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

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

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

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

instance SetTheory.Set.instIntersection : Inter Set where
  inter X Y := X.specify (fun x ↦ x.val ∈ Y)

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.replace (A:Set) {P: A → Object → Prop}
  (hP : ∀ x y y', P x y ∧ P x y' → y = y') : Set := SetTheory.replace A P hP

/-- Ordered pairs -/ 
@[ext]
structure OrderedPair where
  fst : Object
  snd : Object

/-- Ordered pair as an object (using a Kuratowski-style encoding). -/
def OrderedPair.toObject : OrderedPair ↪ Object where
  toFun p := ({ (({p.fst}:Set):Object), (({p.fst, p.snd}:Set):Object) }:Set)
  inj' := by
    sorry

instance OrderedPair.inst_coeObject : Coe OrderedPair Object where
  coe := OrderedPair.toObject

/-- A technical operation `{x} × Y` used to define the full Cartesian product -/
abbrev SetTheory.Set.slice (x:Object) (Y:Set) : Set :=
  Y.replace (P := fun y z ↦ z = (⟨x, y⟩:OrderedPair)) (by
    intro x y y' h
    rcases h with ⟨h1, h2⟩
    simpa [h1, h2]
  )

/-- Cartesian product -/
abbrev SetTheory.Set.cartesian (X Y:Set) : Set :=
  SetTheory.union (X.replace (P := fun x z ↦ z = slice x Y) (by
    intro x z z' h
    rcases h with ⟨h1, h2⟩
    exact Eq.trans h1 h2.symm
  ))

/-- This instance enables the ×ˢ notation for Cartesian product. -/
instance SetTheory.Set.inst_SProd : SProd Set Set Set where
  sprod := SetTheory.Set.cartesian

/-- Exercise 3.5.5 -/
 theorem SetTheory.Set.inter_of_prod (A B C D:Set) :
    (A ×ˢ B) ∩ (C ×ˢ D) = (A ∩ C) ×ˢ (B ∩ D) := by
end Chapter3