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

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

open SetTheory.Set

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

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

@[simp, refl]
theorem SetTheory.Set.subset_self (A:Set) : A ⊆ A := by
  sorry

@[simp]
theorem SetTheory.Set.empty_subset (A:Set) : ∅ ⊆ A := by
  sorry

theorem SetTheory.Set.subset_trans {A B C:Set} (hAB:A ⊆ B) (hBC:B ⊆ C) : A ⊆ C := by
  sorry

theorem SetTheory.Set.subset_antisymm (A B:Set) (hAB:A ⊆ B) (hBA:B ⊆ A) : A = B := 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

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

theorem  SetTheory.Set.union_inter_distrib_left (A B C:Set) :
    A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) := by
  sorry

instance SetTheory.Set.instDistribLattice : DistribLattice Set where
  le := (· ⊆ ·)
  le_refl := subset_self
  le_trans := fun _ _ _ ↦ subset_trans
  le_antisymm := subset_antisymm
  inf := (· ∩ ·)
  sup := (· ∪ ·)
  le_sup_left := by sorry
  le_sup_right := by sorry
  sup_le := by sorry
  inf_le_left := by sorry
  inf_le_right := by sorry
  le_inf := by sorry
  le_sup_inf := by
    intro X Y Z; change (X ∪ Y) ∩ (X ∪ Z) ⊆ X ∪ (Y ∩ Z)
    rw [←union_inter_distrib_left]

instance SetTheory.Set.instOrderBot : OrderBot Set where
  bot := ∅
  bot_le := empty_subset

abbrev axiom_of_universal_specification : Prop :=
  ∀ P : Object → Prop, ∃ A : Set, ∀ x : Object, x ∈ A ↔ P x

theorem Russells_paradox : ¬ axiom_of_universal_specification := by
  sorry

theorem SetTheory.Set.axiom_of_regularity {A:Set} (h: A ≠ ∅) :
    ∃ x:A, ∀ S:Set, x.val = S → Disjoint S A := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.emptyset_exists (h: axiom_of_universal_specification):
    ∃ (X:Set), ∀ x, x ∉ X := by
-- </TARGET_THEOREM>

end Chapter3