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

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

@[simp] abbrev SetTheory.Set.toSubtype (A : Set) := Subtype (fun x ↦ x ∈ A)

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

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 := (SetTheory.specification_axiom A P).1 x h

@[simp] theorem SetTheory.Set.specification_axiom' {A:Set} (P: A → Prop) (x:A) :
    x.val ∈ A.specify P ↔ P x := (SetTheory.specification_axiom A P).2 x

abbrev Nat := SetTheory.nat

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

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

abbrev SetTheory.Set.Fin (n : ℕ) : Set :=
  nat.specify (fun m ↦ (m : ℕ) < 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 :=
  ⟨ SetTheory.Set.EqualCard,
    { refl := SetTheory.Set.EqualCard.refl,
      symm := SetTheory.Set.EqualCard.symm,
      trans := SetTheory.Set.EqualCard.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

-- TARGET THEOREM
/-- Exercise 3.6.7 -/
 theorem SetTheory.Set.injection_iff_card_le {A B:Set} (hA: A.finite) (hB: B.finite) :
    (∃ f:A → B, Function.Injective f) ↔ A.card ≤ B.card := by
end Chapter3