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 (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

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

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

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

instance SetTheory.Set.instNatCast : NatCast Nat where
  natCast n := SetTheory.Set.nat_equiv n

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

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) : SetTheory.Set.EqualCard X X := by
  sorry

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

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

instance SetTheory.Set.EqualCard.inst_setoid : Setoid SetTheory.Set := ⟨ 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
noncomputable def SetTheory.Set.card (X:Set) : ℕ := if h:X.finite then h.choose else 0

noncomputable def SetTheory.Set.pow_fun_equiv {A B : Set} : ↑(A ^ B) ≃ (B → A) where
  toFun := sorry
  invFun := sorry
  left_inv := sorry
  right_inv := sorry

def SetTheory.Set.Permutations (n: ℕ): Set := (SetTheory.Set.Fin n ^ SetTheory.Set.Fin n).specify (fun F ↦
    Function.Bijective (SetTheory.Set.pow_fun_equiv F))
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.Permutations_card (n: ℕ):
    (SetTheory.Set.Permutations n).card = n.factorial := by
-- </TARGET_THEOREM>

end Chapter3