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]

/-- Definition 3.1.1 (objects can be elements of sets) -/
instance SetTheory.objects_mem_sets : Membership Object Set where
  mem X x := SetTheory.mem x X

/-- Axiom 3.1 (Sets are objects)-/
instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := SetTheory.set_to_object X

/-- Axiom 3.1 (Sets are objects)-/
@[simp]
 theorem SetTheory.Set.coe_eq_iff (X Y:Set) : (X: Object) = (Y: Object) ↔  X = Y := by
  constructor
  · intro h; exact SetTheory.set_to_object.inj' h
  · rintro rfl; rfl

/-- Axiom 3.2 (Equality of sets).  The `[ext]` tag allows the `ext` tactic to work for sets. -/
@[ext]
 theorem SetTheory.Set.ext {X Y:Set} (h: ∀ x, x ∈ X ↔ x ∈ Y) : X = Y :=
  SetTheory.extensionality _ _ h

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

@[simp]
 theorem SetTheory.Set.not_mem_empty : ∀ x, x ∉ (∅:Set) := SetTheory.emptyset_mem

/-- This defines the subtype `A.toSubtype` for any `A:Set`. -/
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.instSingleton : Singleton Object Set where
  singleton := SetTheory.singleton

@[simp]
 theorem SetTheory.Set.mem_singleton (x a:Object) : x ∈ ({a}:Set) ↔ x = a :=
  SetTheory.singleton_axiom x a

instance SetTheory.Set.instUnion : Union Set where
  union := SetTheory.union_pair

@[simp]
 theorem SetTheory.Set.mem_union (x:Object) (X Y:Set) : x ∈ (X ∪ Y) ↔ (x ∈ X ∨ x ∈ Y) :=
  SetTheory.union_pair_axiom X Y x

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

abbrev Nat := SetTheory.nat

/-- Axiom 3.8 (Axiom of infinity) -/
 def SetTheory.Set.nat_equiv : ℕ ≃ Nat := SetTheory.nat_equiv

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

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

/-- Definition 3.6.1 (Equal cardinality) -/
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

/-- Proposition 3.6.4 / Exercise 3.6.1 -/
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} ⟩

/-- Definition 3.6.5 -/
abbrev SetTheory.Set.Fin (n : ℕ) : Set :=
  nat.specify (fun m ↦ ((m : ℕ) < n))

/-- Definition 3.6.5 -/
abbrev SetTheory.Set.has_card (X:Set) (n:ℕ) : Prop := X ≈ SetTheory.Set.Fin n

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

open Classical in
/-- It is convenient for Lean purposes to give infinite sets the ``junk`` cardinality of zero. -/
noncomputable def SetTheory.Set.card (X:Set) : ℕ := if h:X.finite then h.choose else 0

/-- Proposition 3.6.14 (a) / Exercise 3.6.4 -/
 theorem SetTheory.Set.card_insert {X:Set} (hX: X.finite) {x:Object} (hx: x ∉ X) :
    (X ∪ {x}).finite ∧ (X ∪ {x}).card = X.card + 1 := by
end Chapter3