import Mathlib.Tactic

/-!
Self-contained slice for Chapter 3, Section 3.4 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)

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`.

  To produce an element `x'` of this subtype, use `⟨ x, hx ⟩`, where `x: Object` and `hx: x ∈ A`.
  The object `x` associated to a subtype element `x'` is recovered as `x'.val`, and
  the property `hx` that `x` belongs to `A` is recovered as `x'.property`.
-/
abbrev SetTheory.Set.toSubtype (A:Set) := Subtype (fun x ↦ x ∈ A)

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

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

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

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

@[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

/-- Definition 3.4.1.  Interestingly, the definition does not require S to be a subset of X. -/
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 3.4.4 (inverse images).
  Again, it is not required that U be a subset of Y.
-/
abbrev SetTheory.Set.preimage {X Y:Set} (f:X → Y) (U: Set) : Set := X.specify (P := fun x ↦ (f x).val ∈ U)

/-- Exercise 3.4.1 -/
theorem SetTheory.Set.preimage_eq_image_of_inv {X Y V:Set} (f:X → Y) (f_inv: Y → X)
  (hf: Function.LeftInverse f_inv f ∧ Function.RightInverse f_inv f) (hV: V ⊆ Y) :
    image f_inv V = preimage f V := by
end Chapter3