import Mathlib.Tactic

/-!
Self-contained slice to typecheck Proposition 3.6.14 (c).
-/

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

/-- 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

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

/-- 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

/-- Re-expose `specify` at the `Set` namespace level. -/
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.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).val

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

/-- Reflexivity of equal cardinality -/
@[refl]
theorem SetTheory.Set.EqualCard.refl (X:Set) : SetTheory.Set.EqualCard X X := by
  sorry

/-- Symmetry of equal cardinality -/
@[symm]
theorem SetTheory.Set.EqualCard.symm {X Y:Set} (h: SetTheory.Set.EqualCard X Y) :
    SetTheory.Set.EqualCard Y X := by
  sorry

/-- Transitivity of equal cardinality -/
@[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,
    ⟨ SetTheory.Set.EqualCard.refl, SetTheory.Set.EqualCard.symm, SetTheory.Set.EqualCard.trans ⟩ ⟩

/-- `Fin n` as a set (Definition 3.6.5 auxiliary) -/
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

/-- `finite` sets (Section 3.6) -/
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 (c) / Exercise 3.6.4 -/
 theorem SetTheory.Set.card_subset {X Y:Set} (hX: X.finite) (hY: Y ⊆ X) :
    Y.finite ∧ Y.card ≤ X.card := by
end Chapter3