introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Basic
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Finset.Interval
import Mathlib.Data.Int.Basic

-- Lightweight stand-in for the project-specific `grind` tactic used in source proofs.
macro "grind" : tactic => `(tactic| intros <;> simp [*])

namespace Chapter7

@[ext]
structure Series where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

/-- Functions from ℕ to ℝ can be thought of as series. -/
instance Series.instCoe : Coe (ℕ → ℝ) Series where
  coe := fun a ↦ {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by grind
  }

/-- Definition 7.2.2 (Convergence of series) -/
abbrev Series.partial (s : Series) (N : ℤ) : ℝ :=
  ∑ n ∈ Finset.Icc s.m N, s.seq n

abbrev Series.convergesTo (s : Series) (L : ℝ) : Prop :=
  Filter.atTop.Tendsto (s.partial) (nhds L)

abbrev Series.converges (s : Series) : Prop := ∃ L, s.convergesTo L

/-- Example 7.4.2 -/
theorem Series.zeta_2_converges :
  (fun n : ℕ ↦ 1 / (n + 1 : ℝ) ^ 2 : Series).converges := by
end Chapter7
import Mathlib
import Analysis.Section_7_3
import Analysis.Tags.Section_7_3

namespace Chapter7

-- <CONTEXT>
@[ext]
structure Series where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

instance Series.instCoe : Coe (ℕ → ℝ) Series where
  coe := fun a ↦ {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by grind
  }

abbrev Series.partial (s : Series) (N : ℤ) : ℝ :=
  ∑ n ∈ Finset.Icc s.m N, s.seq n

abbrev Series.convergesTo (s : Series) (L : ℝ) : Prop :=
  Filter.atTop.Tendsto (s.partial) (nhds L)

abbrev Series.converges (s : Series) : Prop := ∃ L, s.convergesTo L

abbrev Series.nonneg (s : Series) : Prop := ∀ n, s.seq n ≥ 0

open Classical in
noncomputable abbrev Series.sum (s : Series) : ℝ := if h : s.converges then h.choose else 0

theorem Series.sum_eq_sum (b:ℕ → ℝ) {N:ℤ} (hN: N ≥ 0) : ∑ n ∈ .Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ .Iic N.toNat, b n := by
  sorry

theorem Series.converges_of_permute_nonneg {a:ℕ → ℝ} (ha: (a:Series).nonneg) (hconv: (a:Series).converges)
  {f: ℕ → ℕ} (hf: Function.Bijective f) :
    (fun n ↦ a (f n) : Series).converges ∧ (a:Series).sum = (fun n ↦ a (f n) : Series).sum := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Series.zeta_2_converges : (fun n:ℕ ↦ 1/(n+1:ℝ)^2 : Series).converges := by
-- </TARGET_THEOREM>

end Chapter7
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter7

namespace Series

/-- Example 7.4.2: the series `∑ n, 1 / (n + 1)^2` converges. -/
theorem zeta_2_converges :
  Summable (fun n : ℕ ↦ 1 / (n + 1 : ℝ) ^ 2) := by
end Series

end Chapter7

end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Basic

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

open BigOperators

abbrev AbsConvergent' {X:Type} (f: X → ℝ) : Prop := BddAbove ( (fun A ↦ ∑ x ∈ A, |f x|) '' .univ )

/-- Lemma 8.2.5 / Exercise 8.2.2-/
theorem AbsConvergent'.countable_supp {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) :
  AtMostCountable { x | f x ≠ 0 } := by
end Chapter8
import Mathlib
import Analysis.Section_7_2
import Analysis.Tags.Section_7_2
import Analysis.Section_7_3
import Analysis.Tags.Section_7_3
import Analysis.Section_7_4
import Analysis.Tags.Section_7_4
import Analysis.Section_8_1
import Analysis.Tags.Section_8_1

namespace Chapter8

-- <CONTEXT>
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

abbrev AbsConvergent' {X:Type} (f: X → ℝ) : Prop := BddAbove ( (fun A ↦ ∑ x ∈ A, |f x|) '' .univ )

theorem AbsConvergent'.of_finite {X:Type} [Finite X] (f:X → ℝ) : AbsConvergent' f := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem AbsConvergent'.countable_supp {X:Type} {f:X → ℝ} (hf: AbsConvergent' f) :
  AtMostCountable { x | f x ≠ 0 } := by
-- </TARGET_THEOREM>

end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Mathlib translation of Tao's `AbsConvergent'.countable_supp`.
If the family of finite subsums of `|f x|` is bounded above, then the type
of points where `f` is nonzero is at most countable. -/
theorem AbsConvergent'.countable_supp {X : Type _} {f : X → ℝ}
    (hf :
      BddAbove
        ((fun A : Finset X => ∑ x ∈ A, |f x|) ''
          (Set.univ : Set (Finset X)))) :
    Countable {x : X // f x ≠ 0} := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Int.Interval
import Mathlib.Topology.Instances.Real
import Mathlib.Data.Set.Basic
import Mathlib.Algebra.BigOperators.Basic

/-!
Self-contained slice for Chapter 8.2 target theorem.
-/

namespace Chapter7

open BigOperators

/--
  Definition 7.2.1 (Formal infinite series). This is similar to Chapter 6 sequence, but is
  manipulated differently. As with Chapter 5, we will start series from 0 by default.
-/
@[ext]
structure Series where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

/-- Functions from ℕ to ℝ can be thought of as series. -/
instance Series.instCoe : Coe (ℕ → ℝ) Series where
  coe := fun a ↦ {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by grind
  }

@[simp]
theorem Series.eval_coe (a: ℕ → ℝ) (n: ℕ) : (a: Series).seq n = a n := by simp

abbrev Series.mk' {m:ℤ} (a: { n // n ≥ m } → ℝ) : Series where
  m := m
  seq n := if h : n ≥ m then a ⟨n, h⟩ else 0
  vanish := by grind

/-- Definition 7.2.2 (Convergence of series) -/
abbrev Series.partial (s : Series) (N:ℤ) : ℝ := ∑ n ∈ Finset.Icc s.m N, s.seq n

abbrev Series.convergesTo (s : Series) (L:ℝ) : Prop := Filter.atTop.Tendsto (s.partial) (nhds L)

abbrev Series.converges (s : Series) : Prop := ∃ L, s.convergesTo L

/-- Definition 7.2.8 (Absolute convergence) -/
abbrev Series.abs (s:Series) : Series := mk' (m:=s.m) (fun n ↦ |s.seq n|)

abbrev Series.absConverges (s:Series) : Prop := s.abs.converges

end Chapter7

namespace Chapter8

open Chapter7 Chapter7.Series BigOperators Function

/-- Definition 8.2.1 (Series on countable sets).  Note that with this definition, functions defined
on finite sets will not be absolutely convergent; one should use `AbsConvergent'` instead for such
cases.-/
abbrev AbsConvergent {X:Type} (f: X → ℝ) : Prop := ∃ g: ℕ → X, Bijective g ∧ (f ∘ g: Chapter7.Series).absConverges

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

/-- Lemma 8.2.3 / Exercise 8.2.1 -/
 theorem AbsConvergent.iff {X:Type} (hX:CountablyInfinite X) (f : X → ℝ) :
  AbsConvergent f ↔ BddAbove ( (fun A ↦ ∑ x ∈ A, |f x|) '' (.univ : Set (Finset X)) ) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem AbsConvergent.iff {X : Type _} (hX : Nonempty (X ≃ ℕ)) (f : X → ℝ) :
    (∃ g : ℕ → X, Function.Bijective g ∧ Summable (fun n : ℕ => |f (g n)|)) ↔
      BddAbove
        ((fun A : Finset X => ∑ x ∈ A, |f x|) '' (Set.univ : Set (Finset X))) := by
end TaoBench
import Mathlib.Data.Set.Basic
import Mathlib.Tactic

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

/-- Corollary 8.1.9 / Exercise 8.1.5 -/
theorem AtMostCountable.image {X:Type} (hX: CountablyInfinite X) {Y: Type} (f: X → Y) : AtMostCountable (f '' .univ) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Corollary 8.1.9 / Exercise 8.1.5 (Mathlib formulation).

If a type `X` is countably infinite (i.e. there exists a bijection `X → ℕ`),
then for any function `g : X → Y`, the image `g '' univ` is at most countable,
in the sense that it is either in bijection with `ℕ` or finite. -/
theorem AtMostCountable.image
    {X : Type} (hX : ∃ f : X → ℕ, Function.Bijective f)
    {Y : Type} (g : X → Y) :
    (∃ f : (g '' (Set.univ : Set X)) → ℕ, Function.Bijective f) ∨
      Finite (g '' (Set.univ : Set X)) := by
end TaoBench
import Mathlib.Data.Set.Basic

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

/-- Proposition 8.1.8 / Exercise 8.1.4 -/
theorem AtMostCountable.image_nat (Y: Type) (f: ℕ → Y) : AtMostCountable (f '' .univ) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Chapter8.AtMostCountable.image_nat (Y : Type) (f : ℕ → Y) :
    Set.Countable (f '' (Set.univ : Set ℕ)) := by
end TaoBench
import Mathlib.Tactic

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

/-- Corollary 8.1.14 / Exercise 8.1.8 -/
theorem CountablyInfinite.prod {X Y:Type} (hX: CountablyInfinite X) (hY: CountablyInfinite Y) :
  CountablyInfinite (X × Y) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem CountablyInfinite.prod {X Y : Type}
    (hX : ∃ f : X → ℕ, Function.Bijective f)
    (hY : ∃ f : Y → ℕ, Function.Bijective f) :
    ∃ f : X × Y → ℕ, Function.Bijective f := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Set.Lattice

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

/-- Proposition 8.1.10 / Exercise 8.1.7 -/
theorem CountablyInfinite.union {A:Type} {X Y: Set A} (hX: CountablyInfinite X) (hY: CountablyInfinite Y) :
  CountablyInfinite (X ∪ Y: Set A) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem CountablyInfinite.union
    {A : Type _} {X Y : Set A}
    (hX : ∃ f : {x : A // x ∈ X} → ℕ, Function.Bijective f)
    (hY : ∃ f : {y : A // y ∈ Y} → ℕ, Function.Bijective f) :
    ∃ f : {z : A // z ∈ X ∪ Y} → ℕ, Function.Bijective f := by
end TaoBench
import Mathlib.Tactic

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

end Chapter8

open Chapter8

/-- Exercise 8.1.1 -/
theorem Exercise_8_1_1 (X: Type) : Infinite X ↔ ∃ Y : Set X, Y ≠ .univ ∧ EqualCard Y X := by
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Exercise 8.1.1 (Mathlib version):  
A type `X` is infinite iff it has a proper subset with the same cardinality. -/
theorem Exercise_8_1_1 (X : Type) :
    Infinite X ↔ ∃ Y : Set X, Y ≠ Set.univ ∧ ∃ f : Y → X, Function.Bijective f := by
end TaoBench
import Mathlib.Order.WellFounded
import Mathlib.Data.Real.Basic

/-- Exercise 8.1.2 -/
theorem Exercise_8_1_2a : ¬ WellFoundedLT ℤ := by sorry
theorem Exercise_8_1__2b : ¬ WellFoundedLT ℚ := by sorry
theorem Exercise_8_1_2c : ¬ WellFoundedLT ℝ := by
import Mathlib.Tactic

/-!
# Analysis I, Section 8.1: Countability

This snippet extracts just enough from the chapter to typecheck the target theorem.
-/

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

/-- Exercise 8.1.6 -/
theorem Exercise_8_1_6 (A: Type) : AtMostCountable A ↔ ∃ f : A → ℕ, Function.Injective f := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_8_1_6 (A : Type) :
    ((∃ f : A → ℕ, Function.Bijective f) ∨ Finite A) ↔
    ∃ f : A → ℕ, Function.Injective f := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Fintype.Basic

/-!
# Analysis I, Section 8.1: Countability (extracted snippet)

This file extracts the minimal context needed to state and typecheck Exercise 8.1.9.
-/

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

/-- Exercise 8.1.9 -/
theorem Exercise_8_1_9 {I X:Type} (hI: AtMostCountable I) (A: I → Set X) (hA: ∀ i, AtMostCountable (A i)) :
  AtMostCountable (⋃ i, A i) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_8_1_9 {I X : Type}
    (hI : Countable I) (A : I → Set X) (hA : ∀ i, (A i).Countable) :
    (⋃ i, A i).Countable := by
end TaoBench
import Mathlib
import Analysis.Section_8_1
import Analysis.Tags.Section_8_1
import Analysis.Section_8_2
import Analysis.Tags.Section_8_2

namespace Chapter8

-- <CONTEXT>
theorem Uncountable.power_set_nat : Uncountable (Set ℕ) := by
  sorry

open Real in
theorem Uncountable.real : Uncountable ℝ := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Exercise_8_3_1 {X:Type} [Finite X] : Nat.card (Set X) = 2 ^ Nat.card X := by
-- </TARGET_THEOREM>

end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Exercise 8.3.1: For a finite type `X`, the type `Set X` of all subsets of `X`
has `2 ^ Nat.card X` elements. -/
theorem Exercise_8_3_1 {X : Type} [Fintype X] :
  Nat.card (Set X) = 2 ^ Nat.card X := by
end TaoBench
namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev LeCard (X Y: Type) : Prop := ∃ f: X → Y, Function.Injective f

abbrev LtCard (X Y: Type) : Prop := LeCard X Y ∧ ¬ EqualCard X Y

end Chapter8

open Chapter8

/-- Exercise 8.3.4 -/
theorem Exercise_8_3_4a {X:Type} : LtCard X (Set X) := by
import Mathlib
import Analysis.Section_8_1
import Analysis.Tags.Section_8_1
import Analysis.Section_8_2
import Analysis.Tags.Section_8_2

namespace Chapter8

-- <CONTEXT>
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

theorem EqualCard.power_set_false (X:Type) : ¬ EqualCard X (Set X) := by
  sorry

theorem Uncountable.iff (X:Type) : Uncountable X ↔ ¬ AtMostCountable X := by
  sorry

theorem Uncountable.equiv {X Y: Type} (hXY : EqualCard X Y) :
  Uncountable X ↔ Uncountable Y := by
  sorry

theorem Uncountable.power_set_nat : Uncountable (Set ℕ) := by
  sorry

open Real in
theorem Uncountable.real : Uncountable ℝ := by
  sorry

abbrev LeCard (X Y: Type) : Prop := ∃ f: X → Y, Function.Injective f

abbrev LtCard (X Y: Type) : Prop := LeCard X Y ∧ ¬ EqualCard X Y

abbrev CardOrder : Preorder Type := {
  le := LeCard
  lt := LtCard
  le_refl := by
    sorry
  le_trans := by
    sorry
  lt_iff_le_not_ge := by
    sorry
}
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Exercise_8_3_4a {X:Type} : LtCard X (Set X) := by
-- </TARGET_THEOREM>

end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_8_3_4a {X : Type} :
    (∃ f : X → Set X, Function.Injective f) ∧
    ¬ (∃ f : X → Set X, Function.Bijective f)
:= by
end TaoBench
-- Minimal imports; `Set` and basic logic are available from Lean core

namespace Chapter8

/-- The definition of equal cardinality. For simplicity we restrict attention to the Type 0 universe.
This is analogous to `Chapter3.SetTheory.Set.EqualCard`, but we are not using the latter since
the Chapter 3 set theory is deprecated. -/
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

end Chapter8

open Chapter8

/-- Exercise 8.3.5 -/
theorem Exercise_8_3_5 (X:Type) : ¬ CountablyInfinite (Set X) := by
import Mathlib
import Analysis.Section_8_1
import Analysis.Tags.Section_8_1
import Analysis.Section_8_2
import Analysis.Tags.Section_8_2

namespace Chapter8

-- <CONTEXT>
abbrev EqualCard (X Y : Type) : Prop := ∃ f : X → Y, Function.Bijective f

abbrev CountablyInfinite (X : Type) : Prop := EqualCard X ℕ

abbrev AtMostCountable (X : Type) : Prop := CountablyInfinite X ∨ Finite X

theorem EqualCard.power_set_false (X:Type) : ¬ EqualCard X (Set X) := by
  sorry

theorem Uncountable.iff (X:Type) : Uncountable X ↔ ¬ AtMostCountable X := by
  sorry

theorem Uncountable.equiv {X Y: Type} (hXY : EqualCard X Y) :
  Uncountable X ↔ Uncountable Y := by
    sorry

theorem Uncountable.power_set_nat : Uncountable (Set ℕ) := by
  sorry

open Real in
theorem Uncountable.real : Uncountable ℝ := by
  sorry

abbrev LeCard (X Y: Type) : Prop := ∃ f: X → Y, Function.Injective f

abbrev LtCard (X Y: Type) : Prop := LeCard X Y ∧ ¬ EqualCard X Y

abbrev CardOrder : Preorder Type := {
  le := LeCard
  lt := LtCard
  le_refl := by
    sorry
  le_trans := by
    sorry
  lt_iff_le_not_ge := by
    sorry
}
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Exercise_8_3_5 (X:Type) : ¬ CountablyInfinite (Set X) := by
-- </TARGET_THEOREM>

end Chapter8
import Mathlib.Tactic
import Mathlib.Data.Set.Lattice
import Mathlib.Order.WellFounded
import Mathlib.Order.Basic

namespace Chapter8

/-- Definition 8.5.3.  Here we just review the Mathlib `LinearOrder` class. -/
-- We include the definition of `IsTotal` used throughout Section 8.5.
def IsTotal (X:Type) [PartialOrder X] : Prop := ∀ x y:X, x ≤ y ∨ y ≤ x

/-- Exercise 8.5.11 -/
theorem Exercise_8_5_11 {X:Type} [PartialOrder X] {Y Y':Set X}
  (hY: IsTotal Y) (hY': IsTotal Y') (hY_well: WellFoundedLT Y)
  (hY'_well: WellFoundedLT Y') (hYY': IsTotal (Y ∪ Y': Set X)) :
  WellFoundedLT (Y ∪ Y': Set X) := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_8_5_11
  {X : Type*} [PartialOrder X] {Y Y' : Set X}
  (hY : IsTotal {x : X // x ∈ Y} (· ≤ ·))
  (hY' : IsTotal {x : X // x ∈ Y'} (· ≤ ·))
  (hY_well : WellFoundedLT {x : X // x ∈ Y})
  (hY'_well : WellFoundedLT {x : X // x ∈ Y'})
  (hYY' : IsTotal {x : X // x ∈ (Y ∪ Y')} (· ≤ ·)) :
  WellFoundedLT {x : X // x ∈ (Y ∪ Y')} := by
end TaoBench
theorem Exercise_8_5_2a :
    ∃ (X : Type) (h₀ : LE X),
      (∀ x : X, x ≤ x) ∧
      (∀ x y : X, x ≤ y → y ≤ x → x = y) ∧
      ¬ (∀ x y z : X, x ≤ y → y ≤ z → x ≤ z) := by
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_8_5_2a :
    ∃ (X : Type) (r : X → X → Prop),
      (∀ x : X, r x x) ∧
      (∀ x y : X, r x y → r y x → x = y) ∧
      ¬ (∀ x y z : X, r x y → r y z → r x z) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Lattice
import Mathlib.Order.Bounds

namespace Chapter8

/-- Exercise 8.5.4 -/
theorem Exercise_8_5_4 : ¬ ∃ x : {x:ℝ| x > 0}, IsMin x := by
end Chapter8
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_8_5_4 :
    ¬ ∃ x : ℝ, 0 < x ∧ ∀ y : ℝ, 0 < y → x ≤ y := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Exercise 8.5.5 -/
theorem Exercise_8_5_5 {X Y : Type} [PartialOrder Y] (f : X → Y) :
  ∃ h₀ : PartialOrder X, h₀.le = (fun x y => f x < f y ∨ x = y) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Exercise 8.5.7 -/
theorem Exercise_8_5_7a {Y : Type} [PartialOrder Y] {x y : Y}
    (hx : IsBot x) (hy : IsBot y) : x = y := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Order.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.SetTheory.Cardinal.Basic

/-- Exercise 8.5.9 -/
theorem Exercise_8_5_9
  {X : Type}
  [LinearOrder X]
  (hmin : ∀ Y : Set X, Y.Nonempty → ∃ x : Y, IsMin x)
  (hmax : ∀ Y : Set X, Y.Nonempty → ∃ x : Y, IsMax x)
  : Finite X :=
by
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Exercise 8.5.9 (Tao, Analysis I).

If a linearly ordered type `X` has the property that every nonempty subset of `X`
has both a least and a greatest element (with respect to `≤`), then `X` is finite. -/
theorem Exercise_8_5_9
  {X : Type}
  [LinearOrder X]
  (hmin : ∀ Y : Set X, Y.Nonempty → ∃ x : X, IsLeast Y x)
  (hmax : ∀ Y : Set X, Y.Nonempty → ∃ x : X, IsGreatest Y x) :
  Finite X :=
by
end TaoBench