internlm/Lean-Github huggingface dataset:
Theorem: residue_nonneg
Proof state before:
case h.e'_3
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
ε : ℝ
ψ : ℝ → ℝ
h1 : ContDiff ℝ ⊤ ψ
h2 : HasCompactSupport ψ
h3 : closure (Function.support ψ) ⊆ Ioi 0
h4 : (Ico 1 2).indicator 1 ≤ ψ
key : Tendsto (fun x => (∑' (n : ℕ), f n * ψ (↑n / x)) / x) atTop (𝓝 (A * ∫ (y : ℝ) in Ioi 0, ψ y))
l2 : 0 ≤ ψ
l1 : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S ψ x
l3 : 0 ≤ A * ∫ (y : ℝ) in Ioi 0, ψ y
r1 : 0 ≤ᶠ[ae (volume.restrict (Ioi 0))] ψ
r2 : IntegrableOn (fun y => ψ y) (Ioi 0) volume
r3 : Ico 1 2 ⊆ Function.support ψ
r4 : Ico 1 2 ⊆ Function.support ψ ∩ Ioi 0
⊢ 1 = volume.measureOf (Ico 1 2)
Tactic:
norm_num
Proof state after:
no goals
Theorem: WienerIkeharaInterval
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
by_cases hab : a = b
Proof state after:
case pos
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
hab : a = b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))

case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
hab : ¬a = b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case pos
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
hab : a = b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))

case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
hab : ¬a = b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
simp [hab]
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
hab : ¬a = b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a ≤ b
hab : ¬a = b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
replace hb : a < b := lt_of_le_of_ne hb hab
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hab : ¬a = b
hb : a < b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hab : ¬a = b
hb : a < b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
clear hab
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
let S (g : ℝ → ℝ) (x : ℝ) :=  (∑' n, f n * g (n / x)) / x
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
have hSnonneg {g : ℝ → ℝ} (hg : 0 ≤ g) : ∀ᶠ x : ℝ in atTop, 0 ≤ S g x := by
  filter_upwards [eventually_ge_atTop 0] with x hx
  refine div_nonneg ?_ hx
  refine tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
have hA : 0 ≤ A := residue_nonneg hpos hf hcheby hG hG'
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
let Iab : ℝ → ℝ := indicator (Ico a b) 1
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
⊢ Tendsto (fun x => (∑' (n : ℕ), f n * (Ico a b).indicator 1 (↑n / x)) / x) atTop (𝓝 (A * (b - a)))
Tactic:
change Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Tactic:
have hIab : HasCompactSupport Iab := by simpa [Iab, HasCompactSupport, tsupport, hb.ne] using isCompact_Icc
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Tactic:
have Iab_nonneg : ∀ᶠ x : ℝ in atTop, 0 ≤ S Iab x := hSnonneg (indicator_nonneg (by simp))
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
Iab_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S Iab x
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
Iab_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S Iab x
Iab2 : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop (S Iab)
Iab3 : IsBoundedUnder (fun x x_1 => x ≥ x_1) atTop (S Iab)
Iab0 : IsCoboundedUnder (fun x x_1 => x ≥ x_1) atTop (S Iab)
Iab1 : IsCoboundedUnder (fun x x_1 => x ≤ x_1) atTop (S Iab)
sup_le : limsup (S Iab) atTop ≤ A * (b - a)
le_inf : A * (b - a) ≤ liminf (S Iab) atTop
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Tactic:
have : liminf (S Iab) atTop ≤ limsup (S Iab) atTop := liminf_le_limsup Iab2 Iab3
Proof state after:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
Iab_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S Iab x
Iab2 : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop (S Iab)
Iab3 : IsBoundedUnder (fun x x_1 => x ≥ x_1) atTop (S Iab)
Iab0 : IsCoboundedUnder (fun x x_1 => x ≥ x_1) atTop (S Iab)
Iab1 : IsCoboundedUnder (fun x x_1 => x ≤ x_1) atTop (S Iab)
sup_le : limsup (S Iab) atTop ≤ A * (b - a)
le_inf : A * (b - a) ≤ liminf (S Iab) atTop
this : liminf (S Iab) atTop ≤ limsup (S Iab) atTop
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Theorem: WienerIkeharaInterval
Proof state before:
case neg
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
Iab_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S Iab x
Iab2 : IsBoundedUnder (fun x x_1 => x ≤ x_1) atTop (S Iab)
Iab3 : IsBoundedUnder (fun x x_1 => x ≥ x_1) atTop (S Iab)
Iab0 : IsCoboundedUnder (fun x x_1 => x ≥ x_1) atTop (S Iab)
Iab1 : IsCoboundedUnder (fun x x_1 => x ≤ x_1) atTop (S Iab)
sup_le : limsup (S Iab) atTop ≤ A * (b - a)
le_inf : A * (b - a) ≤ liminf (S Iab) atTop
this : liminf (S Iab) atTop ≤ limsup (S Iab) atTop
⊢ Tendsto (S Iab) atTop (𝓝 (A * (b - a)))
Tactic:
refine tendsto_of_liminf_eq_limsup ?_ ?_ Iab2 Iab3 <;> linarith
Proof state after:
no goals
Theorem: WienerIkeharaInterval
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
g : ℝ → ℝ
hg : 0 ≤ g
⊢ ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
Tactic:
filter_upwards [eventually_ge_atTop 0] with x hx
Proof state after:
case h
n : ℕ
A a b c d u x✝ y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
g : ℝ → ℝ
hg : 0 ≤ g
x : ℝ
hx : 0 ≤ x
⊢ 0 ≤ S g x
Theorem: WienerIkeharaInterval
Proof state before:
case h
n : ℕ
A a b c d u x✝ y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
g : ℝ → ℝ
hg : 0 ≤ g
x : ℝ
hx : 0 ≤ x
⊢ 0 ≤ S g x
Tactic:
refine div_nonneg ?_ hx
Proof state after:
case h
n : ℕ
A a b c d u x✝ y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
g : ℝ → ℝ
hg : 0 ≤ g
x : ℝ
hx : 0 ≤ x
⊢ 0 ≤ ∑' (n : ℕ), f n * g (↑n / x)
Theorem: WienerIkeharaInterval
Proof state before:
case h
n : ℕ
A a b c d u x✝ y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
g : ℝ → ℝ
hg : 0 ≤ g
x : ℝ
hx : 0 ≤ x
⊢ 0 ≤ ∑' (n : ℕ), f n * g (↑n / x)
Tactic:
refine tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))
Proof state after:
no goals
Theorem: WienerIkeharaInterval
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
⊢ HasCompactSupport Iab
Tactic:
simpa [Iab, HasCompactSupport, tsupport, hb.ne] using isCompact_Icc
Proof state after:
no goals
Theorem: WienerIkeharaInterval
Proof state before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
ha : 0 < a
hb : a < b
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
hA : 0 ≤ A
Iab : ℝ → ℝ := (Ico a b).indicator 1
hIab : HasCompactSupport Iab
⊢ ∀ a_1 ∈ Ico a b, 0 ≤ 1 a_1
Tactic:
simp
Proof state after:
no goals