import Mathlib
import Analysis.Section_3_1
import Analysis.Tags.Section_3_1

namespace Chapter3

-- <CONTEXT>
universe u v

class SetTheory where
  Set : Type u
  Object : Type v
  set_to_object : Set ↪ Object
  mem : Object → Set → Prop
  extensionality X Y : (∀ x, mem x X ↔ mem x Y) → X = Y
  emptyset: Set
  emptyset_mem x : ¬ mem x emptyset
  singleton : Object → Set
  singleton_axiom x y : mem x (singleton y) ↔ x = y
  union_pair : Set → Set → Set
  union_pair_axiom X Y x : mem x (union_pair X Y) ↔ (mem x X ∨ mem x Y)
  specify A (P: Subtype (mem . A) → Prop) : Set
  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
  replace A (P: Subtype (mem . A) → Object → Prop)
    (hP: ∀ x y y', P x y ∧ P x y' → y = y') : Set
  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
  nat : Set
  nat_equiv : ℕ ≃ Subtype (mem . nat)
  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
  pow : Set → Set → Set
  function_to_object (X: Set) (Y: Set) :
    (Subtype (mem . X) → Subtype (mem . Y)) ↪ Object
  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
  union : Set → Set
  union_axiom A x : mem x (union A) ↔ ∃ S, mem x S ∧ mem (set_to_object S) A

export SetTheory (Set Object)

variable [SetTheory]

instance SetTheory.objects_mem_sets : Membership Object Set where
  mem X x := mem x X

instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := set_to_object X

@[simp]
theorem SetTheory.Set.coe_eq_iff (X Y:Set) : (X: Object) = (Y: Object) ↔  X = Y := by
  sorry

instance SetTheory.Set.instEmpty : EmptyCollection Set where
  emptyCollection := emptyset

@[simp]
theorem SetTheory.Set.not_mem_empty : ∀ x, x ∉ (∅:Set) := by
  sorry

lemma SetTheory.Set.nonempty_def {X:Set} (h: X ≠ ∅) : ∃ x, x ∈ X := by
  sorry

instance SetTheory.Set.instSingleton : Singleton Object Set where
  singleton := singleton

@[simp]
theorem SetTheory.Set.mem_singleton (x a:Object) : x ∈ ({a}:Set) ↔ x = a := by
  sorry

instance SetTheory.Set.instUnion : Union Set where
  union := union_pair

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

instance SetTheory.Set.instInsert : Insert Object Set where
  insert x X := {x} ∪ X

@[simp]
theorem SetTheory.Set.mem_insert (a b: Object) (X: Set) : a ∈ insert b X ↔ a = b ∨ a ∈ X := by
  sorry

@[simp]
theorem SetTheory.Set.mem_pair (x a b:Object) : x ∈ ({a,b}:Set) ↔ (x = a ∨ x = b) := by
  sorry

@[simp]
theorem SetTheory.Set.mem_triple (x a b c:Object) : x ∈ ({a,b,c}:Set) ↔ (x = a ∨ x = b ∨ x = c) := by
  sorry

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

theorem SetTheory.Set.subset_def (X Y:Set) : X ⊆ Y ↔ ∀ x, x ∈ X → x ∈ Y := by
  sorry

abbrev SetTheory.Set.toSubtype (A:Set) := Subtype (fun x ↦ x ∈ A)

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

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

@[simp]
theorem SetTheory.Set.specification_axiom'' {A:Set} (P: A → Prop) (x:Object) :
    x ∈ A.specify P ↔ ∃ h:x ∈ A, P ⟨ x, h ⟩ := by
  sorry

instance SetTheory.Set.instIntersection : Inter Set where
  inter X Y := X.specify (fun x ↦ x.val ∈ Y)

@[simp]
theorem SetTheory.Set.mem_inter (x:Object) (X Y:Set) : x ∈ (X ∩ Y) ↔ (x ∈ X ∧ x ∈ Y) := by
  sorry

instance SetTheory.Set.instSDiff : SDiff Set where
  sdiff X Y := X.specify (fun x ↦ x.val ∉ Y)

@[simp]
theorem SetTheory.Set.mem_sdiff (x:Object) (X Y:Set) : x ∈ (X \ Y) ↔ (x ∈ X ∧ x ∉ Y) := by
  sorry

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 := by
  sorry

abbrev Nat := SetTheory.nat

def SetTheory.Set.nat_equiv : ℕ ≃ Nat := SetTheory.nat_equiv

instance SetTheory.Set.instOfNat {n:ℕ} : OfNat Nat n where
  ofNat := nat_equiv n

instance SetTheory.Set.instNatCast : NatCast Nat where
  natCast n := nat_equiv n

instance SetTheory.Set.toNat : Coe Nat ℕ where
  coe n := nat_equiv.symm n

instance SetTheory.Object.instNatCast : NatCast Object where
  natCast n := (n:Nat).val

instance SetTheory.Object.instOfNat {n:ℕ} : OfNat Object n where
  ofNat := ((n:Nat):Object)

@[simp]
lemma SetTheory.Object.ofnat_eq {n:ℕ} : ((n:Nat):Object) = (n:Object) := by
  sorry

@[simp]
theorem SetTheory.Object.natCast_inj (n m:ℕ) :
    (n : Object) = (m : Object) ↔ n = m := by
  sorry

instance SetTheory.Set.inst_coe_set : Coe Set (_root_.Set Object) where
  coe X := { x | x ∈ X }

export SetTheory (Set Object nat)

variable [SetTheory]

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)

theorem SetTheory.Set.mem_image {X Y:Set} (f:X → Y) (S: Set) (y:Object) :
    y ∈ image f S ↔ ∃ x:X, x.val ∈ S ∧ f x = y := by
  sorry

theorem SetTheory.Set.image_eq_specify {X Y:Set} (f:X → Y) (S: Set) :
    image f S = Y.specify (fun y ↦ ∃ x:X, x.val ∈ S ∧ f x = y) := by
  sorry

theorem SetTheory.Set.image_eq_image {X Y:Set} (f:X → Y) (S: Set):
    (image f S: _root_.Set Object) = Subtype.val '' (f '' {x | x.val ∈ S}) := by
  sorry

theorem SetTheory.Set.image_in_codomain {X Y:Set} (f:X → Y) (S: Set) :
    image f S ⊆ Y := by
  sorry

abbrev f_3_4_2 : nat → nat := fun n ↦ (2*n:ℕ)

theorem SetTheory.Set.image_f_3_4_2 : image f_3_4_2 {1,2,3} = {2,4,6} := by
  sorry

theorem SetTheory.Set.mem_image_of_eval {X Y:Set} (f:X → Y) (S: Set) (x:X) :
    x.val ∈ S → (f x).val ∈ image f S := by
  sorry

theorem SetTheory.Set.mem_image_of_eval_counter :
    ∃ (X Y:Set) (f:X → Y) (S: Set) (x:X), ¬((f x).val ∈ image f S → x.val ∈ S) := by
  sorry

abbrev SetTheory.Set.preimage {X Y:Set} (f:X → Y) (U: Set) : Set := X.specify (P := fun x ↦ (f x).val ∈ U)

@[simp]
theorem SetTheory.Set.mem_preimage {X Y:Set} (f:X → Y) (U: Set) (x:X) :
    x.val ∈ preimage f U ↔ (f x).val ∈ U := by
  sorry

theorem SetTheory.Set.mem_preimage' {X Y:Set} (f:X → Y) (U: Set) (x:Object) :
    x ∈ preimage f U ↔ ∃ x': X, x'.val = x ∧ (f x').val ∈ U := by
  sorry

theorem SetTheory.Set.preimage_eq {X Y:Set} (f:X → Y) (U: Set) :
    ((preimage f U): _root_.Set Object) = Subtype.val '' (f⁻¹' {y | y.val ∈ U}) := by
  sorry

theorem SetTheory.Set.preimage_in_domain {X Y:Set} (f:X → Y) (U: Set) :
    (preimage f U) ⊆ X := by
  sorry

theorem SetTheory.Set.preimage_f_3_4_2 : preimage f_3_4_2 {2,4,6} = {1,2,3} := by
  sorry

theorem SetTheory.Set.image_preimage_f_3_4_2 :
    image f_3_4_2 (preimage f_3_4_2 {1,2,3}) ≠ {1,2,3} := by
  sorry

instance SetTheory.Set.inst_pow : Pow Set Set where
  pow := pow

@[coe]
def SetTheory.Set.coe_of_fun {X Y:Set} (f: X → Y) : Object := function_to_object X Y f

instance SetTheory.Set.inst_coe_of_fun {X Y:Set} : CoeOut (X → Y) Object where
  coe := coe_of_fun

@[simp]
theorem SetTheory.Set.coe_of_fun_inj {X Y:Set} (f g:X → Y) : (f:Object) = (g:Object) ↔ f = g := by
  sorry

@[simp]
theorem SetTheory.Set.powerset_axiom {X Y:Set} (F:Object) :
    F ∈ (X ^ Y) ↔ ∃ f: Y → X, f = F := by
  sorry

abbrev f_3_4_9_a : ({4,7}:Set) → ({0,1}:Set) := fun x ↦ ⟨ 0, by simp ⟩

open Classical in
noncomputable abbrev f_3_4_9_b : ({4,7}:Set) → ({0,1}:Set) :=
  fun x ↦ if x.val = 4 then ⟨ 0, by simp ⟩ else ⟨ 1, by simp ⟩

open Classical in
noncomputable abbrev f_3_4_9_c : ({4,7}:Set) → ({0,1}:Set) :=
  fun x ↦ if x.val = 4 then ⟨ 1, by simp ⟩ else ⟨ 0, by simp ⟩

abbrev f_3_4_9_d : ({4,7}:Set) → ({0,1}:Set) := fun x ↦ ⟨ 1, by simp ⟩

theorem SetTheory.Set.example_3_4_9 (F:Object) :
    F ∈ ({0,1}:Set) ^ ({4,7}:Set) ↔ F = f_3_4_9_a
    ∨ F = f_3_4_9_b ∨ F = f_3_4_9_c ∨ F = f_3_4_9_d := by
  sorry

def SetTheory.Set.powerset (X:Set) : Set :=
  (({0,1} ^ X): Set).replace (P := sorry) (by sorry)

theorem SetTheory.Set.exists_powerset (X:Set) :
   ∃ (Z: Set), ∀ x, x ∈ Z ↔ ∃ Y:Set, x = Y ∧ Y ⊆ X := by
  sorry

theorem SetTheory.Set.powerset_of_triple (a b c x:Object) :
    x ∈ powerset {a,b,c}
    ↔ x = (∅:Set)
    ∨ x = ({a}:Set)
    ∨ x = ({b}:Set)
    ∨ x = ({c}:Set)
    ∨ x = ({a,b}:Set)
    ∨ x = ({a,c}:Set)
    ∨ x = ({b,c}:Set)
    ∨ x = ({a,b,c}:Set) := by
  sorry

theorem SetTheory.Set.union_axiom (A: Set) (x:Object) :
    x ∈ union A ↔ ∃ (S:Set), x ∈ S ∧ (S:Object) ∈ A := by
  sorry

theorem SetTheory.Set.example_3_4_12 :
    union { (({2,3}:Set):Object), (({3,4}:Set):Object), (({4,5}:Set):Object) } = {2,3,4,5} := by
  sorry

theorem SetTheory.Set.union_eq (A: Set) :
    (union A : _root_.Set Object) =
    ⋃₀ { S : _root_.Set Object | ∃ S':Set, S = S' ∧ (S':Object) ∈ A } := by
  sorry

abbrev SetTheory.Set.iUnion (I: Set) (A: I → Set) : Set :=
  union (I.replace (P := fun α S ↦ S = A α) (by intro _ _ _ ⟨h1, h2⟩; exact h1.trans h2.symm))

theorem SetTheory.Set.mem_iUnion {I:Set} (A: I → Set) (x:Object) :
    x ∈ iUnion I A ↔ ∃ α:I, x ∈ A α := by
  sorry

open Classical in
noncomputable abbrev SetTheory.Set.index_example : ({1,2,3}:Set) → Set :=
  fun i ↦ if i.val = 1 then {2,3} else if i.val = 2 then {3,4} else {4,5}

theorem SetTheory.Set.iUnion_example : iUnion {1,2,3} index_example = {2,3,4,5} := by
  sorry

theorem SetTheory.Set.iUnion_eq (I: Set) (A: I → Set) :
    (iUnion I A : _root_.Set Object) = ⋃ α, (A α: _root_.Set Object) := by
  sorry

theorem SetTheory.Set.iUnion_of_empty (A: (∅:Set) → Set) : iUnion (∅:Set) A = ∅ := by
  sorry

noncomputable abbrev SetTheory.Set.nonempty_choose {I:Set} (hI: I ≠ ∅) : I :=
  ⟨(nonempty_def hI).choose, (nonempty_def hI).choose_spec⟩

abbrev SetTheory.Set.iInter' (I:Set) (β:I) (A: I → Set) : Set :=
  (A β).specify (P := fun x ↦ ∀ α:I, x.val ∈ A α)

noncomputable abbrev SetTheory.Set.iInter (I: Set) (hI: I ≠ ∅) (A: I → Set) : Set :=
  iInter' I (nonempty_choose hI) A

theorem SetTheory.Set.mem_iInter {I:Set} (hI: I ≠ ∅) (A: I → Set) (x:Object) :
    x ∈ iInter I hI A ↔ ∀ α:I, x ∈ A α := by
  sorry

def SetTheory.Set.image_of_inter' : Decidable (∀ X Y:Set, ∀ f:X → Y, ∀ A B: Set, image f (A ∩ B) = (image f A) ∩ (image f B)) := by
  sorry

def SetTheory.Set.image_of_diff' : Decidable (∀ X Y:Set, ∀ f:X → Y, ∀ A B: Set, image f (A \ B) = (image f A) \ (image f B)) := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.preimage_image_of_inj {X Y:Set} (f:X → Y) :
    (∀ S, S ⊆ X → preimage f (image f S) = S) ↔ Function.Injective f := by
-- </TARGET_THEOREM>

end Chapter3