introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Nat.induction_from {n : ℕ} {P : ℕ → Prop}
    (hind : ∀ m, P m → P (Nat.succ m)) :
    P n → ∀ m, m ≥ n → P m := by
end TaoBench
namespace Chapter2

inductive Nat where
| zero : Nat
| succ : Nat → Nat
deriving Repr, DecidableEq

instance Nat.instZero : Zero Nat := ⟨ zero ⟩

postfix:100 "++" => Nat.succ

abbrev Nat.recurse (f: Nat → Nat → Nat) (c: Nat) : Nat → Nat := fun n ↦ match n with
| 0 => c
| n++ => f n (recurse f c n)

abbrev Nat.add (n m : Nat) : Nat := Nat.recurse (fun _ sum ↦ sum++) m n

instance Nat.instAdd : Add Nat where
  add := add

abbrev Nat.mul (n m : Nat) : Nat := Nat.recurse (fun _ prod ↦ prod + m) 0 n

instance Nat.instMul : Mul Nat where
  mul := mul

/-- Proposition 2.3.5 (Multiplication is associative) / Exercise 2.3.3
Compare with Mathlib's `Nat.mul_assoc` -/
theorem Nat.mul_assoc (a b c: Nat) : (a * b) * c = a * (b * c) := by
end Chapter2
import Mathlib
import Analysis.Section_2_2
import Analysis.Tags.Section_2_2

namespace Chapter2

-- <CONTEXT>
inductive Nat where
| zero : Nat
| succ : Nat → Nat
deriving Repr, DecidableEq

instance Nat.instZero : Zero Nat := ⟨ zero ⟩

postfix:100 "++" => Nat.succ

instance Nat.instOfNat {n:_root_.Nat} : OfNat Nat n where
  ofNat := _root_.Nat.rec 0 (fun _ n ↦ n++) n

abbrev Nat.recurse (f: Nat → Nat → Nat) (c: Nat) : Nat → Nat := fun n ↦ match n with
| 0 => c
| n++ => f n (recurse f c n)

abbrev Nat.add (n m : Nat) : Nat := Nat.recurse (fun _ sum ↦ sum++) m n

instance Nat.instAdd : Add Nat where
  add := add

def Nat.IsPos (n:Nat) : Prop := n ≠ 0

abbrev Nat.mul (n m : Nat) : Nat := Nat.recurse (fun _ prod ↦ prod + m) 0 n

instance Nat.instMul : Mul Nat where
  mul := mul

theorem Nat.zero_mul (m: Nat) : 0 * m = 0 := by
  sorry

theorem Nat.succ_mul (n m: Nat) : (n++) * m = n * m + m := by
  sorry

theorem Nat.one_mul' (m: Nat) : 1 * m = 0 + m := by
  sorry

theorem Nat.one_mul (m: Nat) : 1 * m = m := by
  sorry

theorem Nat.two_mul (m: Nat) : 2 * m = 0 + m + m := by
  sorry

lemma Nat.mul_zero (n: Nat) : n * 0 = 0 := by
  sorry

lemma Nat.mul_succ (n m:Nat) : n * m++ = n * m + n := by
  sorry

theorem Nat.mul_one (m: Nat) : m * 1 = m := by
  sorry

lemma Nat.pos_mul_pos {n m: Nat} (h₁: n.IsPos) (h₂: m.IsPos) : (n * m).IsPos := by
  sorry

theorem Nat.mul_add (a b c: Nat) : a * (b + c) = a * b + a * c := by
  sorry

theorem Nat.add_mul (a b c: Nat) : (a + b)*c = a*c + b*c := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Nat.mul_assoc (a b c: Nat) : (a * b) * c = a * (b * c) := by
-- </TARGET_THEOREM>

end Chapter2
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter2

/-- Proposition 2.3.5 (Multiplication is associative) / Exercise 2.3.3
Compare with Mathlib's `Nat.mul_assoc` -/
theorem Nat.mul_assoc (a b c : Nat) : (a * b) * c = a * (b * c) := by
end Chapter2

end TaoBench
import Mathlib.Tactic

namespace Chapter2

/--
  Assumption 2.6 (Existence of natural numbers).  Here we use an explicit construction of the
  natural numbers (using an inductive type).  For a more axiomatic approach, see the epilogue to
  this chapter.
-/
inductive Nat where
| zero : Nat
| succ : Nat → Nat
deriving Repr, DecidableEq  -- this allows `decide` to work on `Nat`

/-- Axiom 2.1 (0 is a natural number) -/
instance Nat.instZero : Zero Nat := ⟨ zero ⟩

/-- Axiom 2.2 (Successor of a natural number is a natural number) -/
postfix:100 "++" => Nat.succ

/--
  Definition 2.1.3 (Definition of the numerals 0, 1, 2, etc.). Note: to avoid ambiguity, one may
  need to use explicit casts such as (0:Nat), (1:Nat), etc. to refer to this chapter's version of
  the natural numbers.
-/
instance Nat.instOfNat {n:_root_.Nat} : OfNat Nat n where
  ofNat := _root_.Nat.rec 0 (fun _ n ↦ n++) n

instance Nat.instOne : One Nat := ⟨ 1 ⟩

/--
  Recursion. Analogous to the inbuilt Mathlib method `Nat.rec` associated to
  the Mathlib natural numbers
-/
abbrev Nat.recurse (f: Nat → Nat → Nat) (c: Nat) : Nat → Nat := fun n ↦ match n with
| 0 => c
| n++ => f n (recurse f c n)

/-- Definition 2.2.1. (Addition of natural numbers).
    Compare with Mathlib's `Nat.add` -/
abbrev Nat.add (n m : Nat) : Nat := Nat.recurse (fun _ sum ↦ sum++) m n

/-- This instance allows for the `+` notation to be used for natural number addition. -/
instance Nat.instAdd : Add Nat where
  add := add

/-- Definition 2.3.1 (Multiplication of natural numbers) -/
abbrev Nat.mul (n m : Nat) : Nat := Nat.recurse (fun _ prod ↦ prod + m) 0 n

/-- This instance allows for the `*` notation to be used for natural number multiplication. -/
instance Nat.instMul : Mul Nat where
  mul := mul

/-- Definition 2.3.11 (Exponentiation for natural numbers) -/
abbrev Nat.pow (m n: Nat) : Nat := Nat.recurse (fun _ prod ↦ prod * m) 1 n

instance Nat.instPow : HomogeneousPow Nat where
  pow := Nat.pow

/-- Exercise 2.3.4-/
theorem Nat.sq_add_eq (a b: Nat) :
    (a + b) ^ (2 : Nat) = a ^ (2 : Nat) + 2 * a * b + b ^ (2 : Nat) := by
end Chapter2
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Nat.sq_add_eq (a b : Nat) :
    (a + b) ^ (2 : Nat) = a ^ (2 : Nat) + 2 * a * b + b ^ (2 : Nat) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Nat.strong_induction {m₀ : Nat} {P : Nat → Prop}
  (hind : ∀ m, m ≥ m₀ → (∀ m', m₀ ≤ m' ∧ m' < m → P m') → P m) :
  ∀ m, m ≥ m₀ → P m := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter2

/-- Proposition 2.2.13 (Trichotomy of order for natural numbers) / Exercise 2.2.4.
For any natural numbers `a` and `b`, exactly one of `a < b`, `a = b`, or `a > b` holds. -/
theorem Nat.trichotomous (a b : Nat) : a < b ∨ a = b ∨ a > b := by
end Chapter2

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

namespace TaoBench
theorem Function.comp_bijective
    {X Y Z : Type*} {f : X → Y} {g : Y → Z}
    (hf : Function.Bijective f) (hg : Function.Bijective g) :
    Function.Bijective (g ∘ f) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Function.comp_cancel_left {X Y Z : Type*} {f f' : X → Y} {g : Y → Z}
  (heq : g ∘ f = g ∘ f') (hg : Function.Injective g) : f = f' := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Function.comp_injective {X Y Z : Type*} {f : X → Y} {g : Y → Z}
    (hinj : Function.Injective (g ∘ f)) :
    Function.Injective f := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- If `f` and `g` are injective, then their composition `g ∘ f` is injective.
This is the Mathlib-style translation of Tao's Exercise 3.3.2
(`Function.comp_of_inj`). -/
theorem Function.comp_of_inj {α β γ : Type*} {f : α → β} {g : β → γ}
    (hf : Function.Injective f) (hg : Function.Injective g) :
    Function.Injective (g ∘ f) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Function.inv_of_comp
  {X Y Z : Type*} (f : X ≃ Y) (g : Y ≃ Z) :
    (f.trans g).symm = g.symm.trans f.symm := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Function.inverse_comp_self
  {α β : Type*} {f : α → β} (h : Function.Bijective f) (x : α) :
    (Equiv.ofBijective f h).symm (f x) = x := by
end TaoBench
import Mathlib.Tactic

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 nat)

variable [SetTheory]

/-- Definition 3.5.1 (Ordered pair).  One could also have used `Object × Object` to
define `OrderedPair` here. -/
@[ext]
structure OrderedPair where
  fst: Object
  snd: Object

/--
  Exercise 3.5.3. The spirit here is to avoid direct rewrites (which make all of these claims
  trivial), and instead use `OrderedPair.eq` or `SetTheory.Set.tuple_inj`
-/
theorem OrderedPair.refl (p: OrderedPair) : p = p := by
end Chapter3
import Mathlib.Tactic

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 nat)

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

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

/-- Definition 3.5.1 (Ordered pair).  One could also have used `Object × Object` to
define `OrderedPair` here. -/
@[ext]
structure OrderedPair where
  fst: Object
  snd: Object

/-- Exercise 3.5.1, first part -/
def OrderedPair.toObject : OrderedPair ↪ Object where
  toFun p := ({ (({p.fst}:Set):Object), (({p.fst, p.snd}:Set):Object) }:Set)
  inj' := by
end Chapter3
import Mathlib
import Analysis.Section_3_1
import Analysis.Tags.Section_3_1
import Analysis.Section_3_2
import Analysis.Tags.Section_3_2
import Analysis.Section_3_4
import Analysis.Tags.Section_3_4

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 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

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

open SetTheory.Set

@[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
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
def OrderedPair.toObject : OrderedPair ↪ Object := by
-- </TARGET_THEOREM>

end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

universe u
variable (α : Type u)

/-- Mathlib version of Tao's `OrderedPair.toObject`:
there is an embedding from `α × α` into the type of sets of sets of `α`,
corresponding to the Kuratowski ordered pair construction `{{a}, {a, b}}`. -/
def OrderedPair.toObject : (α × α) ↪ Set (Set α) := by
end TaoBench
import Mathlib.Tactic

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 nat)

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

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

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

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

/-- Definition 3.5.1 (Ordered pair).  One could also have used `Object × Object` to
define `OrderedPair` here. -/
@[ext]
structure OrderedPair where
  fst: Object
  snd: Object

/-- Exercise 3.5.1, second part (requires axiom of regularity) -/
abbrev OrderedPair.toObject' : OrderedPair ↪ Object where
  toFun p := ({ p.fst, (({p.fst, p.snd}:Set):Object) }:Set)
  inj' := by
end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter3

universe u

/-- In standard set-theoretic terms: for any type `α`, one can represent an
ordered pair `(a,b)` as a single set (for instance via the Kuratowski encoding
`{{a}, {a, b}}`), thus obtaining an embedding of `α × α` into `Set (Set α)`. -/
abbrev OrderedPair.toObject' (α : Type u) :
    α × α ↪ Set (Set α) := by
end Chapter3

end TaoBench
import Mathlib
import Analysis.Section_3_5
import Analysis.Tags.Section_3_5

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 nat)

variable [SetTheory]

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

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

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

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

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

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 := SetTheory.Set.nat_equiv n

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

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

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

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

abbrev SetTheory.Set.EqualCard (X Y:Set) : Prop := ∃ f : X → Y, Function.Bijective f
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem SetTheory.Set.Example_3_6_2 : EqualCard {0,1,2} {3,4,5} := by
-- </TARGET_THEOREM>

end Chapter3
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Example_3_6_2 :
    ∃ f : ({0, 1, 2} : Set ℕ) → ({3, 4, 5} : Set ℕ),
      Function.Bijective f := by
end TaoBench