import Mathlib
import Analysis.Section_3_5
import Analysis.Tags.Section_3_5

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 : Set) (P : Subtype (mem . A) → Prop) : Set
  specification_axiom (A : Set) (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 : Set) (P : Subtype (mem . A) → Object → Prop)
    (hP : ∀ x y y', P x y ∧ P x y' → y = y') : Set
  replacement_axiom (A : Set) (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.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 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)

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

theorem SetTheory.Set.Example_3_6_2 : EqualCard {0,1,2} {3,4,5} := by
  sorry

theorem SetTheory.Set.Example_3_6_3 : EqualCard nat (nat.specify (fun x ↦ Even (x:ℕ))) := by
  sorry

@[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 ≈ Fin n

theorem SetTheory.Set.has_card_iff (X:Set) (n:ℕ) :
    X.has_card n ↔ ∃ f: X → Fin n, Function.Bijective f := by
  sorry

theorem SetTheory.Set.Remark_3_6_6 (n:ℕ) :
    (nat.specify (fun x ↦ 1 ≤ (x:ℕ) ∧ (x:ℕ) ≤ n)).has_card n := by
  sorry

theorem SetTheory.Set.Example_3_6_7a (a:Object) : ({a}:Set).has_card 1 := by
  sorry

theorem SetTheory.Set.Example_3_6_7b {a b c d:Object} (hab: a ≠ b) (hac: a ≠ c) (had: a ≠ d)
    (hbc: b ≠ c) (hbd: b ≠ d) (hcd: c ≠ d) : ({a,b,c,d}:Set).has_card 4 := by
  sorry

theorem SetTheory.Set.pos_card_nonempty {n:ℕ} (h: n ≥ 1) {X:Set} (hX: X.has_card n) : X ≠ ∅ := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.has_card_zero {X:Set} : X.has_card 0 ↔ X = ∅ := by
-- </TARGET_THEOREM>

end Chapter3