introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib
import Analysis.Section_6_4
import Analysis.Tags.Section_6_4

namespace Chapter6

-- <CONTEXT>

abbrev _root_.Real.Close (ε x y : ℝ) : Prop := dist x y ≤ ε

@[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

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

abbrev Sequence.from (a:Sequence) (m₁:ℤ) : Sequence := mk' (max a.m m₁) (a ↑·)

instance Sequence.inst_mul : Mul Sequence where
  mul a b := {
    m := min a.m b.m
    seq n := a n * b n
    vanish n hn := by simp [a.vanish n (by grind), b.vanish n (by grind)]
  }

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

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

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

theorem Sequence.lim_of_const (c:ℝ) :  ((fun (_:ℕ) ↦ c):Sequence).TendsTo c := by
  sorry

instance Sequence.inst_pow: Pow Sequence ℕ where
  pow a k := {
    m := a.m
    seq n := if n ≥ a.m then a n ^ k else 0
    vanish := by grind
  }

@[simp]
lemma Sequence.pow_eval {a:Sequence} {k: ℕ} {n: ℤ} (hn : n ≥ a.m): (a ^ k) n = a n ^ k := by
  sorry

@[simp]
lemma Sequence.pow_one (a:Sequence) : a^1 = a := by
  sorry

lemma Sequence.pow_succ (a:Sequence) (k:ℕ): a^(k+1) = a^k * a := by
  sorry

theorem Sequence.lim_of_power_decay {k:ℕ} :
    ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)^(1/(k+1:ℝ))):Sequence).TendsTo 0 := by
  sorry

-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Sequence.lim_of_geometric {x:ℝ} (hx: |x| < 1) : ((fun (n:ℕ) ↦ x^n):Sequence).TendsTo 0 := by
-- </TARGET_THEOREM>

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

namespace TaoBench

namespace Chapter6
namespace Sequence

/-- Lemma 6.5.2 / Exercise 6.5.2 (geometric sequence tends to 0 when `|x| < 1`). -/
theorem lim_of_geometric {x : ℝ} (hx : |x| < 1) :
    Tendsto (fun n : ℕ => x ^ n) atTop (𝓝 (0 : ℝ)) := by
end Sequence
end Chapter6

end TaoBench
import Mathlib.Tactic
import Mathlib.Topology.Instances.Real

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

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

lemma Sequence.eval_mk {n m:ℤ} (a: { n // n ≥ m } → ℝ) (h: n ≥ m) :
    (Sequence.mk' m a) n = a ⟨ n, h ⟩ := by simp [h]

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

/--
  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 ↑·)

lemma Sequence.from_eval (a:Sequence) {m₁ n:ℤ} (hn: n ≥ m₁) :
  (a.from m₁) n = a n := by
  simp [hn]; intros; symm; solve_by_elim [a.vanish]

end Chapter6

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

/-- 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

/-- Lemma 6.5.2 / Exercise 6.5.2 -/
theorem Sequence.lim_of_geometric' {x:ℝ} (hx: x = 1) : ((fun (n:ℕ) ↦ x^n):Sequence).TendsTo 1 := by
end Chapter6
import Mathlib
import Analysis.Section_6_4
import Analysis.Tags.Section_6_4

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

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

lemma Sequence.eval_mk {n m:ℤ} (a: { n // n ≥ m } → ℝ) (h: n ≥ m) :
    (Sequence.mk' m a) n = a ⟨ n, h ⟩ := by
  sorry

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

abbrev Sequence.from (a:Sequence) (m₁:ℤ) : Sequence := mk' (max a.m m₁) (a ↑·)

lemma Sequence.from_eval (a:Sequence) {m₁ n:ℤ} (hn: n ≥ m₁) :
  (a.from m₁) n = a n := by
  sorry

instance Sequence.inst_mul : Mul Sequence where
  mul a b := {
    m := min a.m b.m
    seq n := a n * b n
    vanish n hn := by simp [a.vanish n (by grind), b.vanish n (by grind)]
  }

@[simp]
theorem Sequence.mul_eval {a b: Sequence} (n:ℤ) : (a * b) n = a n * b n := by
  sorry

end Chapter6

abbrev Real.Close (ε x y : ℝ) : Prop := dist x y ≤ ε

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

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

theorem Sequence.lim_of_const (c:ℝ) :  ((fun (_:ℕ) ↦ c):Sequence).TendsTo c := by
  sorry

instance Sequence.inst_pow: Pow Sequence ℕ where
  pow a k := {
    m := a.m
    seq n := if n ≥ a.m then a n ^ k else 0
    vanish := by grind
  }

@[simp]
lemma Sequence.pow_eval {a:Sequence} {k: ℕ} {n: ℤ} (hn : n ≥ a.m): (a ^ k) n = a n ^ k := by
  sorry

@[simp]
lemma Sequence.pow_one (a:Sequence) : a^1 = a := by
  sorry

lemma Sequence.pow_succ (a:Sequence) (k:ℕ): a^(k+1) = a^k * a := by
  sorry

theorem Sequence.lim_of_power_decay {k:ℕ} :
    ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)^(1/(k+1:ℝ))):Sequence).TendsTo 0 := by
  sorry

-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Sequence.lim_of_geometric' {x:ℝ} (hx: x = 1) : ((fun (n:ℕ) ↦ x^n):Sequence).TendsTo 1 := by
-- </TARGET_THEOREM>

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

namespace TaoBench

namespace Chapter6

namespace Sequence
theorem lim_of_geometric' {x : ℝ} (hx : x = 1) :
    Tendsto (fun n : ℕ => x ^ n) atTop (𝓝 1) := by
end Sequence

end Chapter6

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

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

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

lemma Sequence.eval_mk {n m:ℤ} (a: { n // n ≥ m } → ℝ) (h: n ≥ m) :
    (Sequence.mk' m a) n = a ⟨ n, h ⟩ := by simp [h]

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

/--
  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 ↑·)

lemma Sequence.from_eval (a:Sequence) {m₁ n:ℤ} (hn: n ≥ m₁) :
  (a.from m₁) n = a n := by
  simp [hn]; intros; symm; solve_by_elim [a.vanish]

end Chapter6

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

/-- 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

/-- Definition 6.1.8 -/
abbrev Sequence.Divergent (a:Sequence) : Prop := ¬ a.Convergent

/-- Lemma 6.5.2 / Exercise 6.5.2 -/
theorem Sequence.lim_of_geometric'' {x:ℝ} (hx: x = -1 ∨ |x| > 1) :
    ((fun (n:ℕ) ↦ x^n):Sequence).Divergent := by
end Chapter6
import Mathlib
import Analysis.Section_6_4
import Analysis.Tags.Section_6_4

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

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

abbrev Sequence.from (a:Sequence) (m₁:ℤ) : Sequence := mk' (max a.m m₁) (a ↑·)

abbrev _root_.Real.Close (ε x y : ℝ) : Prop := dist x y ≤ ε

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

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

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

abbrev Sequence.Convergent (a:Sequence) : Prop := ∃ L, a.TendsTo L

abbrev Sequence.Divergent (a:Sequence) : Prop := ¬ a.Convergent

instance Sequence.inst_mul : Mul Sequence where
  mul a b := {
    m := min a.m b.m
    seq n := a n * b n
    vanish n hn := by simp [a.vanish n (by grind), b.vanish n (by grind)]
  }

theorem Sequence.lim_of_const (c:ℝ) :  ((fun (_:ℕ) ↦ c):Sequence).TendsTo c := by
  sorry

instance Sequence.inst_pow: Pow Sequence ℕ where
  pow a k := {
    m := a.m
    seq n := if n ≥ a.m then a n ^ k else 0
    vanish := by grind
  }

@[simp]
lemma Sequence.pow_eval {a:Sequence} {k: ℕ} {n: ℤ} (hn : n ≥ a.m): (a ^ k) n = a n ^ k := by
  sorry

@[simp]
lemma Sequence.pow_one (a:Sequence) : a^1 = a := by
  sorry

lemma Sequence.pow_succ (a:Sequence) (k:ℕ): a^(k+1) = a^k * a := by
  sorry

theorem Sequence.lim_of_power_decay {k:ℕ} :
    ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)^(1/(k+1:ℝ))):Sequence).TendsTo 0 := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Sequence.lim_of_geometric'' {x:ℝ} (hx: x = -1 ∨ |x| > 1) :
    ((fun (n:ℕ) ↦ x^n):Sequence).Divergent := by
-- </TARGET_THEOREM>

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

namespace TaoBench
theorem Sequence.lim_of_geometric'' {x : ℝ} (hx : x = -1 ∨ |x| > 1) :
  ¬ ∃ L : ℝ, Tendsto (fun n : ℕ => x ^ n) atTop (𝓝 L) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter6
namespace Sequence

/-- Mathlib version of Tao's Proposition 6.3.8 / Exercise 6.3.3:
A monotone increasing real sequence that is bounded above converges,
and its limit is the supremum of its range. -/
theorem lim_of_monotone
  {a : ℕ → ℝ}
  (hbound : ∃ M, ∀ n, a n ≤ M)
  (hmono : Monotone a) :
  Tendsto a atTop (𝓝 (sSup (Set.range a))) := by
end Sequence
end Chapter6

end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib/Analysis/SpecialFunctions/Pow/Real

/-!
Self-contained extraction for the target theorem from Chapter 6, Section 6.5.
Only minimal prerequisites from Section 6.1 are included, verbatim.
-/

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

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

lemma Sequence.eval_mk {n m:ℤ} (a: { n // n ≥ m } → ℝ) (h: n ≥ m) :
    (Sequence.mk' m a) n = a ⟨ n, h ⟩ := by simp [h]

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

/--
  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 ↑·)

lemma Sequence.from_eval (a:Sequence) {m₁ n:ℤ} (hn: n ≥ m₁) :
  (a.from m₁) n = a n := by
  simp [hn]; intros; symm; solve_by_elim [a.vanish]

end Chapter6

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

/-- 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

/-- Exercise 6.5.1 -/
theorem Sequence.lim_of_rat_power_decay {q:ℚ} (hq: q > 0) :
    (fun (n:ℕ) ↦ 1/((n+1:ℝ)^(q:ℝ)):Sequence).TendsTo 0 := by
end Chapter6
import Mathlib
import Analysis.Section_6_4
import Analysis.Tags.Section_6_4

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

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

abbrev Sequence.from (a:Sequence) (m₁:ℤ) : Sequence := mk' (max a.m m₁) (a ↑·)

abbrev _root_.Real.CloseSeq (ε: ℝ) (a: Chapter6.Sequence) (L:ℝ) : Prop := ∀ n ≥ a.m, dist (a n) L ≤ ε

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

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

instance Sequence.inst_mul : Mul Sequence where
  mul a b := {
    m := min a.m b.m
    seq n := a n * b n
    vanish n hn := by simp [a.vanish n (by grind), b.vanish n (by grind)]
  }

theorem Sequence.lim_of_const (c:ℝ) :  ((fun (_:ℕ) ↦ c):Sequence).TendsTo c := by
  sorry

instance Sequence.inst_pow: Pow Sequence ℕ where
  pow a k := {
    m := a.m
    seq n := if n ≥ a.m then a n ^ k else 0
    vanish := by grind
  }

@[simp]
lemma Sequence.pow_eval {a:Sequence} {k: ℕ} {n: ℤ} (hn : n ≥ a.m): (a ^ k) n = a n ^ k := by
  sorry

@[simp]
lemma Sequence.pow_one (a:Sequence) : a^1 = a := by
  sorry

lemma Sequence.pow_succ (a:Sequence) (k:ℕ): a^(k+1) = a^k * a := by
  sorry

theorem Sequence.lim_of_power_decay {k:ℕ} :
    ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)^(1/(k+1:ℝ))):Sequence).TendsTo 0 := by
  sorry

-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Sequence.lim_of_rat_power_decay {q:ℚ} (hq: q > 0) :
    (fun (n:ℕ) ↦ 1/((n+1:ℝ)^(q:ℝ)):Sequence).TendsTo 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 Chapter6.Sequence.lim_of_rat_power_decay {q : ℚ} (hq : q > 0) :
    Tendsto (fun n : ℕ => 1 / ((n + 1 : ℝ) ^ (q : ℝ))) atTop (𝓝 (0 : ℝ)) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic

/-!
Self-contained extraction for Chapter 6 sequences sufficient to state the target theorem from
Section 6.5. We avoid importing any `Analysis.Section_*` modules and copy only the minimal
infrastructure from Section 6.1 actually needed to parse/typecheck the target statement.
-/

/- Definition 6.1.1 (Distance).  Here we use the Mathlib distance. -/
#check Real.dist_eq

abbrev Real.Close (ε x y : ℝ) : Prop := dist x y ≤ ε

/--
  Definition 6.1.2 (ε-close). This is similar to the previous notion of ε-closeness, but where
  all quantities are real instead of rational.
-/
theorem Real.close_def (ε x y : ℝ) : ε.Close x y ↔ dist x y ≤ ε := by rfl

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

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

lemma Sequence.eval_mk {n m:ℤ} (a: { n // n ≥ m } → ℝ) (h: n ≥ m) :
    (Sequence.mk' m a) n = a ⟨ n, h ⟩ := by simp [h]

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

/--
  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 ↑·)

lemma Sequence.from_eval (a:Sequence) {m₁ n:ℤ} (hn: n ≥ m₁) :
  (a.from m₁) n = a n := by
  simp [hn]; intros; symm; solve_by_elim [a.vanish]

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 -/
theorem Real.closeSeq_def (ε: ℝ) (a: Chapter6.Sequence) (L:ℝ) :
  ε.CloseSeq a L ↔ ∀ n ≥ a.m, dist (a n) L ≤ ε := by rfl

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

/-- Definition 6.1.5 -/
theorem Real.eventuallyClose_def (ε: ℝ) (a: Chapter6.Sequence) (L:ℝ) :
  ε.EventuallyClose a L ↔ ∃ N, (N ≥ a.m) ∧ ε.CloseSeq (a.from N) L := by rfl

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

/-- Definition 6.1.8 -/
abbrev Sequence.Divergent (a:Sequence) : Prop := ¬ a.Convergent

/-- Exercise 6.5.1 -/
theorem Sequence.lim_of_rat_power_growth {q:ℚ} (hq: q > 0) :
    (fun (n:ℕ) ↦ ((n+1:ℝ)^(q:ℝ)):Sequence).Divergent := by
end Chapter6
import Mathlib
import Analysis.Section_6_4
import Analysis.Tags.Section_6_4

namespace Chapter6

-- <CONTEXT>
abbrev _root_.Real.Close (ε x y : ℝ) : Prop := dist x y ≤ ε

@[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 := Sequence.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

abbrev Sequence.from (a:Sequence) (m₁:ℤ) : Sequence := mk' (max a.m m₁) (a ↑·)

abbrev _root_.Real.CloseSeq (ε: ℝ) (a: Sequence) (L:ℝ) : Prop := ∀ n ≥ a.m, ε.Close (a n) L

abbrev _root_.Real.EventuallyClose (ε: ℝ) (a: Sequence) (L:ℝ) : Prop :=
  ∃ N ≥ a.m, ε.CloseSeq (a.from N) L

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

abbrev Sequence.Convergent (a:Sequence) : Prop := ∃ L, a.TendsTo L

abbrev Sequence.Divergent (a:Sequence) : Prop := ¬ a.Convergent

instance Sequence.inst_mul : Mul Sequence where
  mul a b := {
    m := min a.m b.m
    seq n := a n * b n
    vanish n hn := by simp [a.vanish n (by grind), b.vanish n (by grind)]
  }

theorem Sequence.lim_of_const (c:ℝ) :  ((fun (_:ℕ) ↦ c):Sequence).TendsTo c := by
  sorry

instance Sequence.inst_pow: Pow Sequence ℕ where
  pow a k := {
    m := a.m
    seq n := if n ≥ a.m then a n ^ k else 0
    vanish := by grind
  }

@[simp]
lemma Sequence.pow_eval {a:Sequence} {k: ℕ} {n: ℤ} (hn : n ≥ a.m): (a ^ k) n = a n ^ k := by
  sorry

@[simp]
lemma Sequence.pow_one (a:Sequence) : a^1 = a := by
  sorry

lemma Sequence.pow_succ (a:Sequence) (k:ℕ): a^(k+1) = a^k * a := by
  sorry

theorem Sequence.lim_of_power_decay {k:ℕ} :
    ((fun (n:ℕ) ↦ 1/((n:ℝ)+1)^(1/(k+1:ℝ))):Sequence).TendsTo 0 := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Sequence.lim_of_rat_power_growth {q:ℚ} (hq: q > 0) :
    (fun (n:ℕ) ↦ ((n+1:ℝ)^(q:ℝ)):Sequence).Divergent := by
-- </TARGET_THEOREM>

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

namespace TaoBench

namespace Chapter6
namespace Sequence

/-- Mathlib translation of Tao's Exercise 6.5.1: the sequence `(n + 1)^q` diverges for `q > 0`. -/
theorem lim_of_rat_power_growth {q : ℚ} (hq : q > 0) :
    ¬ ∃ L : ℝ, Tendsto (fun n : ℕ => ((n + 1 : ℝ) ^ (q : ℝ))) atTop (nhds L) := by
end Sequence
end Chapter6

end TaoBench