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)

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

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

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 } ∪ X

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

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

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

namespace OrderedPair

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

instance inst_coeObject : Coe OrderedPair Object where
  coe := toObject

end OrderedPair

abbrev SetTheory.Set.slice (x : Object) (Y : Set) : Set :=
  Y.replace (P := fun y z ↦ z = (⟨x, y⟩ : OrderedPair)) (by
    intro x y y'
    intro h
    exact h.1.trans h.2.symm)

abbrev SetTheory.Set.cartesian (X Y : Set) : Set :=
  SetTheory.union (X.replace (P := fun x z ↦ z = slice x Y) (by
    intro x y y'
    intro h
    exact h.1.trans h.2.symm))

infixl:70 " ×ˢ " => SetTheory.Set.cartesian

abbrev Nat := SetTheory.nat

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

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

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

@[refl]
theorem SetTheory.Set.EqualCard.refl (X : Set) : EqualCard X X := by
  sorry

@[symm]
theorem SetTheory.Set.EqualCard.symm {X Y : Set} (h : EqualCard X Y) : EqualCard Y X := by
  sorry

@[trans]
theorem SetTheory.Set.EqualCard.trans {X Y Z : Set} (h1 : EqualCard X Y) (h2 : EqualCard Y Z) : EqualCard X Z := by
  sorry

instance SetTheory.Set.EqualCard.inst_setoid : Setoid SetTheory.Set := ⟨ EqualCard, {refl, symm, trans} ⟩

abbrev SetTheory.Set.has_card (X : Set) (n : ℕ) : Prop := X ≈ (SetTheory.Set.Fin n)

abbrev SetTheory.Set.finite (X : Set) : Prop := ∃ n : ℕ, X.has_card n

open Classical in
noncomputable def SetTheory.Set.card (X : Set) : ℕ := if h : X.finite then h.choose else 0
theorem SetTheory.Set.card_prod {X Y:Set} (hX: X.finite) (hY: Y.finite) :
    (X ×ˢ Y).finite ∧ (X ×ˢ Y).card = X.card * Y.card := by
end Chapter3