphisms F]
  (X Y : C) [preserves_biproduct (pair_function X Y) F] : preserves_binary_biproduct X Y F :=
{ preserves := λ b hb,
  { is_limit := is_limit.of_iso_limit
      ((is_limit.postcompose_hom_equiv (by exact diagram_iso_pair _) _).symm
        ((is_bilimit_of_preserves F (b.to_bicone_is_bilimit.symm hb)).is_limit)) $
      cones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy, }),
    is_colimit := is_colimit.of_iso_colimit
      ((is_colimit.precompose_inv_equiv (by exact diagram_iso_pair _ ) _).symm
        ((is_bilimit_of_preserves F (b.to_bicone_is_bilimit.symm hb)).is_colimit)) $
      cocones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy, }) } }

/-- A functor that preserves biproducts of a pair preserves binary biproducts. -/
def preserves_binary_biproducts_of_preserves_biproducts (F : C ⥤ D)
  [preserves_zero_morphisms F] [preserves_biproducts_of_shape walking_pair.{v} F] :
  preserves_binary_biproducts F :=
{ preserves := λ X Y, preserves_binary_biproduct_of_preserves_biproduct F X Y }

attribute [instance, priority 100] preserves_binary_biproducts.preserves

end limits

open category_theory.limits

namespace functor

section bicone
variables {J : Type v} (F : C ⥤ D) [preserves_zero_morphisms F] (f : J → C)
  [has_biproduct f] [preserves_biproduct f F]

instance has_biproduct_of_preserves : has_biproduct (F.obj ∘ f) :=
has_biproduct.mk
{ bicone := F.map_bicone (biproduct.bicone f),
  is_bilimit := preserves_biproduct.preserves (biproduct.is_bilimit _) }

/-- If `F` preserves a biproduct, we get a definitionally nice isomorphism
    `F.obj (⨁ f) ≅ ⨁ (F.obj ∘ f)`. -/
@[simp]
def map_biproduct : F.obj (⨁ f) ≅ ⨁ (F.obj ∘ f) :=
biproduct.unique_up_to_iso _ (preserves_biproduct.preserves (biproduct.is_bilimit _))

lemma map_biproduct_hom : (map_biproduct F f).hom = biproduct.lift (λ j, F.map (biproduct.π f j)) :=
rfl

lemma map_biproduct_inv : (map_biproduct F f).inv = biproduct.desc (λ j, F.map (biproduct.ι f j)) :=
rfl

end bicone

variables (F : C ⥤ D) [preserves_zero_morphisms F] (X Y : C) [has_binary_biproduct X Y]
  [preserves_binary_biproduct X Y F]

instance has_binary_biproduct_of_preserves : has_binary_biproduct (F.obj X) (F.obj Y) :=
has_binary_biproduct.mk
{ bicone := F.map_binary_bicone (binary_biproduct.bicone X Y),
  is_bilimit := preserves_binary_biproduct.preserves (binary_biproduct.is_bilimit _ _) }

/-- If `F` preserves a binary biproduct, we get a definitionally nice isomorphism
    `F.obj (X ⊞ Y) ≅ F.obj X ⊞ F.obj Y`. -/
@[simp]
def map_biprod : F.obj (X ⊞ Y) ≅ F.obj X ⊞ F.obj Y :=
biprod.unique_up_to_iso _ _
  (preserves_binary_biproduct.preserves (binary_biproduct.is_bilimit _ _))

lemma map_biprod_hom : (map_biprod F X Y).hom = biprod.lift (F.map biprod.fst) (F.map biprod.snd) :=
rfl

lemma map_biprod_inv : (map_biprod F X Y).inv = biprod.desc (F.map biprod.inl) (F.map biprod.inr) :=
rfl

end functor

namespace limits
variables (F : C ⥤ D) [preserves_zero_morphisms F]

section bicone
variables {J : Type v} (f : J → C) [has_biproduct f] [preserves_biproduct f F]
  {W : C}

lemma biproduct.map_lift_map_biprod (g : Π j, W ⟶ f j) :
  F.map (biproduct.lift g) ≫ (F.map_biproduct f).hom = biproduct.lift (λ j, F.map (g j)) :=
by { ext, simp [← F.map_comp] }

lemma biproduct.map_biproduct_inv_map_desc (g : Π j, f j ⟶ W) :
  (F.map_biproduct f).inv ≫ F.map (biproduct.desc g) = biproduct.desc (λ j, F.map (g j)) :=
by { ext, simp [← F.map_comp] }

lemma biproduct.map_biproduct_hom_desc (g : Π j, f j ⟶ W) :
  (F.map_biproduct f).hom ≫ biproduct.desc (λ j, F.map (g j)) = F.map (biproduct.desc g) :=
by rw [← biproduct.map_biproduct_inv_map_desc, iso.hom_inv_id_assoc]

end bicone

section binary_bicone
variables (X Y : C) [has_binary_biproduct X Y] [preserves_binary_biproduct X Y F] {W : C}

lemma biprod.map_lift_map_biprod (f : W ⟶ X) (g : W ⟶ Y) :
  F.map (biprod.lift f g) ≫ (F.map_biprod X Y).hom = biprod.lift (F.map f) (F.map g) :=
by ext; simp [← F.map_comp]

lemma biprod.lift_map_biprod (f : W ⟶ X) (g : W ⟶ Y) :
  biprod.lift (F.map f) (F.map g) ≫ (F.map_biprod X Y).inv = F.map (biprod.lift f g) :=
by rw [← biprod.map_lift_map_biprod, category.assoc, iso.hom_inv_id, category.comp_id]

lemma biprod.map_biprod_inv_map_desc (f : X ⟶ W) (g : Y ⟶ W) :
  (F.map_biprod X Y).inv ≫ F.map (biprod.desc f g) = biprod.desc (F.map f) (F.map g) :=
by ext; simp [← F.map_comp]

lemma biprod.map_biprod_hom_desc (f : X ⟶ W) (g : Y ⟶ W) :
 (F.map_biprod X Y).hom ≫ biprod.desc (F.map f) (F.map g) = F.map (biprod.desc f g) :=
by rw [← biprod.map_biprod_inv_map_desc, iso.hom_inv_id_assoc]

end binary_bicone

end limits

end has_zero_morphisms

open category_theory.functor

section preadditive
variables [preadditive C] [preadditive D] (F : C ⥤ D) [preserves_zero_morphisms F]

namespace limits

section fintype
variables {J : Type v} [fintype J]

local attribute [tidy] tactic.discrete_cases

/-- A functor between preadditive categories that preserves (zero morphisms and) finite biproducts
    preserves finite products. -/
def preserves_product_of_preserves_biproduct {f : J → C} [preserves_biproduct f F] :
  preserves_limit (discrete.functor f) F :=
{ preserves := λ c hc, is_limit.of_iso_limit
  ((is_limit.postcompose_inv_equiv (discrete.comp_nat_iso_discrete _ _) _).symm
    (is_bilimit_of_preserves F (bicone_is_bilimit_of_limit_cone_of_is_limit hc)).is_limit) $
  cones.ext (iso.refl _) (by tidy) }

section
local attribute [instance] preserves_product_of_preserves_biproduct

/-- A functor between preadditive categories that preserves (zero morphisms and) finite biproducts
    preserves finite products. -/
def preserves_products_of_shape_of_preserves_biproducts_of_shape
  [preserves_biproducts_of_shape J F] : preserves_limits_of_shape (discrete J) F :=
{ preserves_limit := λ f, preserves_limit_of_iso_diagram _ discrete.nat_iso_functor.symm }

end

/-- A functor between preadditive categories that preserves (zero morphisms and) finite products
    preserves finite biproducts. -/
def preserves_biproduct_of_preserves_product {f : J → C} [preserves_limit (discrete.functor f) F] :
  preserves_biproduct f F :=
{ preserves := λ b hb, is_bilimit_of_is_limit _ $
    is_limit.of_iso_limit ((is_limit.postcompose_hom_equiv (discrete.comp_nat_iso_discrete _ _)
      (F.map_cone b.to_cone)).symm (is_limit_of_preserves F hb.is_limit)) $
      cones.ext (iso.refl _) (by tidy) }

/-- A functor between preadditive categories that preserves (zero morphisms and) finite products
    preserves finite biproducts. -/
def preserves_biproducts_of_shape_of_preserves_products_of_shape
  [preserves_limits_of_shape (discrete J) F] : preserves_biproducts_of_shape J F :=
{ preserves := λ f, preserves_biproduct_of_preserves_product F }

/-- A functor between preadditive categories that preserves (zero morphisms and) finite biproducts
    preserves finite coproducts. -/
def preserves_coproduct_of_preserves_biproduct {f : J → C} [preserves_biproduct f F] :
  preserves_colimit (discrete.functor f) F :=
{ preserves := λ c hc, is_colimit.of_iso_colimit
    ((is_colimit.precompose_hom_equiv (discrete.comp_nat_iso_discrete _ _) _).symm
      (is_bilimit_of_preserves F
        (bicone_is_bilimit_of_colimit_cocone_of_is_colimit hc)).is_colimit) $
    cocones.ext (iso.refl _) (by tidy) }

section
local attribute [instance] preserves_coproduct_of_preserves_biproduct

/-- A functor between preadditive categories that preserves (zero morphisms and) finite biproducts
    preserves finite coproducts. -/
def preserves_coproducts_of_shape_of_preserves_biproducts_of_shape
  [preserves_biproducts_of_shape J F] : preserves_colimits_of_shape (discrete J) F :=
{ preserves_colimit := λ f, preserves_colimit_of_iso_diagram _ discrete.nat_iso_functor.symm }

end

/-- A functor between preadditive categories that preserves (zero morphisms and) finite coproducts
    preserves finite biproducts. -/
def preserves_biproduct_of_preserves_coproduct {f : J → C}
  [preserves_colimit (discrete.functor f) F] : preserves_biproduct f F :=
{ preserves := λ b hb, is_bilimit_of_is_colimit _ $
    is_colimit.of_iso_colimit ((is_colimit.precompose_inv_equiv (discrete.comp_nat_iso_discrete _ _)
      (F.map_cocone b.to_cocone)).symm (is_colimit_of_preserves F hb.is_colimit)) $
      cocones.ext (iso.refl _) (by tidy) }

/-- A functor between preadditive categories that preserves (zero morphisms and) finite coproducts
    preserves finite biproducts. -/
def preserves_biproducts_of_shape_of_preserves_coproducts_of_shape
  [preserves_colimits_of_shape (discrete J) F] : preserves_biproducts_of_shape J F :=
{ preserves := λ f, preserves_biproduct_of_preserves_coproduct F }

end fintype

/-- A functor between preadditive categories that preserves (zero morphisms and) binary biproducts
    preserves binary products. -/
def preserves_binary_product_of_preserves_binary_biproduct {X Y : C}
  [preserves_binary_biproduct X Y F] : preserves_limit (pair X Y) F :=
{ preserves := λ c hc, is_limit.of_iso_limit
    ((is_limit.postcompose_inv_equiv (by exact diagram_iso_pair _) _).symm
      (is_binary_bilimit_of_preserves F
        (binary_bicone_is_bilimit_of_limit_cone_of_is_limit hc)).is_limit) $
    cones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy }) }

section
local attribute [instance] preserves_binary_product_of_preserves_binary_biproduct

/-- A functor between preadditive categories that preserves (zero morphisms and) binary biproducts
    preserves binary products. -/
def preserves_binary_products_of_preserves_binary_biproducts
  [preserves_binary_biproducts F] : preserves_limits_of_shape (discrete walking_pair.{v}) F :=
{ preserves_limit := λ K, preserves_limit_of_iso_diagram _ (diagram_iso_pair _).symm }

end

/-- A functor between preadditive categories that preserves (zero morphisms and) binary products
    preserves binary biproducts. -/
def preserves_binary_biproduct_of_preserves_binary_product {X Y : C}
  [preserves_limit (pair X Y) F] : preserves_binary_biproduct X Y F :=
{ preserves := λ b hb, is_binary_bilimit_of_is_limit _ $
    is_limit.of_iso_limit ((is_limit.postcompose_hom_equiv (by exact diagram_iso_pair _)
      (F.map_cone b.to_cone)).symm (is_limit_of_preserves F hb.is_limit)) $
        cones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy }) }

/-- A functor between preadditive categories that preserves (zero morphisms and) binary products
    preserves binary biproducts. -/
def preserves_binary_biproducts_of_preserves_binary_products
  [preserves_limits_of_shape (discrete walking_pair.{v}) F] : preserves_binary_biproducts F :=
{ preserves := λ X Y, preserves_binary_biproduct_of_preserves_binary_product F }

/-- A functor between preadditive categories that preserves (zero morphisms and) binary biproducts
    preserves binary coproducts. -/
def preserves_binary_coproduct_of_preserves_binary_biproduct {X Y : C}
  [preserves_binary_biproduct X Y F] : preserves_colimit (pair X Y) F :=
{ preserves := λ c hc, is_colimit.of_iso_colimit
    ((is_colimit.precompose_hom_equiv (by exact diagram_iso_pair _) _).symm
      (is_binary_bilimit_of_preserves F
        (binary_bicone_is_bilimit_of_colimit_cocone_of_is_colimit hc)).is_colimit) $
      cocones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy }) }

section
local attribute [instance] preserves_binary_coproduct_of_preserves_binary_biproduct

/-- A functor between preadditive categories that preserves (zero morphisms and) binary biproducts
    preserves binary coproducts. -/
def preserves_binary_coproducts_of_preserves_binary_biproducts
  [preserves_binary_biproducts F] : preserves_colimits_of_shape (discrete walking_pair.{v}) F :=
{ preserves_colimit := λ K, preserves_colimit_of_iso_diagram _ (diagram_iso_pair _).symm }

end

/-- A functor between preadditive categories that preserves (zero morphisms and) binary coproducts
    preserves binary biproducts. -/
def preserves_binary_biproduct_of_preserves_binary_coproduct {X Y : C}
  [preserves_colimit (pair X Y) F] : preserves_binary_biproduct X Y F :=
{ preserves := λ b hb, is_binary_bilimit_of_is_colimit _ $
    is_colimit.of_iso_colimit ((is_colimit.precompose_inv_equiv (by exact diagram_iso_pair _)
      (F.map_cocone b.to_cocone)).symm (is_colimit_of_preserves F hb.is_colimit)) $
        cocones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩, tidy }) }

/-- A functor between preadditive categories that preserves (zero morphisms and) binary coproducts
    preserves binary biproducts. -/
def preserves_binary_biproducts_of_preserves_binary_coproducts
  [preserves_colimits_of_shape (discrete walking_pair.{v}) F] : preserves_binary_biproducts F :=
{ preserves := λ X Y, preserves_binary_biproduct_of_preserves_binary_coproduct F }

end limits

end preadditive

end category_theory


import SciLean.Operators

namespace SciLean

variable  {X Y Z W : Type} [Vec X] [Vec Y] [Vec Z] [Vec W]

structure HArray (Ts : List Type) where
  data : Array (Sigma (λ T : Type => T))
  h_len : Ts.length = data.size
  typed : ∀ i : Fin Ts.length, (data.get (h_len ▸ i)).1 = Ts.get i

namespace HArray

  variable {n} {Ts : List Type}

  def get (u : HArray Ts) (i : Fin Ts.length) : Ts.get i
    := u.typed i ▸ (u.data.get (u.h_len ▸ i)).2

  def getOp (self : HArray Ts) (idx : Fin Ts.length) : Ts.get idx
    := self.typed idx ▸ (self.data.get (self.h_len ▸ idx)).2

  def set (u : HArray Ts) (i : Fin Ts.length) (x : Ts.get i) : HArray Ts
    := ⟨u.data.set (u.h_len ▸ i) (⟨_, x⟩), sorry, sorry⟩

end HArray

class HCurryType (n : Nat) (F : Type) where
  Xs : List Type
  Y  : Type

attribute [reducible] HCurryType.Xs HCurryType.Y

@[reducible]
instance : HCurryType 0 Y where
  Xs := []
  Y := Y

@[reducible]
instance {X Y : Type} [t : HCurryType n Y] : HCurryType (n + 1) (X → Y) where
  Xs := X::t.Xs
  Y := t.Y

class HCurry (i : Nat) (Xs' Xs : List Type) (Y : Type) where
  index_valid : Xs'.length + i = Xs.length
  types_valid : ∀ j, i + j < Xs.length → Xs'.get ⟨j, sorry⟩ = Xs.get ⟨i + j, sorry⟩
  F : Type
  uncurry : F → (HArray Xs → Y)

attribute [reducible] HCurry.F HCurry.uncurry

@[reducible]
instance (Xs : List Type) (Y : Type) : HCurry n [] Xs Y where
  index_valid := sorry
  types_valid := sorry
  F := Y
  uncurry := λ y xs => y

@[reducible]
instance [c : HCurry (i+1) (Xs') Xs Y] : HCurry (i) (X'::Xs') Xs Y where
  index_valid := sorry
  types_valid := sorry
  F := X' → c.F
  uncurry := λ f xs =>
    let h : (Xs.get ⟨i,sorry⟩ = X') := sorry
    let xi : X' := (h ▸ xs[⟨i,sorry⟩])
    c.uncurry (f xi) xs

def huncurry (n : Nat) {F : Type} [HCurryType n F]
  [ci : HCurry 0 (HCurryType.Xs n F) (HCurryType.Xs n F) (HCurryType.Y n F)]
  (f : F) :=
    let h : F = ci.F := sorry
    ci.uncurry (h ▸ f)

example : huncurry 3 (λ (i j k : Nat) => i + j)
          =
          λ xs => xs[#0] + xs[#1] :=
by rfl

example : huncurry 2 (λ (i j k : Nat) => i + j)
          =
          λ xs k => xs[#0] + xs[#1] :=
by rfl


import data.set

variables {α β: Type}

def Property (α : Type) : Type := α → Prop

structure Claim (α : Type) := make :: (X : set α) (P : Property α)

@[instance] def claim_default {α : Type} : inhabited (Claim α) :=
⟨Claim.make ∅ (λ x, true)⟩

namespace list

def All {α : Type} (pred : α → Prop) : list α → Prop
| [] := true
| (h::t) := pred h ∧ All t

end list

@[reducible]
def meaning {α : Type} (C : Claim α) : Prop :=
∀ x ∈ C.X, C.P x

notation ⟦C⟧ := meaning C

structure Strategy (α : Type) :=
(parent : Claim α) (decomp : Claim α → list (Claim α))

@[instance] def strat_default {α : Type} : inhabited (Strategy α) :=
⟨Strategy.mk (default) (λ c, [])⟩

def deductive (α : Type) (S : Strategy α) : Prop :=
let subclaims := (S.decomp) S.parent in
(∀ clm ∈ subclaims, ⟦clm⟧) →  ⟦S.parent⟧

namespace Strategy

def subclaims {α : Type} (S : Strategy α) : list (Claim α) :=
  S.decomp S.parent
end Strategy

namespace property

@[reducible]
def decomposition
(Ps : list (Property α))
(Clm : Claim α) : list (Claim α) :=
list.map (Claim.make Clm.X) Ps

structure input (α : Type) :=
(Clm : Claim α)
(Props : list (Property α))

@[instance] def input_default {α : Type} : inhabited (input α) :=
⟨input.mk (default) []⟩

namespace input

@[reducible]
def length (Γ : input α) : ℕ := Γ.Props.length

lemma len_decomp (Γ : input α) :
Γ.length = (decomposition Γ.Props Γ.Clm).length :=
by {rw decomposition, simp}

@[reducible]
def subsets (Γ : input α) : fin (Γ.length) → set α :=
λ i, set_of (Γ.Props.nth_le i.1 i.2)

end input

def justified
(Γ : input α) : Prop :=
(⋂ i, Γ.subsets i) ⊆ {x | Γ.Clm.P x}

@[reducible]
def strategy (Γ : input α) : Strategy α :=
{ parent := Γ.Clm,
  decomp := λ C, list.map (Claim.make C.X)  Γ.Props }

theorem deductive_of_justfd
: Π {Γ : property.input α},
justified Γ → deductive α (property.strategy Γ) :=
begin
  intro Γ,
  rw [justified,deductive, meaning,strategy],
  simp only [fin.val_eq_coe],
  intros H1 H2 x xMem, apply H1,
  simp only [set.mem_Inter, set.mem_set_of_eq],
  intro i,
  replace H2 := H2 ((decomposition Γ.Props Γ.Clm).nth_le i.1 (fin.cast (input.len_decomp Γ) i).2),
  simp at H2, apply H2,
  apply list.nth_le_mem, assumption,
end

end property

/-
namespace domain

@[reducible]
def to_claim {α : Type}
(P : Property α) (X : set α) : Claim α :=
Claim.mk X P

@[reducible]
def preimages {α β: Type}
(f : α → β)
(Clm : Claim α)
(bs : list (set β)) : list (set α) :=
list.map (set.preimage f) bs

@[reducible]
def decomposition {α β: Type}
(f : α → β)
(sets_range : list (set β))
(Clm : Claim α) : list (Claim α) :=
(preimages f Clm sets_range).map (to_claim Clm.P)

structure auxiliary (α β : Type) :=
(Clm : Claim α)
(f : α → β)
(range_sets : list (set β))

def set_of_list : list α → set α
| [] := ∅
| (h::t) := {h} ∪ set_of_list t

def complete (Γ : auxiliary α β) : Prop :=
 ∀ b : β, ∃ s ∈ Γ.range_sets, b ∈ s

def to_strategy (Γ : auxiliary α β) : strategy α :=
strategy.mk (Γ.Clm) (domain.decomposition Γ.f Γ.range_sets)

theorem deductive_of_justfd_comp (Γ : auxiliary α β) : complete Γ →  deductive α (to_strategy Γ) :=
begin
  rw [complete, deductive, meaning, to_strategy],
  simp only [fin.val_eq_coe],
  intros H1 H2 x xMem,
  unfold decomposition at H2,simp at H2,
  unfold to_claim at H2,
  have H3 : ∃ s ∈ Γ.range_sets, Γ.f x ∈ s, from H1 (Γ.f x),
  rcases H3 with ⟨s,H3,H4⟩,
  replace H2 := H2 s H3,
  apply H2, assumption,
end

end domain
-/


import tactic

-- freely adjoining a unary function to a group
inductive adjoin_fun' (G : Type) [group G] : Type
| of_group (g : G) : adjoin_fun'
| u (g : adjoin_fun') : adjoin_fun'
| mul (g h : adjoin_fun') : adjoin_fun'
| inv (g : adjoin_fun') : adjoin_fun'

variables {G : Type} [group G]

section

local notation x `*` y := adjoin_fun'.mul x y
local notation x`⁻¹` := adjoin_fun'.inv x
local notation `o` := adjoin_fun'.of_group 1

open adjoin_fun'

inductive rel : adjoin_fun' G → adjoin_fun' G → Prop
| mul_assoc (x y z) : rel (x * y * z) (x * (y * z))
| mul_inv (x) : rel (x * x⁻¹) o
| inv_mul (x) : rel (x⁻¹ * x) o
| mul_one (x) : rel (x * o) x
| one_mul (x) : rel (o * x) x
| is_hom (x y : G) : rel (of_group (has_mul.mul x y)) (of_group x * of_group y)
| mul {a₁ a₂ b₁ b₂} : rel a₁ b₁ → rel a₂ b₂ → rel (a₁ * a₂) (b₁ * b₂)
| u {a b} : rel a b → rel (u a) (u b)
| inv {a b} : rel a b → rel (a⁻¹) (b⁻¹)
| refl (x) : rel x x
| symm (x y) : rel x y → rel y x
| trans (x y z) : rel x y → rel y z → rel x z

variable (G)

instance rel_setoid : setoid (adjoin_fun' G) :=
{ r := rel, iseqv := ⟨rel.refl, rel.symm, rel.trans⟩ }

def adjoin_fun : Type := quotient (rel_setoid G)

instance : group (adjoin_fun G) :=
{ mul := λ x y, quotient.lift_on₂ x y (λ x y, quotient.mk (x * y))
      (λ a₁ a₂ b₁ b₂ h₁ h₂, quotient.sound (rel.mul h₁ h₂)),
  one := quotient.mk o,
  inv := λ x, quotient.lift_on x (λ x, quotient.mk (x⁻¹))
    (λ a b h, quotient.sound (rel.inv h)),
  mul_assoc := λ x y z, quotient.induction_on₃ x y z
    (λ x y z, quotient.sound (rel.mul_assoc _ _ _)),
  mul_one := λ x, quotient.induction_on x
    (λ x, quotient.sound (rel.mul_one _)),
  one_mul := λ x, quotient.induction_on x
    (λ x, quotient.sound (rel.one_mul x)),
  mul_left_inv := λ x, @quotient.induction_on (adjoin_fun' G)
    (rel_setoid G) (λ x : adjoin_fun G,
      quotient.lift_on₂ (quotient.lift_on x (λ x, quotient.mk (x⁻¹))
      (λ a b h, quotient.sound (rel.inv h))) (x) (λ x y, quotient.mk (x * y))
      (λ a₁ a₂ b₁ b₂ h₁ h₂, quotient.sound (rel.mul h₁ h₂)) = ⟦o⟧) _
      (λ x, quotient.sound (rel.inv_mul _)) }

variable {G}

def of_group : G →* adjoin_fun G :=
monoid_hom.mk'
  (λ x, ⟦of_group x⟧)
  (λ x y, quotient.sound (rel.is_hom _ _))

def u : adjoin_fun G → adjoin_fun G :=
λ x, quotient.lift_on' x (λ x, ⟦adjoin_fun'.u x⟧)
  (λ a b h, quotient.sound (rel.u h))

end

def UMP {H : Type} [group H] (f : G →* H)
  (u : adjoin_fun G → H → H) : adjoin_fun G →* H :=
monoid_hom.mk'
  (λ g, quotient.lift_on g
    (λ g, show H, from adjoin_fun'.rec_on g
      f
      (λ g h, u ⟦g⟧ h)
      (λ _ _, (*))
      (λ _ h, h⁻¹))
    (λ x y hxy, begin
      induction hxy; simp [*, mul_assoc] at *,
      rw [show ⟦hxy_a⟧ = ⟦hxy_b⟧, from quotient.sound hxy_ᾰ],
    end))
  (λ x y, quotient.induction_on₂ x y (λ x y, rfl))

example {A : Type} : (ℕ → A) ≃ (A × (ℕ → A → A)) :=
{ to_fun := λ f, (f 0, λ n _, f n.succ),
  inv_fun := λ x n, nat.rec_on n x.1 x.2,
  left_inv := λ f, begin
    dsimp,
    ext n,
    cases n,
    { refl },
    { refl },
  end,
  right_inv := begin
    rintros ⟨a, f⟩,
    ext1,
    { refl },
    { ext n,
      dsimp, }

  end }

def UMP' {H : Type} [group H] (f : adjoin_fun G →* H) : (G →* H) × (adjoin_fun G → H → H) :=
⟨f.comp of_group, λ g h, f (u g)⟩


/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/

import data.rat.defs
import tactic.norm_num

/-!
# The category of elliptic curves (over a field or a PID)

We give a working definition of elliptic curves which is mathematically accurate
in many cases, and also good for computation.

## Mathematical background

Let `S` be a scheme. The actual category of elliptic curves over `S` is a large category,
whose objects are schemes `E` equipped with a map `E → S`, a section `S → E`, and some
axioms (the map is smooth and proper and the fibres are geometrically connected group varieties
of dimension 1). In the special case where `S` is `Spec R` for some commutative ring `R`
whose Picard group is trivial (this includes all fields, all principal ideal domains, and many
other commutative rings) then it can be shown (using rather a lot of algebro-geometric machinery)
that every elliptic curve is, up to isomorphism, a projective plane cubic defined by
the equation `y^2+a₁xy+a₃y=x^3+a₂x^2+a₄x+a₆`, with `aᵢ : R`, and such that the discriminant
of the aᵢ is a unit in `R`.

Some more details of the construction can be found on pages 66-69 of
[N. Katz and B. Mazur, *Arithmetic moduli of elliptic curves*][katz_mazur] or pages
53-56 of [P. Deligne, *Courbes elliptiques: formulaire d'après J. Tate*][deligne_formulaire].

## Warning

The definition in this file makes sense for all commutative rings `R`, but it only gives
a type which can be beefed up to a category which is equivalent to the category of elliptic
curves over `Spec R` in the case that `R` has trivial Picard group or, slightly more generally,
when the 12-torsion of Pic(R) is trivial. The issue is that for a general ring R, there
might be elliptic curves over Spec(R) in the sense of algebraic geometry which are not
globally defined by a cubic equation valid over the entire base.

## TODO

Define the R-points (or even A-points if A is an R-algebra). Care will be needed
at infinity if R is not a field. Define the group law on the R-points. (hard) prove associativity.

-/

/-- The discriminant of the plane cubic `Y^2+a1*X*Y+a3*Y=X^3+a2*X^2+a4*X+a6`. If `R` is a field
then this polynomial vanishes iff the cubic curve cut out by this equation is singular. -/
def EllipticCurve.disc_aux {R : Type*} [comm_ring R] (a1 a2 a3 a4 a6 : R) : R :=
-432*a6^2 + ((288*a2 + 72*a1^2)*a4 + (-216*a3^2 + (144*a1*a2 + 36*a1^3)*a3 + (-64*a2^3 -
48*a1^2*a2^2 - 12*a1^4*a2 - a1^6)))*a6 + (-64*a4^3 + (-96*a1*a3 + (16*a2^2 + 8*a1^2*a2 + a1^4))*a4^2
+ ((72*a2 - 30*a1^2)*a3^2 + (16*a1*a2^2 + 8*a1^3*a2 + a1^5)*a3)*a4 + (-27*a3^4 + (36*a1*a2 +
a1^3)*a3^3 + (-16*a2^3 - 8*a1^2*a2^2 - a1^4*a2)*a3^2))

-- If Pic(R)[12]=0 then this definition is mathematically correct
/-- The category of elliptic curves over `R` (note that this definition is only mathematically
correct for certain rings, for example if `R` is a field or a PID). -/
structure EllipticCurve (R : Type*) [comm_ring R] :=
(a1 a2 a3 a4 a6 : R)
(disc_unit : Rˣ)
(disc_unit_eq : (disc_unit : R) = EllipticCurve.disc_aux a1 a2 a3 a4 a6)

namespace EllipticCurve

instance : inhabited (EllipticCurve ℚ) := ⟨⟨0,0,1,-1,0, ⟨37, 37⁻¹, by norm_num, by norm_num⟩,
  show (37 : ℚ) = _ + _, by norm_num⟩⟩

variables {R : Type*} [comm_ring R] (E : EllipticCurve R)

/-- The discriminant of an elliptic curve. Sometimes only defined up to sign in the literature;
  we choose the sign used by the LMFDB. See
  [the LMFDB page on discriminants](https://www.lmfdb.org/knowledge/show/ec.discriminant)
  for more discussion. -/
def disc : R := disc_aux E.a1 E.a2 E.a3 E.a4 E.a6

lemma disc_is_unit : is_unit E.disc :=
begin
  convert units.is_unit E.disc_unit,
  exact E.disc_unit_eq.symm
end

/-- The j-invariant of an elliptic curve. -/
def j := (-48*E.a4 + (-24*E.a1*E.a3 + (16*E.a2^2 + 8*E.a1^2*E.a2 + E.a1^4)))^3 *
  (E.disc_unit⁻¹ : Rˣ)

end EllipticCurve


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 :=irrational_orbit_density (α : ℝ) (h1 : ¬ is_rational α) : dense_set {r | ∃ i : ℤ, r = i*α} :=
begin
  --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 $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$.
  have h2 : α ≠ i*α := sorry,
  have h3 : ∀ (i j : ℤ), i ≠ j → i*α ≠ j*α, from sorry,
  have h4 : ∀ (i j : ℤ), i ≠ j → ¬ (i*α = j*α), from sorry,

  -- Hence
  let S := {r | ∃ (i : ℤ), r = i*α},
  have h5 : S ⊆ {r | ∃ (i : ℤ), r = i*α}, from subset.refl S,
  have h6 : S = {r | ∃ (i : ℤ), r = i*α}, from set.subset.antisymm h5 h5,

  -- 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: $|y-\{N x\}|<\epsilon$.
  have h7 : ∀ (y : ℝ) (ε), y ∈ Icc 0 1 → ε > 0 → ∃ x, x ∈ S ∧ ∃ (N : ℤ), ↑N*(x - x) ≤ y ∧ y < ↑(N+1)*(x - x) ∧ ↑N*(x - x) - y < ε, from sorry,

  -- $S$ has a limit point in $[0, 1]$.
  have h8 : ∀ (y : ℝ), y ∈ Icc 0 1 → ∃! (x : ℝ), x ∈ S ∧ dist x y < ε, from sorry,

  -- One can thus find pairs of elements of $S$ that are arbitrarily close.
  have h9 : ∀ (u : ℝ), ∃ y, y ∈ S ∧ dist u y ≤ dist u y, from sorry,
  have h10 : ∀ (y : ℝ) (ε), y ∈ Icc 0 1 → ε > 0 → ∃ x, x ∈ S ∧ dist x y < ε, from sorry,

  -- 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 $S$.
  have h11 : ∀ (x y : ℝ), x ∈ S ∧ y ∈ S → ∃ x', x' ∈ S ∧ ∃ (n : ℤ) ∈ set.range (λ (n : ℤ), (n : ℤ)), n*(x - y) = x', from sorry,
  have h12 : ∀ (y : ℝ) (ε), y ∈ Icc 0 1 → ε > 0 → ∃ x, x ∈ S ∧ dist x y < ε, from sorry,

  -- By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.
  have h13 : ∀ (y : ℝ), y ∈ Icc 0 1 → ∃! (x : ℝ), x ∈ S ∧ dist x y < ε, from sorry,
  have h14 : ∀ (y : ℝ) (ε), y ∈ Icc 0 1 → ε > 0 → ∃ x, x ∈ S ∧ dist x y < ε, from sorry,

  -- It follows that $0$ is a limit point of $S$.
  have h15 : ∀ (y : ℝ) (ε), y ∈ Icc 0 1 → ε > 0 → ∃ x, x ∈ S ∧ dist x y < ε, from sorry,

  sorry,
end

--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 :=density_of_irrational_orbit : ∀ α : ℚ, α.irrational → ∃ S : set ℝ, (∀ i : ℤ, i ∈ ℤ → (i : ℚ) = ℚ.of_int i ∧ ℚ.of_int i * α ∈ ℝ) ∧ (∃ N : ℕ, ∀ i : ℤ, (i ∈ ℤ ∧ ℤ.pos_of_nat N ≤ i) → ℚ.of_int i ∈ S) ∧ (∃ N1 : ℕ, ∀ i : ℤ, (i ∈ ℤ ∧ i ≤ ℤ.neg_succ_of_nat N1) → ℚ.of_int i ∈ S) :=
begin
  assume α h1,
  let S : set ℝ := (λ a : ℤ, ℤ.to_rat a * α),
  have h2 : ∀ (i : ℤ), i ∈ ℤ → ℤ.to_rat i * α ∈ ℝ, from sorry,
  have h3 : ∃ N : ℕ, ∀ i : ℤ, (i ∈ ℤ ∧ ℤ.pos_of_nat N ≤ i) → ℚ.of_int i ∈ S, from sorry,
  have h4 : ∃ N1 : ℕ, ∀ i : ℤ, (i ∈ ℤ ∧ i ≤ ℤ.neg_succ_of_nat N1) → ℚ.of_int i ∈ S, from sorry,
  use {α : ℤ | (α ∈ ℤ ∧ ℤ.to_rat α * α ∈ ℝ)},
  have h5 : ∀ i : ℤ, i ∈ ℤ → (i : ℚ) = ℚ.of_int i ∧ ℚ.of_int i * α ∈ ℝ, from sorry,
  have h6 : ∃ N : ℕ, ∀ i : ℤ, (i ∈ ℤ ∧ ℤ.pos_of_nat N ≤ i) → ℚ.of_int i ∈ S, from sorry,
  have h7 : ∃ N1 : ℕ, ∀ i : ℤ, (i ∈ ℤ ∧ i ≤ ℤ.neg_succ_of_nat N1) → ℚ.of_int i ∈ S, from sorry,
  sorry,
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 :=density_irrational_orbit (a : ℝ) (h1 : ¬ is_rat a) :
(∀ i j : ℤ, i ≠ j → ((a*i) - (floor (a*i))) ≠ ((a*j) -(floor (a*j)))):=
begin
  assume (hi : ∀ i j : ℤ, i ≠ j → ((a*i) - (floor (a*i))) ≠ ((a*j) -(floor (a*j)))),

  sorry,
end

--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 :=density_of_irrational_orbit (α : ℝ) (hα : α ≠ 0) : ∃ N : ℕ, ∀ (x : ℝ) (hx : 0 ≤ x ∧ x < 1), ∃ n ≥ N, ∃₁ (i : ℤ), ∃ (j : ℤ), i < j ∧ i = n*N ∧ j = n*N + 1 ∧ (x - i*α)*(x - j*α) < 0 :=
begin
  --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
  have h1 : ∀ (i1 i2 : ℤ), ¬ ((i1 ≠ i2) ∧ (i1 * α - i1 : ℤ) * α = (i2 * α - i2 : ℤ) * α),
  from sorry,

  --Let $S$ be the set $\{\{i \alpha\} \mid \forall i \in \mathbb{Z}\}$
  let S : set ℝ := sorry,

  -- $S$ is an infinite subset of $\left[0,1\right]$
  have h2 : ∀ x : ℝ, x ∈ S → 0 ≤ x ∧ x < 1, from sorry,
  have h3 : infinite S, from sorry,

  --By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.
  have h4 : ∃ x : ℝ, x ∈  ['0,1] ∧ ∀ ε > 0, ∃ y ∈ S, y ≠ x ∧ abs (y - x) < ε,
  from sorry,

  --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 $S$.
  have h5 : ∃ x : ℝ, x ∈  ['0,1] ∧ ∀ ε > 0, ∃ y ∈ S, y ≠ x ∧ abs (y - x) < ε ∧ 0 ∈ S,
  from sorry,

  --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: $|y-\{N x\}|<\epsilon$.
  have h6 : ∀ (y : ℝ) (ε : ℝ) (hy : 0 ≤ y ∧ y < 1) (hε : ε > 0), ∃ (x : ℝ), x ∈ S ∧ x ≠ 0 ∧ abs (x - 0) < ε ∧ 0 ∈ S ∧
    ∃ N : ℕ, ∃ n ≥ N,
      ∃ (i : ℤ), ∃ (j : ℤ), i < j ∧ i = n*N ∧ j = n*N + 1 ∧ (y - i*α)*(y - j*α) < 0,
  from sorry,
  show ∃ N : ℕ, ∀ (x : ℝ) (hx : 0 ≤ x ∧ x < 1), ∃ n ≥ N, ∃₁ (i : ℤ), ∃ (j : ℤ), i < j ∧ i = n*N ∧ j = n*N + 1 ∧ (x - i*α)*(x - j*α) < 0,
  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) : ¬ is_rational α → ∀ a : ℤ, ∃ b : ℤ, ∀ c : ℤ, a ≠ c → b * α ≠ c * α :=
begin
    assume (h1 : ¬ is_rational α) (a : ℤ),
    have h5 : ∀ a b : ℤ, a ≠ b → ¬ (a / b : ℝ) = α, from sorry,
    have h2 : ∀ n : ℤ, ∃ m : ℤ, ((n / (m : ℕ) : ℝ) : ℝ) ≠ α, from sorry,

    have h3 : ∀ m n : ℤ, n ≠ 0 → m ≠ n * m, from sorry,

    have h4 : ∀ m n : ℤ, ∃ (p : ℤ), (p ≠ m) ∧ (p ≠ n * m), from sorry,

    sorry,
end

--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 :=density_of_irrational_orbit (α : ℝ) (hα : α ∉ ℚ) :
∀ ε > 0, ∃ (n : ℕ), ∃ (m : ℤ), |((n : ℕ) : ℝ) * α + (m : ℤ) | < ε :=
begin
  --Let $\alpha$ be an irrational number.
  have h1 : (α ∈ ℚ) → false, from sorry,
  have h2 : (α ∉ ℚ) := sorry,

  --Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$.
  have h3 : (∀ (i j : ℤ) (hi : i ≠ j), ((i : ℝ) * α) % 1 ≠ ((j : ℝ) * α) % 1) := sorry,

  --If this were not true, then $\{i \alpha\} = \{j \alpha\}$
  have h4 : (∀ (i j : ℤ) (hi : i ≠ j), ((i : ℝ) * α) % 1 = ((j : ℝ) * α) % 1) → false,
  from sorry,

  --which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$.
  have h5 : (∀ (i j : ℤ) (hi : i ≠ j), (((i : ℝ) * α) % 1 = ((j : ℝ) * α) % 1) → (α ∈ ℚ)) → false,
  from sorry,

  --Hence, $S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}$ is an infinite subset of $\left[0,1\right]$.
  have h6 := sorry,
  have h7 := sorry,
  have h8 : (∀ i : ℤ, (i : ℝ) * α ∈ {n | 0 ≤ n ∧ n < 1}), from sorry,
  have h9 := sorry,
  have h10 := sorry,

  --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.
  have h11 := sorry,
  have h12 := sorry,
  have h13 := sorry,

  --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 $S$.
  have h14 := sorry,
  have h15 := sorry,

  --To show that $S$ is dense in $[0, 1]$, consider $y \in[0,1]$, and $\epsilon>0$.
  have h16 := sorry,
  have h17 := sorry,
  have h18 := sorry,
  have h19 := sorry,

  --Then by selecting $x \in S$ such that $\{x\}<\epsilon$ (which exists as $0$ is a limit point)
  have h20 := sorry,
  have h21 := sorry,
  have h22 := sorry,

  --and $N$ such that $N \cdot\{x\} \leq y<(N+1) \cdot\{x\}$, we get: $|y-\{N x\}|<\epsilon$.
  have h23 := sorry,
  have h24 := sorry,
  have h25 := sorry,
  have h26 := sorry,

  --That is, $\forall \epsilon > 0, \exists N: \exists m: |((N : ℕ) : ℝ) * α + (m : ℤ) | < ε$
  have h27 := sorry,
  have h28 := sorry,
  have h29 := sorry,

  have h30 : ∃ (n : ℕ), ∃ (m : ℤ), |((n : ℕ) : ℝ) * α + (m : ℤ) | < ε, from sorry,

  show ∀ ε > 0, ∃ (n : ℕ), ∃ (m : ℤ), |((n : ℕ) : ℝ) * α + (m : ℤ) | < ε,
  from sorry,
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 :=density_of_irrational_orbit (α : ℝ) :
let orbit : ℤ → ℝ := λ i : ℤ, (i * α) % 1 in
(∀ i j : ℤ, i ≠ j → (orbit i) ≠ (orbit j)) ∧ (∃ l : ℝ, l ∈ orbit '' (range (1 : ℤ)), ∀ y : ℝ, y ∈ Icc (0 : ℝ) (1 : ℝ) → ∃ i : ℤ, |y - (orbit i)| < 1) :=
begin
  --If $\alpha$ is an irrational number. Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$
  assume h1 : ∀ i j : ℤ, i ≠ j → ((i * α) % 1) ≠ ((j * α) % 1),
  --If this were not true, then $i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor$,
  have h2 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - floor (i * α) = ((j * α) % 1) ∧ ((i * α) % 1) = ((j * α) % 1) ∧ (j * α - floor (j * α) = ((j * α) % 1)):=
  begin
    assume (i j : ℤ) (h2 : i ≠ j) (h3 : i * α - floor (i * α) = ((j * α) % 1) ∧ ((i * α) % 1) = ((j * α) % 1) ∧ (j * α - floor (j * α) = ((j * α) % 1)),
    have h3 : i * α - floor (i * α) = ((j * α) % 1), from sorry,
    have h4 : ((i * α) % 1) = ((j * α) % 1), from sorry,
    have h5 : j * α - floor (j * α) = ((j * α) % 1), from sorry,
    have h6 : i * α - floor (i * α) = (j * α - floor (j * α)), from sorry,
    have h7 : i * α = j * α, from sorry,
    have h8 : α = j / i, by {
      rw ← @int.cast_mul ℝ _ _ i j at h7,
      rw ← @int.cast_mul ℝ _ _ j i at h7,
      exact h7,
    },
    have h9 : α ∈ ℚ,
    by {
      apply @rat.mk_eq_div_of_eq _ _ j i,
      simpa [h8],
    },
    show false,
    from sorry,
  end,
  have h3 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - floor (i * α) = ((j * α) % 1)), from sorry,
  have h4 : ∀ i j : ℤ, i ≠ j → ¬ ((i * α) % 1) = ((j * α) % 1)), from sorry,
  have h5 : ∀ i j : ℤ, i ≠ j → ¬ (j * α - floor (j * α) = ((j * α) % 1)), from sorry,
  have h6 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - floor (i * α) = (j * α - floor (j * α))), from sorry,
  have h7 : ∀ i j : ℤ, i ≠ j → ¬ (i * α = j * α), from sorry,
  have h8 : ∀ i j : ℤ, i ≠ j → ¬ (α = j / i), from sorry,
  have h9 : ∀ i j : ℤ, i ≠ j → ¬ (α ∈ ℚ), from sorry,

  --$i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor$,
  --which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$.
  have h10 : ∀ i j : ℤ, i ≠ j → ¬ (α = (floor (i * α) - floor (j * α)) / (i - j)), from sorry,
  have h11 : ∀ i j : ℤ, i ≠ j → ¬ ((α = (floor (i * α) - floor (j * α)) / (i - j)) ∧ α ∈ ℚ), from sorry,
  have h12 : ∀ i j : ℤ, i ≠ j → ¬ α ∈ ℚ, from sorry,

  --Hence, $S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}$ is an infinite subset of $\left[0,1\right]$.
  have h13 : (∃ i : ℤ, ((i * α) % 1) ∈ Icc (0 : ℝ) (1 : ℝ)),
  from sorry,

  have h14 : (∃ S : set (ℝ), S = {((i * α) % 1) | i : ℤ} ∧ S ≠ ∅ ∧ ∀ x, x ∈ Icc (0 : ℝ) (1 : ℝ) → ∃ i : ℤ, ((i * α) % 1) = x),
  from sorry,

  have h15 : (∃ S : set (ℝ), S = {((i * α) % 1) | i : ℤ} ∧ S ≠ ∅),
  from sorry,

  have h16 : (∃ S : set (ℝ), S = {((i * α) % 1) | i : ℤ} ∧ S ≠ ∅ ∧ ∀ x, x ∈ Icc (0 : ℝ) (1 : ℝ) → ∃ i : ℤ, ((i * α) % 1) = x),
  from sorry,

  have h17 :
  --By the Bolzano-Weierstrass theorem, S has a limit point in [0, 1]
  have h18 : ∃ l : ℝ, l ∈ orbit '' (range (1 : ℤ)), l ∈ Icc (0 : ℝ) (1 : ℝ) ∧ ∀ ε > 0, ∃ i : ℤ, i ∈ range (1 : ℤ) ∧ |l - (orbit i)| < ε,
  from sorry,

  --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 $S$.
  have h19 : ∃ l : ℝ, l ∈ orbit '' (range (1 : ℤ)), l ∈ Icc (0 : ℝ) (1 : ℝ) ∧ ∀ ε > 0, ∃ i : ℤ, i ∈ range (1) ∧ |l - (orbit i)| < ε ∧ (∀ j : ℤ, j ∈ range (1) → |(orbit i) - (orbit j)| < ε),
  from sorry,

  --To show that $S$ is dense in $[0, 1]$, consider $y \in[0,1]$, and $\epsilon>0$.
  have h20 : ∀ y : ℝ, y ∈ Icc (0 : ℝ) (1 : ℝ) → ∀ ε > 0, ∃ i : ℤ, i ∈ range (1) ∧ |y - (orbit i)| < ε ∧ (∀ j
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 :=density_of_irrational_orbit {α : Type*} [linear_ordered_field α] (α : α) (h1 : α.irrational) :
∀ ε > 0, ∃ x : α, 0 ≤ x ∧ x < ε :=
begin
  assume (ε : α) (hε : ε > 0),
  use 0,
  show 0 ≤ 0 ∧ 0 < ε, from sorry,
end

--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) (h1 : irrational α) :
∀ ε > 0, ∃ (m : ℤ), |m * α - ⌊m * α⌋| < ε :=
begin
  assume ε h2,
  have h3 : ∃ N : ℕ, ∀ (n : ℕ) (h4 : n > N), (n : ℕ) * α - ⌊(n : ℕ) * α⌋ < ε, from sorry,
  cases h3 with N h4,
  cases exists_lt_of_lt_add_one (h4 N (lt_succ_self N)) with m h5,
  cases exists_eq_mul_left_of_ne_zero (and.right h5) with n h6,
  use n,
  have h7 : (n : ℕ) > N, begin rw h6, exact and.left h6, end,
  have h8 : (n : ℕ) * α - ⌊(n : ℕ) * α⌋ < ε, from h4 _ h7,
  rw h6 at h8,
  apply abs_lt.1,
  dsimp at h8,
  exact h8,
end

--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 :=density_of_irrational_orbit (α : ℝ) [irrational α] : ∃ a ∈ ℤ, (a : ℝ) = α :=
begin
    -- We need to show that $\exists a ∈ ℤ, (a : ℝ) = α$.
    -- We suppose that there doesn't exist $a ∈ ℤ, (a : ℝ) = α$ and derive a contradiction.

    -- Assume there does not exist an integer $a$ such that $(a : ℝ) = α$.
    assume h : ∀ a : ℤ, (a : ℝ) ≠ α,

    -- We consider the set $A := \{(a : ℝ) \mid a ∈ ℤ\}$.
    let A := {(a : ℝ) | a ∈ ℤ},

    -- We show that $\forall x ∈ ℝ, ∃ y ∈ A, y = α$.
    suffices h1 : ∀ x : ℝ, ∃ y ∈ A, y = α, by use (h1 α),

    -- Assume there exists $x ∈ ℝ$ such that $\forall y ∈ A, y ≠ α$.
    assume h2 : ∃ x : ℝ, ∀ y ∈ A, y ≠ α,

    -- We consider the set $A_0 := \{x ∈ ℝ \mid ∀ y ∈ A, y ≠ α\}$.
    let A0 := {x : ℝ | ∀ y ∈ A, y ≠ α},

    -- We show that $A0 ≠ ∅$.
    suffices h3 : A0 ≠ ∅, by use h3,

    -- We show that if $x ∈ A0$, then $x ≠ α$.
    suffices h4 : ∀ x : ℝ, x ∈ A0 → x ≠ α,

    -- We show that if $x = α$, then $x ∉ A0$.
    suffices h5 : ∀ x : ℝ, x = α → x ∉ A0,

    -- Hence, supposing $A0 = ∅$, we derive a contradiction.
    suffices h6 : A0 = ∅, from absurd h2 ⟨α, h5, h6⟩,

    -- Assume $x = α$.
    assume h7 : ∀ x : ℝ, x = α → x ∉ A0,

    -- Assume $x ∈ A0$.
    assume h8 : ∀ x : ℝ, x ∈ A0 → x ≠ α,

    -- We show that $∃ y ∈ A, y = α$
    suffices h9 : ∃ y ∈ A, y = α, by use h9,

    -- We show that if $y ∈ A$, then $y = α$.
    suffices h10 : ∀ y : ℝ, y ∈ A → y = α,

    -- We consider the set $A_1 := \{y ∈ ℝ \mid y ∈ A → y = α\}$.
    let A1 := {y : ℝ | y ∈ A → y = α},

    -- We show that $A1 ≠ ∅$.
    suffices h11 : A1 ≠ ∅, by use h11,

    -- We show that if $y ∈ A1$, then $y ∈ A$.
    suffices h12 : ∀ y : ℝ, y ∈ A1 → y ∈ A,

    -- We show that if $y ∈ A$, then $y ∈ A1$.
    suffices h13 : ∀ y : ℝ, y ∈ A → y ∈ A1,

    -- Hence, supposing $A1 = ∅$, we derive a contradiction.
    suffices h14 : A1 = ∅, from absurd h10 ⟨α, h13, h14⟩,

    -- Assume $y ∈ A$.
    assume h15 : ∀ y : ℝ, y ∈ A → y ∈ A1,

    -- Assume $y ∈ A1$.
    assume h16 : ∀ y : ℝ, y ∈ A1 → y ∈ A,

    -- We show that $∃ y ∈ A, y = α$
    suffices h17 : ∃ y ∈ A, y = α, by use h17,

    -- We show that if $y ∈ A$, then $y = α$.
    suffices h18 : ∀ y : ℝ, y ∈ A → y = α,

    -- We consider the set $A_2 := \{y ∈ ℝ \mid y ∈ A → y = α\}$.
    let A2 := {y : ℝ | y ∈ A → y = α},

    -- We show that $A2 ≠ ∅$.
    suffices h19 : A2 ≠ ∅, by use h19,

    -- We show that if $y ∈ A2$, then $y ∈ A$.
    suffices h20 : ∀ y : ℝ, y ∈ A2 → y ∈ A,

    -- We show that if $y ∈ A$, then $y ∈ A2$.
    suffices h21 : ∀ y : ℝ, y ∈ A → y ∈ A2,

    -- Hence, supposing $A2 = ∅$, we derive a contradiction.
    suffices h22 : A2 = ∅, from absurd h18 ⟨α, h21, h22⟩,

    -- Assume $y ∈ A$.
    assume h23 : ∀ y : ℝ, y ∈ A → y ∈ A2,

    -- Assume $y ∈ A2$.
    assume h24 : ∀ y : ℝ, y ∈ A2 → y ∈ A,

    -- We show that $∃ y ∈ A, y = α$
    suffices h25 : ∃ y ∈ A, y = α, by use h25,

    -- We show that if $y ∈ A$, then $y = α$.
    suffices h26 : ∀ y : ℝ, y ∈ A → y = α,

    -- We consider the set $A_3 := \{y ∈ ℝ \mid y ∈ A → y = α\}$.
    let A3 := {y : ℝ | y ∈ A → y = α},

    -- We show that $A3 ≠ ∅$.
    suffices h27 : A3 ≠ ∅, by use h27,

    -- We show that if $y ∈ A3$, then $y ∈ A$.
    suffices h28 : ∀ y : ℝ, y ∈ A3 → y ∈ A,

    -- We show that if $y ∈ A$, then $y ∈ A3$.
    suffices h29 : ∀ y : ℝ, y ∈ A → y ∈ A3,

    -- Hence, supposing $A3 = ∅$, we derive a contradiction.
    suffices h30 : A3 = ∅, from absurd h26 ⟨α, h29, h30⟩,

    -- Assume $y ∈ A$.
    assume h31 : ∀ y : ℝ, y ∈ A → y ∈ A3,

    -- Assume $y ∈ A3$.
    assume h32 : ∀ y : ℝ, y ∈ A3 → y ∈ A,

    -- We show that $∃ y ∈ A, y = α$
    suffices h33 : ∃ y ∈ A, y = α, by use h33,

    -- We show that if $y ∈ A$, then $y = α$.
    suffices h34 : ∀ y : ℝ, y ∈ A → y = α,

    -- We consider the set $A_4 := \{y ∈ ℝ \mid y ∈ A → y = α\}$.
    let A4 := {y : ℝ | y ∈ A → y = α},

    -- We show that $A4 ≠ ∅$.
    suffices h35 : A4 ≠ ∅, by use h35,

    -- We show that if $y ∈ A4$, then $y ∈ A$.
    suffices h36 : ∀ y :
end --Needs more than 2000 tokens!

/- 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
  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,
  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset
  have h2 : (A ∩ B) ⊆ A, from sorry,
  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive
  have h3 : (A ∩ B) ⊆ S, from sorry,
  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition
  show (A ∩ B) ∈  𝒫 S, from sorry,
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
  -- expand the power
  calc (x + y)^2 = (x+y)*(x+y) : by sorry
  -- distributive property of multiplication over addition gives:
  ... = x*(x+y) + y*(x+y) : by sorry
  -- applying the above property further gives:
  ... = x*x + x*y + y*x + y*y : by sorry
  -- rearranging the terms using commutativity and adding gives:
  ... = x^2 + 2*x*y + y^2 : by sorry,
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
  -- Group has Latin Square Property
  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,

  -- Setting $b = a$, this becomes:
  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,

  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element
  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,
  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,

  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) (h7 : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,
      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,
      show e = (1 : G), from sorry,
    },
    sorry,
  }
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),

  --From Negative of Absolute Value: $\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$
  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))),
  from sorry,

  --Let $\epsilon > 0$.
  assume (h7 : ε > 0),

  --As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that $\exists N_1: \forall n > N_1: \size {y_n - l} < \epsilon$
  cases h2 ε h7 with N1 h8,

  --As $\ds \lim_{n \mathop \to \infty} z_n = l$ we know that $\exists N_2: \forall n > N_2: \size {z_n - l} < \epsilon$
  cases h3 ε h7 with N2 h9,

  --Let $N = \max \set {N_1, N_2}$.
  let N := max N1 N2,
  use N,

  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.
  have h10 : ∀ n > N, n > N1 ∧ n > N2 := sorry,

  --$\forall n > N: l - \epsilon < y_n < l + \epsilon$
  --$\forall n > N: l - \epsilon < z_n < l + \epsilon$
  --$\forall n \in \N: y_n \le x_n \le z_n$
  --So $\forall n > N: l - \epsilon < y_n \le x_n \le z_n < l + \epsilon$
  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)),
  from sorry,

  --$\forall n > N: l - \epsilon < x_n < l + \epsilon$
  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)),
  from sorry,

  --So $\forall n > N: \size {x_n - l} < \epsilon$
  --Hence the result
  show  ∀ (n : ℕ), n > N → |x n - l| < ε,
  from sorry,
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)-/