import Mathlib.Tactic

/-!
Self-contained slice for Chapter 3 Section 3.6 target theorem.
-/

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

/-- 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 := A.toSubtype

/-- Definition 3.1.14.   -/
instance SetTheory.Set.instSubset : HasSubset Set where
  Subset X Y := ∀ x, x ∈ X → x ∈ Y

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

/-- Axiom 3.7 (Axiom of replacement) -/
@[simp]
theorem SetTheory.Set.replacement_axiom {A:Set} {P: A → Object → Prop}
  (hP: ∀ x y y', P x y ∧ P x y' → y = y') (y:Object) :
    y ∈ A.replace hP ↔ ∃ x, P x y := SetTheory.replacement_axiom A P hP y

/-- Abbreviation for specification (from Section 3.1). -/
abbrev SetTheory.Set.specify (A:Set) (P: A → Prop) : Set := SetTheory.specify A P

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

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

/-- Image of a set under a function between sets (Section 3.6). -/
abbrev SetTheory.Set.image {X Y : Set} (f : X → Y) (S : Set) : Set :=
  X.replace (P := fun x y ↦ f x = y ∧ x.val ∈ S) (by simp_all)

/-- 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) : 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

/-- Proposition 3.6.4 / Exercise 3.6.1 -/
instance SetTheory.Set.EqualCard.inst_setoid : Setoid SetTheory.Set := ⟨ EqualCard, {refl, symm, trans} ⟩

/-- Finite cardinals as sets: `Fin n` -/
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 ≈ Fin n

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

/-- Exercise 3.6.11 -/
theorem SetTheory.Set.two_to_two_iff {X Y:Set} (f: X → Y): Function.Injective f ↔
    ∀ S ⊆ X, S.card = 2 → (image f S).card = 2 := by
end Chapter3