import Mathlib.Tactic

/-
Provide a tiny `grind` tactic alias so copied textbook proofs using `by grind`
compile without importing the textbook tactic module.
-/
macro "grind" : tactic => `(tactic| aesop)

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

@[simp]
 theorem SetTheory.Set.coe_eq {X Y:Set} (h: (X: Object) = (Y: Object)) : X = Y :=
  SetTheory.set_to_object.inj' h

@[simp]
 theorem SetTheory.Set.coe_eq_iff (X Y:Set) : (X: Object) = (Y: Object) ↔  X = Y :=
  ⟨ SetTheory.Set.coe_eq, by rintro rfl; rfl ⟩

/-- 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

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

@[simp]
 theorem SetTheory.Set.mem_union (x:Object) (X Y:Set) : x ∈ (X ∪ Y) ↔ (x ∈ X ∨ x ∈ Y) :=
  SetTheory.union_pair_axiom X Y 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

/-- Elements of a set (implicitly coerced to a subtype) are also elements of the set. -/
lemma SetTheory.Set.subtype_property (A:Set) (x:A) : x.val ∈ A := x.property

/-- Axiom 3.6 (axiom of specification) -/
abbrev SetTheory.Set.specify (A:Set) (P: A → Prop) : Set := SetTheory.specify A P

/-- Axiom 3.6 (axiom of specification) -/
 theorem SetTheory.Set.specification_axiom {A:Set} {P: A → Prop} {x:Object} (h: x ∈ A.specify P) :
    x ∈ A := (SetTheory.specification_axiom A P).1 x h

/-- Axiom 3.6 (axiom of specification) -/
 theorem SetTheory.Set.specification_axiom' {A:Set} (P: A → Prop) (x:A) :
    x.val ∈ A.specify P ↔ P x := (SetTheory.specification_axiom A P).2 x

/-- Axiom 3.6 (axiom of specification) -/
@[simp]
 theorem SetTheory.Set.specification_axiom'' {A:Set} (P: A → Prop) (x:Object) :
    x ∈ A.specify P ↔ ∃ h:x ∈ A, P ⟨ x, h ⟩ := by
  constructor
  · intro h; use SetTheory.Set.specification_axiom (A:=A) (P:=P) h
    simp [←SetTheory.Set.specification_axiom' (A:=A) (P:=P), h]
  intro ⟨ h, hP ⟩
  simpa [←SetTheory.Set.specification_axiom' (A:=A) (P:=P)] using 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 ∈ SetTheory.replace A P hP ↔ ∃ x, P x y :=
  SetTheory.replacement_axiom A P hP 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 3.12 (Union) -/
@[simp]
 theorem SetTheory.Set.union_axiom (A: Set) (x:Object) :
    x ∈ SetTheory.union A ↔ ∃ (S:Set), x ∈ S ∧ (S:Object) ∈ A :=
  SetTheory.union_axiom A x

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

@[simp]
 theorem OrderedPair.eq (x y x' y' : Object) :
    (⟨ x, y ⟩ : OrderedPair) = (⟨ x', y' ⟩ : OrderedPair) ↔ x = x' ∧ y = y' := by aesop

/-- Exercise 3.5.1, first part -/
-- We faithfully copy the definition; the original file contains a `sorry` here.
 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

/-- slice: needed for Cartesian product -/
abbrev SetTheory.Set.slice (x:Object) (Y:Set) : Set :=
  Y.replace (P := fun y z ↦ z = (⟨x, y⟩:OrderedPair)) (by grind)

@[simp]
 theorem SetTheory.Set.mem_slice (x z:Object) (Y:Set) :
    z ∈ (SetTheory.Set.slice x Y) ↔ ∃ y:Y, z = (⟨x, y⟩:OrderedPair) := by
  -- unfold slice and apply replacement axiom
  change z ∈ Y.replace (P := fun y z0 ↦ z0 = (⟨x, y⟩:OrderedPair)) (by grind) ↔ _
  -- re-prove the uniqueness condition explicitly to match the axiom input
  have hP : ∀ (y : Y) (z z' : Object),
      (z = (⟨x, y⟩:OrderedPair)) ∧ (z' = (⟨x, y⟩:OrderedPair)) → z = z' := by
    intro y z z' h; exact h.1.trans h.2.symm
  simpa using (
    SetTheory.replacement_axiom (A:=Y)
      (P:=fun y z0 ↦ z0 = (⟨x, y⟩:OrderedPair)) hP z)

/-- Cartesian product via replacement/union -/
abbrev SetTheory.Set.cartesian (X Y:Set) : Set :=
  SetTheory.union (X.replace (P := fun x z ↦ z = slice x Y)
    (by intro _ _ _ ⟨h1, h2⟩; exact h1.trans 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

@[simp]
 theorem SetTheory.Set.mem_cartesian (z:Object) (X Y:Set) :
    z ∈ X ×ˢ Y ↔ ∃ x:X, ∃ y:Y, z = (⟨x, y⟩:OrderedPair) := by
  simp only [SProd.sprod, SetTheory.Set.union_axiom]; constructor
  · intro ⟨ S, hz, hS ⟩; rw [SetTheory.Set.replacement_axiom] at hS; obtain ⟨ x, hx ⟩ := hS
    use x; simp_all
  rintro ⟨ x, y, rfl ⟩; use slice x Y; refine ⟨ by simp, ?_ ⟩
  rw [SetTheory.Set.replacement_axiom]; use x

noncomputable abbrev SetTheory.Set.fst {X Y:Set} (z:X ×ˢ Y) : X :=
  ((SetTheory.Set.mem_cartesian _ _ _).mp z.property).choose

noncomputable abbrev SetTheory.Set.snd {X Y:Set} (z:X ×ˢ Y) : Y :=
  (exists_comm.mp ((SetTheory.Set.mem_cartesian _ _ _).mp z.property)).choose

/-- Graph of a function X → Y as a subset of X ×ˢ Y -/
abbrev SetTheory.Set.graph {X Y:Set} (f: X → Y) : Set :=
  (X ×ˢ Y).specify (fun p ↦ (f (SetTheory.Set.fst p) = SetTheory.Set.snd p))

/-- Exercise 3.5.10 -/
 theorem SetTheory.Set.graph_inj {X Y:Set} (f f': X → Y) :
    SetTheory.Set.graph f = SetTheory.Set.graph f' ↔ f = f' := by
end Chapter3