import Mathlib.Tactic

namespace Chapter3

universe u v

/-- The axioms of Zermelo-Frankel theory with atoms.  -/
class SetTheory where
  Set : Type u -- Axiom 3.1
  Object : Type v -- Axiom 3.1
  set_to_object : Set ↪ Object -- Axiom 3.1
  mem : Object → Set → Prop -- Axiom 3.1
  extensionality X Y : (∀ x, mem x X ↔ mem x Y) → X = Y -- Axiom 3.2
  emptyset: Set -- Axiom 3.3
  emptyset_mem x : ¬ mem x emptyset -- Axiom 3.3
  singleton : Object → Set -- Axiom 3.4
  singleton_axiom x y : mem x (singleton y) ↔ x = y -- Axiom 3.4
  union_pair : Set → Set → Set -- Axiom 3.5
  union_pair_axiom X Y x : mem x (union_pair X Y) ↔ (mem x X ∨ mem x Y) -- Axiom 3.5
  specify A (P: Subtype (mem . A) → Prop) : Set -- Axiom 3.6
  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 -- Axiom 3.6
  replace A (P: Subtype (mem . A) → Object → Prop)
    (hP: ∀ x y y', P x y ∧ P x y' → y = y') : Set -- Axiom 3.7
  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 -- Axiom 3.7
  nat : Set -- Axiom 3.8
  nat_equiv : ℕ ≃ Subtype (mem . nat) -- Axiom 3.8
  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 -- Axiom 3.9
  pow : Set → Set → Set -- Axiom 3.11
  function_to_object (X: Set) (Y: Set) :
    (Subtype (mem . X) → Subtype (mem . Y)) ↪ Object -- Axiom 3.11
  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 -- Axiom 3.11
  union : Set → Set -- Axiom 3.12
  union_axiom A x : mem x (union A) ↔ ∃ S, mem x S ∧ mem (set_to_object S) A -- Axiom 3.12

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

/-- Define a coercion so any set can be treated as a type, as its subtype of elements. -/
abbrev SetTheory.Set.toSubtype (A : Set) := Subtype (fun x ↦ x ∈ A)

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

namespace SetTheory.Set

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

/-- Axioms of infinity: expose a type alias and coercions between our `Nat` set and `ℕ`. -/
abbrev Nat := SetTheory.nat

def nat_equiv : ℕ ≃ Nat := SetTheory.nat_equiv

instance instNatCast : NatCast Nat where
  natCast n := nat_equiv n

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

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

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

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

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

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

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

/-- Proposition 3.6.4 / Exercise 3.6.1 -/
instance EqualCard.inst_setoid : Setoid SetTheory.Set := ⟨ EqualCard, {refl, symm, trans} ⟩

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

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

open Classical in
noncomputable def card (X:Set) : ℕ := if h:X.finite then h.choose else 0

/-- Proposition 3.6.14 (f) / Exercise 3.6.4 -/
 theorem card_pow {X Y:Set} (hY: Y.finite) (hX: X.finite) :
    (Y ^ X).finite ∧ (Y ^ X).card = Y.card ^ X.card := by
end SetTheory.Set

end Chapter3