set_option maxHeartbeats 0

import Mathlib.Tactic

/-!
Self-contained extract for Chapter 3 Section 3.3 target theorem
-*/

/-
  Minimal tools for ExistsUnique (as used in Section 3.3).
-/

/-- This implements the axiom of unique choice. -/
noncomputable def ExistsUnique.choose {α: Sort*} {p: α → Prop} (h : ∃! x, p x) : α := h.exists.choose

theorem ExistsUnique.choose_spec {α: Sort*} {p: α → Prop} (h : ∃! x, p x) :
  p h.choose := h.exists.choose_spec

theorem ExistsUnique.choose_eq {α: Sort*} {p: α → Prop} (h : ∃! x, p x) {x : α} (hx : p x) :
  h.choose = x := h.unique h.choose_spec hx

theorem ExistsUnique.choose_iff {α: Sort*} {p: α → Prop} (h : ∃! x, p x) (x : α):
  p x ↔ x = h.choose :=
  ⟨ by intro hx; exact (h.choose_eq hx).symm, by rintro rfl; exact h.choose_spec ⟩


/-
  Section 3.1 minimal context
-/

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

-- This enables one to use `Set` and `Object` instead of `SetTheory.Set` and `SetTheory.Object`.
export SetTheory (Set Object)

-- This instance implicitly imposes the axioms of Zermelo-Frankel set theory with atoms.
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`.
  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 := A.toSubtype

/-
  Section 3.3 (relevant fragments)
-/

/--
  Definition 3.3.1. `Function X Y` is the structure of functions from `X` to `Y`.
  Analogous to the Mathlib type `X → Y`.
-/
@[ext]
structure Function (X Y: Set) where
  P : X → Y → Prop
  unique : ∀ x: X, ∃! y: Y, P x y

#check Function.mk

/--
  Converting a Chapter 3 function `f: Function X Y` to a Mathlib function `f: X → Y`.
  The Chapter 3 definition of a function was nonconstructive, so we have to use the
  axiom of choice here.
-/
noncomputable def Function.to_fn {X Y: Set} (f: Function X Y) : X → Y :=
  fun x ↦ (f.unique x).choose

noncomputable instance Function.inst_coefn (X Y: Set) : CoeFun (Function X Y) (fun _ ↦ X → Y) where
  coe := Function.to_fn

@[simp] theorem Function.to_fn_eval {X Y: Set} (f: Function X Y) (x:X) : f.to_fn x = f x := rfl

/-- Converting a Mathlib function to a Chapter 3 `Function` -/
abbrev Function.mk_fn {X Y: Set} (f: X → Y) : Function X Y :=
  Function.mk (fun x y ↦ y = f x) (by simp)

/-- Definition 3.3.1 -/
@[simp] theorem Function.eval {X Y: Set} (f: Function X Y) (x: X) (y: Y) : y = f x ↔ f.P x y := by
  convert ((f.unique x).choose_iff y).symm

@[simp]
theorem Function.eval_of {X Y: Set} (f: X → Y) (x:X) : (Function.mk_fn f) x = f x := by
  symm; rw [eval]

/-- Definition 3.3.13 (Composition) -/
noncomputable abbrev Function.comp {X Y Z: Set} (g: Function Y Z) (f: Function X Y) :
    Function X Z :=
  Function.mk_fn (fun x ↦ g (f x))

-- `∘` is already taken in Mathlib for the composition of Mathlib functions,
-- so we use `○` here instead to avoid ambiguity.
infix:90 "○" => Function.comp

@[simp] theorem Function.comp_eval {X Y Z: Set} (g: Function Y Z) (f: Function X Y) (x: X) :
    (g ○ f) x = g (f x) := Function.eval_of _ _

abbrev Function.one_to_one {X Y: Set} (f: Function X Y) : Prop := ∀ x x': X, x ≠ x' → f x ≠ f x'

theorem Function.one_to_one_iff {X Y: Set} (f: Function X Y) :
    f.one_to_one ↔ ∀ x x': X, f x = f x' → x = x' := by sorry

/-- Definition 3.3.20 (Onto functions) -/
abbrev Function.onto {X Y: Set} (f: Function X Y) : Prop := ∀ y: Y, ∃ x: X, f x = y

/-- Definition 3.3.23 (Bijective functions) -/
abbrev Function.bijective {X Y: Set} (f: Function X Y) : Prop := f.one_to_one ∧ f.onto

/--
  We cannot use the notation `f⁻¹` for the inverse because in Mathlib's `Inv` class, the inverse
  of `f` must be exactly of the same type of `f`, and `Function Y X` is a different type from
  `Function X Y`.
-/
abbrev Function.inverse {X Y: Set} (f: Function X Y) (h: f.bijective) :
    Function Y X :=
  Function.mk (fun y x ↦ f x = y) (by
    intros
    apply existsUnique_of_exists_of_unique
    . aesop
    intro _ _ hx hx'; simp at hx hx'
    rw [←hx'] at hx
    apply f.one_to_one_iff.mp h.1
    simp [hx]
  )

@[simp] theorem Function.inverse_eval {X Y: Set} {f: Function X Y} (h: f.bijective) (y: Y) (x: X) :
    x = (f.inverse h) y ↔ f x = y := Function.eval _ _ _

/-- Composition preserves bijectivity. -/
theorem Function.comp_bijective {X Y Z:Set} {f: Function X Y} {g : Function Y Z} (hf: f.bijective)
  (hg: g.bijective) : (g ○ f).bijective := by sorry

/-- TARGET THEOREM (Exercise 3.3.7) -/
theorem Function.inv_of_comp {X Y Z:Set} {f: Function X Y} {g : Function Y Z}
  (hf: f.bijective) (hg: g.bijective) :
    (g ○ f).inverse (Function.comp_bijective hf hg) = (f.inverse hf) ○ (g.inverse hg) := by
end Chapter3