import Mathlib.Tactic

namespace Chapter3

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 nat)

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

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

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

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

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

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

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 := SetTheory.replacement_axiom A P hP y

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

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

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

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 := OrderedPair.toObject

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

abbrev SetTheory.Set.cartesian (X Y : Set) : Set :=
  SetTheory.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 := SetTheory.Set.cartesian

abbrev SetTheory.Set.EqualCard (X Y:Set) : Prop := ∃ f : X → Y, Function.Bijective f

/-- Exercise 3.6.6. You may find `SetTheory.Set.curry_equiv` useful. -/
 theorem SetTheory.Set.pow_pow_EqualCard_pow_prod (A B C:Set) :
    EqualCard ((A ^ B) ^ C) (A ^ (B ×ˢ C)) := by
end Chapter3