introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib.Tactic
import Mathlib.Data.Nat.Parity

namespace Chapter3

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 nat)

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

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

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

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

abbrev Nat := SetTheory.nat

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

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

abbrev SetTheory.Set.EqualCard (X Y:Set) : Prop := ∃ f : X → Y, Function.Bijective f
theorem SetTheory.Set.Example_3_6_3 : EqualCard nat (nat.specify (fun x ↦ Even (x:ℕ))) := by
end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory
namespace Set
theorem Example_3_6_3 :
    ∃ f : ℕ → { n : ℕ // Even n }, Function.Bijective f := by
end Set
end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.Example_3_6_7a {Object : Type _} (a : Object) :
    ∃ f : {x : Object // x ∈ ({a} : Set Object)} → Fin 1, Function.Bijective f := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.Permutations_finite (n : ℕ) :
    ({f : Fin n → Fin n | Function.Bijective f} : Set (Fin n → Fin n)).Finite := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory
namespace Set

/-- Exercise 3.6.12 (i), second part -/
theorem Permutations_ih (n : ℕ) :
    Fintype.card (Equiv.Perm (Fin (n + 1))) =
      (n + 1) * Fintype.card (Equiv.Perm (Fin n)) := by
end Set
end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter3
namespace SetTheory
namespace Set
namespace Tuple

variable {α : Type*} {n : ℕ}

/-- Equality of `n`-tuples (functions `Fin n → α`) is pointwise equality. -/
theorem eq (t t' : Fin n → α) :
    t = t' ↔ ∀ i : Fin n, t i = t' i := by
end Tuple
end Set
end SetTheory
end Chapter3

end TaoBench
import Mathlib.Tactic

/-!
Self-contained extraction for Chapter 3 minimal context to typecheck the 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 := mem x X

/-- Axiom 3.1 (Sets are objects)-/
instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := set_to_object 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 := A.toSubtype

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

abbrev Nat := SetTheory.nat

/-- Axiom 3.8 (Axiom of infinity) -/
def SetTheory.Set.nat_equiv : ℕ ≃ Nat := SetTheory.nat_equiv

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

/-- Abbreviation for the set of naturals less than `n` (as a set in our set theory). -/
abbrev SetTheory.Set.Fin (n : ℕ) : Set := nat.specify (fun m ↦ (m : ℕ) < n)

export SetTheory (nat)

/-- Exercise 3.6.3, phrased using Mathlib natural numbers -/
theorem SetTheory.Set.bounded_on_finite {n:ℕ} (f: Fin n → nat) : ∃ M, ∀ i, (f i:ℕ) ≤ M := by
end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory
namespace Set

/-- Exercise 3.6.3 (Mathlib version):
Any function from a finite type `Fin n` to `ℕ` is bounded above. -/
theorem bounded_on_finite {n : ℕ} (f : Fin n → ℕ) :
    ∃ M, ∀ i, f i ≤ M := by
end Set
end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory
namespace Set

/--
Tao 3.6 (translated to Mathlib):

For a finite “set” (here: a finite type `α`), its cardinal is zero
if and only if it has no elements.
-/
theorem card_eq_zero {α : Type*} [Fintype α] :
    Fintype.card α = 0 ↔ IsEmpty α := by
end Set
end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem card_image {α β : Type*} {s : Set α} (hs : s.Finite) (f : α → β) :
    (f '' s).Finite ∧ Nat.card (f '' s) ≤ Nat.card s := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory
namespace Set

/-- Mathlib translation of `Chapter3.SetTheory.Set.card_image_inj`.

If `α` is a finite type and `f : α → β` is injective, then the
(junk-extended-to-`ℕ`) cardinality of the image of `f` equals that of `α`.

Here `Cardinal.mk` is the cardinality of a type, and `Cardinal.toNat` sends a
cardinal to its finite size when finite, and to `0` when infinite, matching
the behavior of Tao's `Set.card`. -/
theorem card_image_inj
  {α β : Type*} [Fintype α] {f : α → β}
  (hf : Function.Injective f) :
  Cardinal.toNat (Cardinal.mk (Set.range f)) =
    Cardinal.toNat (Cardinal.mk α) := by
end Set
end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.card_insert
    {α : Type*} {X : Set α} (hX : X.Finite) {x : α} (hx : x ∉ X) :
    (X ∪ {x}).Finite ∧ (X ∪ {x}).ncard = X.ncard + 1 := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem card_pow {α β : Type*} [Finite α] [Finite β] :
  Finite (α → β) ∧ Nat.card (α → β) = Nat.card β ^ Nat.card α := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.card_prod {α β : Type*} {s : Set α} {t : Set β}
    (hs : s.Finite) (ht : t.Finite) :
    (s ×ˢ t).Finite ∧ Nat.card (s ×ˢ t) = Nat.card s * Nat.card t := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.card_ssubset {α : Type*} {X Y : Set α}
    (hX : X.Finite) (hY : Y ⊂ X) :
    Y.ncard < X.ncard := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.card_subset {α : Type*} {X Y : Set α}
    (hX : X.Finite) (hY : Y ⊆ X) :
    Y.Finite ∧ Y.ncard ≤ X.ncard := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

variable {α : Type*}

-- Translation of Tao's Proposition 3.6.14(b) / Exercise 3.6.4:
-- For finite sets X and Y, the union X ∪ Y is finite and its
-- cardinality is at most the sum of the cardinalities of X and Y.
theorem SetTheory.Set.card_union {X Y : Set α}
    (hX : Set.Finite X) (hY : Set.Finite Y) :
    Set.Finite (X ∪ Y) ∧ (X ∪ Y).ncard ≤ X.ncard + Y.ncard := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory

namespace Set

/-- Exercise 3.6.9 (Mathlib version): For finite sets `A` and `B`,
`card A + card B = card (A ∪ B) + card (A ∩ B)`.
Here `card` is interpreted as `Set.ncard`. -/
theorem card_union_add_card_inter {α : Type*} {A B : Set α}
    (hA : A.Finite) (hB : B.Finite) :
    A.ncard + B.ncard = (A ∪ B).ncard + (A ∩ B).ncard := by
end Set

end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.card_union_disjoint
  {α : Type*} {X Y : Set α}
  (hX : X.Finite) (hY : Y.Finite)
  (hdisj : Disjoint X Y) :
  (X ∪ Y).ncard = X.ncard + Y.ncard := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter3
namespace SetTheory
namespace Set

variable {α ι : Type _}

/-- De Morgan's law for indexed intersections of sets, corresponding to Tao Exercise 3.4.11. -/
theorem compl_iInter [Nonempty ι] (X : Set α) (A : ι → Set α) :
    X \ ⋂ i, A i = ⋃ i, X \ A i := by
end Set
end SetTheory
end Chapter3

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- De Morgan's law for relative complements: the complement in `X` of an
indexed union is the intersection of the complements in `X`. This corresponds
to Tao's Exercise 3.4.11. -/
theorem SetTheory.Set.compl_iUnion
  {α ι : Type*} [Nonempty ι] (X : Set α) (A : ι → Set α) :
  X \ ⋃ i, A i = ⋂ i, X \ A i := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace SetTheory
namespace Set
def diff_of_prod :
    Decidable (∀ (α : Type*) (A B C D : Set α),
      (A ×ˢ B) \ (C ×ˢ D) = (A \ C) ×ˢ (B \ D)) := by
end Set
end SetTheory

end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.diff_prod {α β : Type*} (A B : Set α) (C : Set β) :
  (A \ B) ×ˢ C = (A ×ˢ C) \ (B ×ˢ C) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem SetTheory.Set.direct_sum {X Y Z : Type _} (f : Z → X) (g : Z → Y) :
    ∃! h : Z → X × Y, Prod.fst ∘ h = f ∧ Prod.snd ∘ h = g := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

-- Translation of Chapter 3, Exercise 3.6.2
theorem SetTheory.Set.empty_iff_card_eq_zero
    {α : Type _} {X : Set α} :
    X = (∅ : Set α) ↔ X.Finite ∧ Nat.card X = 0 := by
end TaoBench
import Mathlib.Tactic

/-!
Self-contained slice for Chapter 3, Sections 3.1–3.2, sufficient to typecheck
the target theorem `Chapter3.SetTheory.Set.emptyset_exists`.
-/

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 := mem x X

/-- Axiom 3.8 (Universal specification) -/
abbrev axiom_of_universal_specification : Prop :=
  ∀ P : Object → Prop, ∃ A : Set, ∀ x : Object, x ∈ A ↔ P x

/--
  Exercise 3.2.1.  The spirit of the exercise is to establish these results without using either
  Russell's paradox, or the empty set.
-/
theorem SetTheory.Set.emptyset_exists (h: axiom_of_universal_specification):
    ∃ (X:Set), ∀ x, x ∉ X := by
end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

universe u

namespace SetTheory
namespace Set
theorem emptyset_exists {α : Type u}
    (h : ∀ P : α → Prop, ∃ A : _root_.Set α, ∀ x : α, x ∈ A ↔ P x) :
    ∃ X : _root_.Set α, ∀ x : α, x ∉ X := by
end Set
end SetTheory

end TaoBench
import Mathlib.Tactic

namespace Chapter3

universe u v

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 := mem x X

/-- Axiom 3.1 (Sets are objects)-/
instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := set_to_object X

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

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

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

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

abbrev SetTheory.Set.empty : Set := ∅
abbrev SetTheory.Set.singleton_empty : Set := {(empty: Object)}
abbrev SetTheory.Set.pair_empty : Set := {(empty: Object), (singleton_empty: Object)}

/-- Exercise 3.1.2 -/
theorem SetTheory.Set.emptyset_neq_pair : empty ≠ pair_empty := by
end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter3

/-- Mathlib translation of Tao's Exercise 3.1.2.
In any type `α`, the empty set is not equal to a pair `{a, b}`. -/
theorem SetTheory.Set.emptyset_neq_pair {α : Type*} (a b : α) :
    (∅ : Set α) ≠ ({a, b} : Set α) := by
end Chapter3

end TaoBench
import Mathlib.Logic.Equiv.Basic

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]

/-- Axiom 3.1 (Sets are objects)-/
instance SetTheory.sets_are_objects : Coe Set Object where
  coe X := set_to_object X

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

-- Make everything we define in `SetTheory.Set.*` accessible directly.
open SetTheory.Set

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

abbrev SetTheory.Set.empty : Set := ∅
abbrev SetTheory.Set.singleton_empty : Set := {(empty: Object)}

/-- Exercise 3.1.2 -/
theorem SetTheory.Set.emptyset_neq_singleton : empty ≠ singleton_empty := by
end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter3
namespace SetTheory

/-- Mathlib translation of Tao's Exercise 3.1.2: ∅ ≠ {∅}. -/
theorem Set.emptyset_neq_singleton :
  (∅ : Set (Set ℕ)) ≠ { (∅ : Set ℕ) } := by
end SetTheory
end Chapter3

end TaoBench