import Mathlib
import Analysis.Section_3_1
import Analysis.Tags.Section_3_1
import Analysis.Section_3_2
import Analysis.Tags.Section_3_2
import Analysis.Section_3_4
import Analysis.Tags.Section_3_4

namespace Chapter3

-- <CONTEXT>
universe u v

class SetTheory where
  Set : Type u
  Object : Type v
  set_to_object : Set ↪ Object
  mem : Object → Set → Prop
  extensionality X Y : (∀ x, mem x X ↔ mem x Y) → X = Y
  emptyset: Set
  emptyset_mem x : ¬ mem x emptyset
  singleton : Object → Set
  singleton_axiom x y : mem x (singleton y) ↔ x = y
  union_pair : Set → Set → Set
  union_pair_axiom X Y x : mem x (union_pair X Y) ↔ (mem x X ∨ mem x Y)
  specify A (P: Subtype (mem . A) → Prop) : Set
  specification_axiom A (P: Subtype (mem . A) → Prop) :
    (∀ x, mem x (specify A P) → mem x A) ∧ ∀ x, mem x.val (specify A P) ↔ P x
  replace A (P: Subtype (mem . A) → Object → Prop)
    (hP: ∀ x y y', P x y ∧ P x y' → y = y') : Set
  replacement_axiom A (P: Subtype (mem . A) → Object → Prop)
    (hP: ∀ x y y', P x y ∧ P x y' → y = y') : ∀ y, mem y (replace A P hP) ↔ ∃ x, P x y
  nat : Set
  nat_equiv : ℕ ≃ Subtype (mem . nat)
  regularity_axiom A (hA : ∃ x, mem x A) :
    ∃ x, mem x A 
      ∧ ∀ S, x = set_to_object S → ¬ ∃ y, mem y A ∧ mem y S
  pow : Set → Set → Set
  function_to_object (X: Set) (Y: Set) :
    (Subtype (mem . X) → Subtype (mem . Y)) ↪ Object
  powerset_axiom (X: Set) (Y: Set) (F:Object) :
    mem F (pow X Y) ↔ ∃ f: Subtype (mem . Y) → Subtype (mem . X),
    function_to_object Y X f = F
  union : Set → Set
  union_axiom A x : mem x (union A) ↔ ∃ S, mem x S ∧ mem (set_to_object S) A

export SetTheory (Set Object)

variable [SetTheory]

instance SetTheory.objects_mem_sets : Membership Object Set where
  mem X x := SetTheory.mem x X

instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := SetTheory.set_to_object X

@[simp]
theorem SetTheory.Set.coe_eq_iff (X Y:Set) : (X: Object) = (Y: Object) ↔  X = Y := by
  sorry

@[ext]
theorem SetTheory.Set.ext {X Y:Set} (h: ∀ x, x ∈ X ↔ x ∈ Y) : X = Y := by
  exact SetTheory.extensionality _ _ h

instance SetTheory.Set.instEmpty : EmptyCollection Set where
  emptyCollection := SetTheory.emptyset

open SetTheory.Set

@[simp]
theorem SetTheory.Set.not_mem_empty : ∀ x, x ∉ (∅:Set) := by
  exact SetTheory.emptyset_mem

instance SetTheory.Set.instSingleton : Singleton Object Set where
  singleton := SetTheory.singleton

@[simp]
theorem SetTheory.Set.mem_singleton (x a:Object) : x ∈ ({a}:Set) ↔ x = a := by
  exact SetTheory.singleton_axiom x a

instance SetTheory.Set.instUnion : Union Set where
  union := SetTheory.union_pair

@[simp]
theorem SetTheory.Set.mem_union (x:Object) (X Y:Set) : x ∈ (X ∪ Y) ↔ (x ∈ X ∨ x ∈ Y) := by
  exact SetTheory.union_pair_axiom X Y x

instance SetTheory.Set.instInsert : Insert Object Set where
  insert x X := {x} ∪ X

@[simp]
theorem SetTheory.Set.mem_insert (a b: Object) (X: Set) : a ∈ insert b X ↔ a = b ∨ a ∈ X := by
  sorry

@[simp]
theorem SetTheory.Set.mem_pair (x a b:Object) : x ∈ ({a,b}:Set) ↔ (x = a ∨ x = b) := by
  sorry

@[simp]
theorem SetTheory.Set.mem_triple (x a b c:Object) : x ∈ ({a,b,c}:Set) ↔ (x = a ∨ x = b ∨ x = c) := by
  sorry

instance SetTheory.Set.instSubset : HasSubset Set where
  Subset X Y := ∀ x, x ∈ X → x ∈ Y

instance SetTheory.Set.instSSubset : HasSSubset Set where
  SSubset X Y := X ⊆ Y ∧ X ≠ Y

abbrev SetTheory.Set.toSubtype (A:Set) := Subtype (fun x ↦ x ∈ A)

instance : CoeSort (Set) (Type v) where
  coe A := A.toSubtype

lemma SetTheory.Set.subtype_property (A:Set) (x:A) : x.val ∈ A := by
  exact x.property

abbrev SetTheory.Set.specify (A:Set) (P: A → Prop) : Set := SetTheory.specify A P

theorem SetTheory.Set.specification_axiom {A:Set} {P: A → Prop} {x:Object} (h: x ∈ A.specify P) :
    x ∈ A := by
  exact (SetTheory.specification_axiom A P).1 x h

theorem SetTheory.Set.specification_axiom' {A:Set} (P: A → Prop) (x:A) :
    x.val ∈ A.specify P ↔ P x := by
  exact (SetTheory.specification_axiom A P).2 x

@[simp]
theorem SetTheory.Set.specification_axiom'' {A:Set} (P: A → Prop) (x:Object) :
    x ∈ A.specify P ↔ ∃ h:x ∈ A, P ⟨ x, h ⟩ := by
  sorry

instance SetTheory.Set.instIntersection : Inter Set where
  inter X Y := X.specify (fun x ↦ x.val ∈ Y)

@[simp]
theorem SetTheory.Set.mem_inter (x:Object) (X Y:Set) : x ∈ (X ∩ Y) ↔ (x ∈ X ∧ x ∈ Y) := by
  sorry

instance SetTheory.Set.instSDiff : SDiff Set where
  sdiff X Y := X.specify (fun x ↦ x.val ∉ Y)

@[simp]
theorem SetTheory.Set.mem_sdiff (x:Object) (X Y:Set) : x ∈ (X \ Y) ↔ (x ∈ X ∧ x ∉ Y) := by
  sorry

abbrev SetTheory.Set.replace (A:Set) {P: A → Object → Prop}
  (hP : ∀ x y y', P x y ∧ P x y' → y = y') : Set := SetTheory.replace A P hP

@[simp]
theorem SetTheory.Set.replacement_axiom {A:Set} {P: A → Object → Prop}
  (hP: ∀ x y y', P x y ∧ P x y' → y = y') (y:Object) :
    y ∈ A.replace hP ↔ ∃ x, P x y := by
  exact SetTheory.replacement_axiom A P hP y

abbrev Nat := SetTheory.nat

def SetTheory.Set.nat_equiv : ℕ ≃ Nat := SetTheory.nat_equiv

instance SetTheory.Set.instOfNat {n:ℕ} : OfNat Nat n where
  ofNat := nat_equiv n

instance SetTheory.Set.instNatCast : NatCast Nat where
  natCast n := nat_equiv n

instance SetTheory.Set.toNat : Coe Nat ℕ where
  coe n := nat_equiv.symm n

instance SetTheory.Object.instNatCast : NatCast Object where
  natCast n := (n:Nat).val

instance SetTheory.Object.instOfNat {n:ℕ} : OfNat Object n where
  ofNat := ((n:Nat):Object)

@[simp]
lemma SetTheory.Object.ofnat_eq {n:ℕ} : ((n:Nat):Object) = (n:Object) := by
  rfl

lemma SetTheory.Object.ofnat_eq' {n:ℕ} : (ofNat(n):Object) = (n:Object) := by
  rfl

@[simp]
lemma SetTheory.Object.ofnat_eq'' {n:Nat} : ((n:ℕ):Object) = (n: Object) := by
  sorry

@[simp]
lemma SetTheory.Object.ofnat_eq''' {n:ℕ} {hn} : ((⟨(n:Object), hn⟩: nat): ℕ) = n := by
  sorry

@[simp]
theorem SetTheory.Object.natCast_inj (n m:ℕ) :
    (n : Object) = (m : Object) ↔ n = m := by
  sorry

instance SetTheory.Set.inst_coe_set : Coe Set (_root_.Set Object) where
  coe X := { x | x ∈ X }

instance SetTheory.Set.inst_pow : Pow Set Set where
  pow := SetTheory.pow

@[coe]
def SetTheory.Set.coe_of_fun {X Y:Set} (f: X → Y) : Object := SetTheory.function_to_object X Y f

instance SetTheory.Set.inst_coe_of_fun {X Y:Set} : CoeOut (X → Y) Object where
  coe := coe_of_fun

@[simp]
theorem SetTheory.Set.coe_of_fun_inj {X Y:Set} (f g:X → Y) : (f:Object) = (g:Object) ↔ f = g := by
  sorry

@[simp]
theorem SetTheory.Set.powerset_axiom {X Y:Set} (F:Object) :
    F ∈ (X ^ Y) ↔ ∃ f: Y → X, f = F := by
  exact SetTheory.powerset_axiom X Y F

@[simp]
theorem SetTheory.Set.union_axiom (A: Set) (x:Object) :
    x ∈ SetTheory.union A ↔ ∃ (S:Set), x ∈ S ∧ (S:Object) ∈ A := by
  exact SetTheory.union_axiom A x

abbrev SetTheory.Set.iUnion (I: Set) (A: I → Set) : Set :=
  SetTheory.union (I.replace (P := fun α S ↦ S = A α) (by intro _ _ _ ⟨h1, h2⟩; exact h1.trans h2.symm))

theorem SetTheory.Set.mem_iUnion {I:Set} (A: I → Set) (x:Object) :
    x ∈ iUnion I A ↔ ∃ α:I, x ∈ A α := by
  sorry

export SetTheory (Set Object nat)

variable [SetTheory]

open SetTheory.Set

@[ext]
structure OrderedPair where
  fst: Object
  snd: Object

@[simp]
theorem OrderedPair.eq (x y x' y' : Object) :
    (⟨ x, y ⟩ : OrderedPair) = (⟨ x', y' ⟩ : OrderedPair) ↔ x = x' ∧ y = y' := by
  sorry

lemma SetTheory.Set.pair_eq_singleton_iff {a b c: Object} : {a, b} = ({c}: Set) ↔
    a = c ∧ b = c := by
  sorry

def OrderedPair.toObject : OrderedPair ↪ Object where
  toFun p := ({ (({p.fst}:Set):Object), (({p.fst, p.snd}:Set):Object) }:Set)
  inj' := by sorry

instance OrderedPair.inst_coeObject : Coe OrderedPair Object where
  coe := toObject

abbrev SetTheory.Set.slice (x:Object) (Y:Set) : Set :=
  Y.replace (P := fun y z ↦ z = (⟨x, y⟩:OrderedPair)) (by grind)

@[simp]
theorem SetTheory.Set.mem_slice (x z:Object) (Y:Set) :
    z ∈ (SetTheory.Set.slice x Y) ↔ ∃ y:Y, z = (⟨x, y⟩:OrderedPair) := by
  sorry

abbrev SetTheory.Set.cartesian (X Y:Set) : Set :=
  union (X.replace (P := fun x z ↦ z = slice x Y) (by intro _ _ _ ⟨h1, h2⟩; exact h1.trans h2.symm))

instance SetTheory.Set.inst_SProd : SProd Set Set Set where
  sprod := cartesian

@[simp]
theorem SetTheory.Set.mem_cartesian (z:Object) (X Y:Set) :
    z ∈ X ×ˢ Y ↔ ∃ x:X, ∃ y:Y, z = (⟨x, y⟩:OrderedPair) := by
  sorry

noncomputable abbrev SetTheory.Set.fst {X Y:Set} (z:X ×ˢ Y) : X :=
  ((mem_cartesian _ _ _).mp z.property).choose

noncomputable abbrev SetTheory.Set.snd {X Y:Set} (z:X ×ˢ Y) : Y :=
  (exists_comm.mp ((mem_cartesian _ _ _).mp z.property)).choose

theorem SetTheory.Set.pair_eq_fst_snd {X Y:Set} (z:X ×ˢ Y) :
    z.val = (⟨ fst z, snd z ⟩:OrderedPair) := by
  sorry

def SetTheory.Set.mk_cartesian {X Y:Set} (x:X) (y:Y) : X ×ˢ Y :=
  ⟨(⟨ x, y ⟩:OrderedPair), by simp⟩

@[simp]
theorem SetTheory.Set.fst_of_mk_cartesian {X Y:Set} (x:X) (y:Y) :
    fst (mk_cartesian x y) = x := by
  sorry

@[simp]
theorem SetTheory.Set.snd_of_mk_cartesian {X Y:Set} (x:X) (y:Y) :
    snd (mk_cartesian x y) = y := by
  sorry

@[simp]
theorem SetTheory.Set.mk_cartesian_fst_snd_eq {X Y: Set} (z: X ×ˢ Y) :
    (mk_cartesian (fst z) (snd z)) = z := by
  sorry

noncomputable abbrev SetTheory.Set.prod_equiv_prod (X Y:Set) :
    ((X ×ˢ Y):_root_.Set Object) ≃ (X:_root_.Set Object) ×ˢ (Y:_root_.Set Object) where
  toFun z := ⟨(fst z, snd z), by simp⟩
  invFun z := mk_cartesian ⟨z.val.1, z.prop.1⟩ ⟨z.val.2, z.prop.2⟩
  left_inv _ := by simp
  right_inv _ := by simp

noncomputable abbrev SetTheory.Set.prod_commutator (X Y:Set) : X ×ˢ Y ≃ Y ×ˢ X where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

noncomputable abbrev SetTheory.Set.curry_equiv {X Y Z:Set} : (X → Y → Z) ≃ (X ×ˢ Y → Z) where
  toFun f z := f (fst z) (snd z)
  invFun f x y := f ⟨ (⟨ x, y ⟩:OrderedPair), by simp ⟩
  left_inv _ := by simp
  right_inv _ := by simp [←pair_eq_fst_snd]

abbrev SetTheory.Set.tuple {I:Set} {X: I → Set} (x: ∀ i, X i) : Object :=
  ((fun i ↦ ⟨ x i, by rw [mem_iUnion]; use i; exact (x i).property ⟩):I → iUnion I X)

abbrev SetTheory.Set.iProd {I: Set} (X: I → Set) : Set :=
  ((iUnion I X)^I).specify (fun t ↦ ∃ x : ∀ i, X i, t = tuple x)

theorem SetTheory.Set.mem_iProd {I: Set} {X: I → Set} (t:Object) :
    t ∈ iProd X ↔ ∃ x: ∀ i, X i, t = tuple x := by
  sorry

theorem SetTheory.Set.tuple_mem_iProd {I: Set} {X: I → Set} (x: ∀ i, X i) :
    tuple x ∈ iProd X := by
  sorry

@[simp]
theorem SetTheory.Set.tuple_inj {I:Set} {X: I → Set} (x y: ∀ i, X i) :
    tuple x = tuple y ↔ x = y := by
  sorry

noncomputable abbrev SetTheory.Set.prod_associator (X Y Z:Set) : (X ×ˢ Y) ×ˢ Z ≃ X ×ˢ (Y ×ˢ Z) where
  toFun p := mk_cartesian (fst (fst p)) (mk_cartesian (snd (fst p)) (snd p))
  invFun p := mk_cartesian (mk_cartesian (fst p) (fst (snd p))) (snd (snd p))
  left_inv _ := by simp
  right_inv _ := by simp

noncomputable abbrev SetTheory.Set.singleton_iProd_equiv (i:Object) (X:Set) :
    iProd (fun _:({i}:Set) ↦ X) ≃ X where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

abbrev SetTheory.Set.empty_iProd_equiv (X: (∅:Set) → Set) : iProd X ≃ Unit where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

noncomputable abbrev SetTheory.Set.iProd_of_const_equiv (I:Set) (X: Set) :
    iProd (fun _:I ↦ X) ≃ (I → X) where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

noncomputable abbrev SetTheory.Set.iProd_equiv_prod (X: ({0,1}:Set) → Set) :
    iProd X ≃ (X ⟨ 0, by simp ⟩) ×ˢ (X ⟨ 1, by simp ⟩) where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

noncomputable abbrev SetTheory.Set.iProd_equiv_prod_triple (X: ({0,1,2}:Set) → Set) :
    iProd X ≃ (X ⟨ 0, by simp ⟩) ×ˢ (X ⟨ 1, by simp ⟩) ×ˢ (X ⟨ 2, by simp ⟩) where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

noncomputable abbrev SetTheory.Set.iProd_equiv_pi (I:Set) (X: I → Set) :
    iProd X ≃ Set.pi .univ (fun i:I ↦ ((X i):_root_.Set Object)) where
  toFun t := ⟨fun i ↦ ((mem_iProd _).mp t.property).choose i, by simp⟩
  invFun x :=
    ⟨tuple fun i ↦ ⟨x.val i, by have := x.property i; simpa⟩, by apply tuple_mem_iProd⟩
  left_inv t := by ext; rw [((mem_iProd _).mp t.property).choose_spec, tuple_inj]
  right_inv x := by
    ext; dsimp
    generalize_proofs _ h
    rw [←(tuple_inj _ _).mp h.choose_spec]

abbrev SetTheory.Set.Fin (n:ℕ) : Set := nat.specify (fun m ↦ (m:ℕ) < n)

theorem SetTheory.Set.mem_Fin (n:ℕ) (x:Object) : x ∈ Fin n ↔ ∃ m, m < n ∧ x = m := by
  sorry

abbrev SetTheory.Set.Fin_mk (n m:ℕ) (h: m < n): Fin n := ⟨ m, by rw [mem_Fin]; use m ⟩

theorem SetTheory.Set.mem_Fin' {n:ℕ} (x:Fin n) : ∃ m, ∃ h : m < n, x = Fin_mk n m h := by
  sorry

@[coe]
noncomputable abbrev SetTheory.Set.Fin.toNat {n:ℕ} (i: Fin n) : ℕ := (mem_Fin' i).choose

noncomputable instance SetTheory.Set.Fin.inst_coeNat {n:ℕ} : CoeOut (Fin n) ℕ where
  coe := toNat

theorem SetTheory.Set.Fin.toNat_spec {n:ℕ} (i: Fin n) :
    ∃ h : i < n, i = Fin_mk n i h := by
  sorry

theorem SetTheory.Set.Fin.toNat_lt {n:ℕ} (i: Fin n) : i < n := by
  sorry

@[simp]
theorem SetTheory.Set.Fin.coe_toNat {n:ℕ} (i: Fin n) : ((i:ℕ):Object) = (i:Object) := by
  sorry

@[simp low]
lemma SetTheory.Set.Fin.coe_inj {n:ℕ} {i j: Fin n} : i = j ↔ (i:ℕ) = (j:ℕ) := by
  sorry

@[simp]
theorem SetTheory.Set.Fin.coe_eq_iff {n:ℕ} (i: Fin n) {j:ℕ} : (i:Object) = (j:Object) ↔ i = j := by
  sorry

@[simp]
theorem SetTheory.Set.Fin.coe_eq_iff' {n m:ℕ} (i: Fin n) (hi : ↑i ∈ Fin m) : ((⟨i, hi⟩ : Fin m):ℕ) = (i:ℕ) := by
  sorry

@[simp]
theorem SetTheory.Set.Fin.toNat_mk {n:ℕ} (m:ℕ) (h: m < n) : (Fin_mk n m h : ℕ) = m := by
  sorry

abbrev SetTheory.Set.Fin_embed (n N:ℕ) (h: n ≤ N) (i: Fin n) : Fin N := ⟨ i.val, by
  have := i.property; rw [mem_Fin] at *; grind
⟩

noncomputable abbrev SetTheory.Set.Fin.Fin_equiv_Fin (n:ℕ) : Fin n ≃ _root_.Fin n where
  toFun m := _root_.Fin.mk m (toNat_lt m)
  invFun m := Fin_mk n m.val m.isLt
  left_inv m := (toNat_spec m).2.symm
  right_inv m := by simp

theorem SetTheory.Set.finite_choice {n:ℕ} {X: Fin n → Set} (h: ∀ i, X i ≠ ∅) : iProd X ≠ ∅ := by
  sorry

abbrev OrderedPair.toObject' : OrderedPair ↪ Object where
  toFun p := ({ p.fst, (({p.fst, p.snd}:Set):Object) }:Set)
  inj' := by sorry

structure SetTheory.Set.Tuple (n:ℕ) where
  X: Set
  x: Fin n → X
  surj: Function.Surjective x

@[ext]
lemma SetTheory.Set.Tuple.ext {n:ℕ} {t t':Tuple n}
    (hX : t.X = t'.X)
    (hx : ∀ n : Fin n, ((t.x n):Object) = ((t'.x n):Object)) :
    t = t' := by
  sorry

noncomputable abbrev SetTheory.Set.iProd_equiv_tuples (n:ℕ) (X: Fin n → Set) :
    iProd X ≃ { t:Tuple n // ∀ i, (t.x i:Object) ∈ X i } where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

def SetTheory.Set.union_of_prod :
  Decidable (∀ (A B C D:Set), (A ×ˢ B) ∪ (C ×ˢ D) = (A ∪ C) ×ˢ (B ∪ D)) := by
  sorry

def SetTheory.Set.diff_of_prod :
  Decidable (∀ (A B C D:Set), (A ×ˢ B) \ (C ×ˢ D) = (A \ C) ×ˢ (B \ D)) := by
  sorry

def SetTheory.Set.prod_subset_prod' :
  Decidable (∀ (A B C D:Set), A ×ˢ B ⊆ C ×ˢ D ↔ A ⊆ C ∧ B ⊆ D) := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.direct_sum {X Y Z:Set} (f: Z → X) (g: Z → Y) :
    ∃! h: Z → X ×ˢ Y, fst ∘ h = f ∧ snd ∘ h = g := by
-- </TARGET_THEOREM>

end Chapter3