introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry

theorem EReal.sup_of_bounded_nonempty {E: Set ℝ} (hbound: BddAbove E) (hnon: E.Nonempty) :
    sSup ((fun (x:ℝ) ↦ (x:EReal)) '' E) = sSup E := by
  sorry

theorem EReal.sup_of_unbounded_nonempty {E: Set ℝ} (hunbound: ¬ BddAbove E) (hnon: E.Nonempty) :
    sSup ((fun (x:ℝ) ↦ (x:EReal)) '' E) = ⊤ := by
  sorry

theorem EReal.sup_of_empty : sSup (∅:Set EReal) = ⊥ := by
  sorry

theorem EReal.sup_of_infty_mem {E: Set EReal} (hE: ⊤ ∈ E) : sSup E = ⊤ := by
  sorry

theorem EReal.sup_of_neg_infty_mem {E: Set EReal} : sSup E = sSup (E \ {⊥}) := by
  sorry

theorem EReal.inf_eq_neg_sup (E: Set EReal) : sInf E = - sSup (-E) := by
  sorry

abbrev Example_6_2_7 : Set EReal := { x | ∃ n:ℕ, x = -((n+1):EReal)} ∪ {⊥}

abbrev Example_6_2_8 : Set EReal := { x | ∃ n:ℕ, x = (1 - (10:ℝ)^(-(n:ℤ)-1):Real)}

abbrev Example_6_2_9 : Set EReal := { x | ∃ n:ℕ, x = n+1}
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.mem_ge_inf (E: Set EReal) {x:EReal} (hx: x ∈ E) : sInf E ≤ x := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Theorem 6.2.11 (a) / Exercise 6.2.2 -/
theorem EReal.mem_ge_inf (E : Set EReal) {x : EReal} (hx : x ∈ E) :
    sInf E ≤ x := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry

theorem EReal.sup_of_bounded_nonempty {E: Set ℝ} (hbound: BddAbove E) (hnon: E.Nonempty) :
    sSup ((fun (x:ℝ) ↦ (x:EReal)) '' E) = sSup E := by
  sorry

theorem EReal.sup_of_unbounded_nonempty {E: Set ℝ} (hunbound: ¬ BddAbove E) (hnon: E.Nonempty) :
    sSup ((fun (x:ℝ) ↦ (x:EReal)) '' E) = ⊤ := by
  sorry

theorem EReal.sup_of_empty : sSup (∅:Set EReal) = ⊥ := by
  sorry

theorem EReal.sup_of_infty_mem {E: Set EReal} (hE: ⊤ ∈ E) : sSup E = ⊤ := by
  sorry

theorem EReal.sup_of_neg_infty_mem {E: Set EReal} : sSup E = sSup (E \ {⊥}) := by
  sorry

theorem EReal.inf_eq_neg_sup (E: Set EReal) : sInf E = - sSup (-E) := by
  sorry

abbrev Example_6_2_7 : Set EReal := { x | ∃ n:ℕ, x = -((n+1):EReal)} ∪ {⊥}

abbrev Example_6_2_8 : Set EReal := { x | ∃ n:ℕ, x = (1 - (10:ℝ)^(-(n:ℤ)-1):Real)}

abbrev Example_6_2_9 : Set EReal := { x | ∃ n:ℕ, x = n+1}
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem EReal.mem_le_sup (E: Set EReal) {x:EReal} (hx: x ∈ E) : x ≤ sSup E := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Theorem 6.2.11 (a) / Exercise 6.2.2 -/
theorem EReal.mem_le_sup (E : Set EReal) {x : EReal} (hx : x ∈ E) : x ≤ sSup E := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.neg_of_lt {x y:EReal} (hxy : x ≤ y): -y ≤ -x := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 6.2.5(d) / Exercise 6.2.1.
If `x ≤ y` in the extended reals `EReal`, then `-y ≤ -x`. -/
theorem EReal.neg_of_lt {x y : EReal} (hxy : x ≤ y) : -y ≤ -x := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.not_gt_and_eq (x y:EReal) : ¬ (x > y ∧ x = y) := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 6.2.5(b) / Exercise 6.2.1 -/
theorem EReal.not_gt_and_eq (x y : EReal) : ¬ (x > y ∧ x = y) := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

open EReal

-- <CONTEXT>
theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.not_lt_and_eq (x y:EReal) : ¬ (x < y ∧ x = y) := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 6.2.5(b) / Exercise 6.2.1.
Mathematically: in the extended real numbers `EReal`, one cannot have
both `x < y` and `x = y` at the same time. -/
theorem EReal.not_lt_and_eq (x y : EReal) : ¬ (x < y ∧ x = y) := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.not_lt_and_gt (x y:EReal) : ¬ (x < y ∧ x > y) := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 6.2.5(b) / Exercise 6.2.1 -/
theorem EReal.not_lt_and_gt (x y : EReal) : ¬ (x < y ∧ x > y) := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem EReal.refl (x:EReal) : x ≤ x := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
namespace Chapter6

/-- Proposition 6.2.5(a) / Exercise 6.2.1: reflexivity of `≤` on `EReal`
(the extended real numbers). -/
theorem EReal.refl (x : EReal) : x ≤ x := by
end Chapter6
end TaoBench
import Mathlib.Data.Real.EReal
import Mathlib.Order.Bounds
import Mathlib.Order.Suprema

/-- Theorem 6.2.11 (b) / Exercise 6.2.2 -/
theorem EReal.sup_le_upper (E: Set EReal) {M:EReal} (hM: M ∈ upperBounds E) : sSup E ≤ M := by
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry

theorem EReal.sup_of_bounded_nonempty {E: Set ℝ} (hbound: BddAbove E) (hnon: E.Nonempty) :
    sSup ((fun (x:ℝ) ↦ (x:EReal)) '' E) = sSup E := by
  sorry

theorem EReal.sup_of_unbounded_nonempty {E: Set ℝ} (hunbound: ¬ BddAbove E) (hnon: E.Nonempty) :
    sSup ((fun (x:ℝ) ↦ (x:EReal)) '' E) = ⊤ := by
  sorry

theorem EReal.sup_of_empty : sSup (∅:Set EReal) = ⊥ := by
  sorry

theorem EReal.sup_of_infty_mem {E: Set EReal} (hE: ⊤ ∈ E) : sSup E = ⊤ := by
  sorry

theorem EReal.sup_of_neg_infty_mem {E: Set EReal} : sSup E = sSup (E \ {⊥}) := by
  sorry

theorem EReal.inf_eq_neg_sup (E: Set EReal) : sInf E = - sSup (-E) := by
  sorry

abbrev Example_6_2_7 : Set EReal := { x | ∃ n:ℕ, x = -((n+1):EReal)} ∪ {⊥}

abbrev Example_6_2_8 : Set EReal := { x | ∃ n:ℕ, x = (1 - (10:ℝ)^(-(n:ℤ)-1):Real)}

abbrev Example_6_2_9 : Set EReal := { x | ∃ n:ℕ, x = n+1}
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem EReal.sup_le_upper (E: Set EReal) {M:EReal} (hM: M ∈ upperBounds E) : sSup E ≤ M := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Theorem 6.2.11 (b) / Exercise 6.2.2 -/
theorem EReal.sup_le_upper (E : Set EReal) {M : EReal}
    (hM : M ∈ upperBounds E) : sSup E ≤ M := by
end TaoBench
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

open EReal

-- <CONTEXT>
theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.trans {x y z:EReal} (hxy : x ≤ y) (hyz: y ≤ z) : x ≤ z := by
-- </TARGET_THEOREM>
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5
import Analysis.Section_5_epilogue
import Analysis.Tags.Section_5_epilogue

-- <CONTEXT>
open EReal

theorem EReal.def (x:EReal) : (∃ (y:Real), y = x) ∨ x = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.real_neq_infty (x:ℝ) : (x:EReal) ≠ ⊤ := by
  sorry

theorem EReal.real_neq_neg_infty (x:ℝ) : (x:EReal) ≠ ⊥ := by
  sorry

theorem EReal.infty_neq_neg_infty : (⊤:EReal) ≠ (⊥:EReal) := by
  sorry

abbrev EReal.IsFinite (x:EReal) : Prop := ∃ (y:Real), y = x

abbrev EReal.IsInfinite (x:EReal) : Prop := x = ⊤ ∨ x = ⊥

theorem EReal.infinite_iff_not_finite (x:EReal): x.IsInfinite ↔ ¬ x.IsFinite := by
  sorry

theorem EReal.neg_of_real (x:Real) : -(x:EReal) = (-x:ℝ) := by
  sorry

theorem EReal.le_iff (x y:EReal) :
    x ≤ y ↔ (∃ (x' y':Real), x = x' ∧ y = y' ∧ x' ≤ y') ∨ y = ⊤ ∨ x = ⊥ := by
  sorry

theorem EReal.lt_iff (x y:EReal) : x < y ↔ x ≤ y ∧ x ≠ y := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Chapter6.EReal.trichotomy (x y:EReal) : x < y ∨ x = y ∨ x > y := by
-- </TARGET_THEOREM>
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Proposition 6.2.5(b) / Exercise 6.2.1 -/
theorem EReal.trichotomy (x y : EReal) :
    x < y ∨ x = y ∨ x > y := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.EReal

namespace Chapter6

/--
  Definition 6.1.3 (Sequence). This is similar to the Chapter 5 sequence, except that now the
  sequence is real-valued. As with Chapter 5, we start sequences from 0 by default.
-/
@[ext]
structure Sequence where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

/-- Sequences can be thought of as functions from ℤ to ℝ. -/
instance Sequence.instCoeFun : CoeFun Sequence (fun _ ↦ ℤ → ℝ) where
  coe a := a.seq

@[coe]
abbrev Sequence.ofNatFun (a:ℕ → ℝ) : Sequence :=
 {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by simp_all
 }

/-- Functions from ℕ to ℝ can be thought of as sequences. -/
instance Sequence.instCoe : Coe (ℕ → ℝ) Sequence where
  coe := ofNatFun

/-- Definition 6.3.1 -/
noncomputable abbrev Sequence.sup (a:Sequence) : EReal := sSup { x | ∃ n ≥ a.m, x = a n }

/-- Example 6.3.4 / Exercise 6.3.1 -/
theorem Exercise_6_3_1a : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).sup = 1 := by
end Chapter6
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_6_3_1a :
    sSup (Set.range (fun n : ℕ => (↑(1 / ((n : ℝ) + 1)) : EReal))) = (1 : EReal) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Instances.EReal

namespace Chapter6

/--
  Definition 6.1.3 (Sequence). This is similar to the Chapter 5 sequence, except that now the
  sequence is real-valued. As with Chapter 5, we start sequences from 0 by default.
-/
@[ext]
structure Sequence where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

/-- Sequences can be thought of as functions from ℤ to ℝ. -/
instance Sequence.instCoeFun : CoeFun Sequence (fun _ ↦ ℤ → ℝ) where
  coe a := a.seq

@[coe]
abbrev Sequence.ofNatFun (a:ℕ → ℝ) : Sequence :=
 {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by simp_all
 }

/-- Functions from ℕ to ℝ can be thought of as sequences. -/
instance Sequence.instCoe : Coe (ℕ → ℝ) Sequence where
  coe := ofNatFun

@[simp]
lemma Sequence.eval_coe (n:ℕ) (a: ℕ → ℝ) : (a:Sequence) n = a n := by simp

/-- Definition 6.3.1 -/
noncomputable abbrev Sequence.inf (a:Sequence) : EReal := sInf { x | ∃ n ≥ a.m, x = a n }

/-- Example 6.3.4 / Exercise 6.3.1 -/
theorem Exercise_6_3_1b : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).inf = 0 := by
end Chapter6
import Mathlib
import Analysis.Section_6_1
import Analysis.Tags.Section_6_1
import Analysis.Section_6_2
import Analysis.Tags.Section_6_2

namespace Chapter6

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

instance Sequence.instCoeFun : CoeFun Sequence (fun _ ↦ ℤ → ℝ) where
  coe a := a.seq

@[coe]
abbrev Sequence.ofNatFun (a:ℕ → ℝ) : Sequence :=
 {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by simp_all
 }

instance Sequence.instCoe : Coe (ℕ → ℝ) Sequence where
  coe := ofNatFun

@[simp]
lemma Sequence.eval_coe (n:ℕ) (a: ℕ → ℝ) : (a:Sequence) n = a n := by
  sorry

noncomputable abbrev Sequence.sup (a:Sequence) : EReal := sSup { x | ∃ n ≥ a.m, x = a n }

noncomputable abbrev Sequence.inf (a:Sequence) : EReal := sInf { x | ∃ n ≥ a.m, x = a n }
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Exercise_6_3_1b : ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)):Sequence).inf = 0 := by
-- </TARGET_THEOREM>

end Chapter6
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Exercise_6_3_1b :
    sInf { x : EReal | ∃ n : ℕ, x = (1 / ((n : ℝ) + 1) : ℝ) } = 0 := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic

/-!
Self-contained extraction for the target theorem in Section 6.7.
We reproduce only the minimal Chapter 6 (real-valued) sequence framework
and the Chapter6.Real.rpow definition needed for the target statement.
-/

/- Definition 6.1.1 (Distance). Using Mathlib distance. -/
abbrev Real.Close (ε x y : ℝ) : Prop := dist x y ≤ ε

namespace Chapter6

/-- Definition 6.1.3 (Sequence). -/
@[ext]
structure Sequence where
  m : ℤ
  seq : ℤ → ℝ
  vanish : ∀ n < m, seq n = 0

/-- Sequences can be thought of as functions from ℤ to ℝ. -/
instance Sequence.instCoeFun : CoeFun Sequence (fun _ ↦ ℤ → ℝ) where
  coe a := a.seq

@[coe]
abbrev Sequence.ofNatFun (a:ℕ → ℝ) : Sequence :=
 {
    m := 0
    seq n := if n ≥ 0 then a n.toNat else 0
    vanish := by simp_all
 }

/-- Functions from ℕ to ℝ can be thought of as sequences. -/
instance Sequence.instCoe : Coe (ℕ → ℝ) Sequence where
  coe := ofNatFun

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

/--
  a.from n₁ starts `a:Sequence` from `n₁`.  It is intended for use when `n₁ ≥ n₀`, but returns
  the "junk" value of the original sequence `a` otherwise.
-/
abbrev Sequence.from (a:Sequence) (m₁:ℤ) : Sequence := mk' (max a.m m₁) (a ↑·)

end Chapter6

/-- Definition 6.1.5 -/
abbrev Real.CloseSeq (ε: ℝ) (a: Chapter6.Sequence) (L:ℝ) : Prop := ∀ n ≥ a.m, ε.Close (a n) L

/-- Definition 6.1.5 -/
abbrev Real.EventuallyClose (ε: ℝ) (a: Chapter6.Sequence) (L:ℝ) : Prop :=
  ∃ N ≥ a.m, ε.CloseSeq (a.from N) L

namespace Chapter6

abbrev Sequence.TendsTo (a:Sequence) (L:ℝ) : Prop :=
  ∀ ε > (0:ℝ), ε.EventuallyClose a L

/-- Definition 6.1.8 -/
abbrev Sequence.Convergent (a:Sequence) : Prop := ∃ L, a.TendsTo L

open Classical in
/--
  Definition 6.1.8.  We give the limit of a sequence the junk value of 0 if it is not convergent.
-/
noncomputable abbrev lim (a:Sequence) : ℝ := if h: a.Convergent then h.choose else 0

namespace Real

/-- Chapter 6: existence of a rational sequence converging to a given real. -/
-- Proof elided to keep the slice dependency-minimal; the statement matches the source.
 theorem eq_lim_of_rat (α : ℝ) : ∃ q : ℕ → ℚ, ((fun n ↦ (q n : ℝ)) : Sequence).TendsTo α := by
  sorry

/-- Chapter 6 definition of real exponentiation via limits of rational powers. -/
noncomputable abbrev rpow (x : ℝ) (α : ℝ) : ℝ :=
  lim ((fun n ↦ x ^ ((eq_lim_of_rat α).choose n : ℝ)) : Sequence)

/-- Proposition 6.7.3(d) / Exercise 6.7.1 -/
 theorem ratPow_mono {x y : ℝ} (hx : x > 0) (hy : y > 0) {q : ℝ} (h : q > 0) :
    x > y ↔ rpow x q > rpow y q := by
end Real

end Chapter6