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]

/-- 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

/--
  This defines the subtype `A.toSubtype` for any `A:Set`.
  Note that `A.toSubtype` gives you a type, similar to how `Object` or `Set` are types.
  A value `x'` of type `A.toSubtype` combines some `x: Object` with a proof that `hx: x ∈ A`.
-/
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

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 Nat := SetTheory.nat

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

-- Below are some API for handling coercions. This may not be the optimal way to set things up.
instance SetTheory.Set.instOfNat {n:ℕ} : OfNat Nat n where
  ofNat := SetTheory.Set.nat_equiv n

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

instance SetTheory.Object.instNatCast : NatCast Object where
  natCast n := ((n:Nat):Object)

instance SetTheory.Object.instOfNat {n:ℕ} : OfNat Object n where
  ofNat := ((n:Nat):Object)

namespace SetTheory.Set

/-- Definition 3.6.1 (Equal cardinality) -/
abbrev EqualCard (X Y:Set) : Prop := ∃ f : X → Y, Function.Bijective f

/-- Example 3.6.2 -/
theorem Example_3_6_2 : EqualCard {0,1,2} {3,4,5} := by
end SetTheory.Set

end Chapter3