Datasets:
internlm
/
Lean-Github

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https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
have Hx := Hz _ hx
case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
refine DifferentiableAt.differentiableWithinAt ?_
case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
replace hf := ((hf x Hx).congr (fun _ hz ↦ H hz) (H Hx)).differentiableAt (Metric.isOpen_ball.mem_nhds Hx) |>.comp_ofReal
case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
simp_rw [hd, ← ofReal_sub, ← ofReal_natCast, ← ofReal_inv, ← ofReal_pow, ← ofReal_mul, ← ofReal_tsum] at hf
case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
exact DifferentiableAt.ofReal_comp_iff.mp hf
case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
simp only [Function.comp_apply, ← H (Hz _ hx), hd, ofReal_tsum]
case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
push_cast
case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
rfl
case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
have H (z : ℂ) := taylorSeries_eq_of_entire' c z hf
f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
simp_rw [hd] at H
f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
refine ⟨fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ * (D n) * (x - c) ^ n, ?_, ?_⟩
f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ f ∘ ofReal' = ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
have := hf.comp_ofReal
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
simp_rw [← H, ← ofReal_sub, ← ofReal_natCast, ← ofReal_inv, ← ofReal_pow, ← ofReal_mul, ← ofReal_tsum] at this
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
exact Differentiable.ofReal_comp_iff.mp this
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
ext x
case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ f ∘ ofReal' = ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
simp only [Function.comp_apply, ofReal_eq_coe, ← H, ofReal_tsum]
case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
push_cast
case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
rfl
case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
have H := taylorSeries_eq_of_entire' 0 z hf
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z ⊢ 0 ≤ f z
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊢ 0 ≤ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
have hz' := eq_re_of_ofReal_le hz
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊢ 0 ≤ f z
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊢ 0 ≤ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
rw [hz'] at hz H ⊢
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊢ 0 ≤ f z
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ 0 ≤ f ↑z.re
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ n, 0 ≤ D n ∧ iteratedDeriv n f 0 = D n
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ 0 ≤ f ↑z.re
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ ∃ D, ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ f ↑z.re
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
simp_rw [← H, hD, ← ofReal_natCast, sub_zero, ← ofReal_pow, ← ofReal_inv, ← ofReal_mul, ← ofReal_tsum]
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ f ↑z.re
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
norm_cast
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a)
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
refine tsum_nonneg fun n ↦ ?_
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
norm_cast at hz
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
have := (hD n).1
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re this : 0 ≤ D n ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
positivity
case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re this : 0 ≤ D n ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
refine ⟨fun n ↦ (iteratedDeriv n f 0).re, fun n ↦ ⟨?_, ?_⟩⟩
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ ∃ D, ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n)
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
have := eq_re_of_ofReal_le (h n) ▸ h n
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ this : 0 ≤ ↑(iteratedDeriv n f 0).re ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
norm_cast at this
case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ this : 0 ≤ ↑(iteratedDeriv n f 0).re ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
rw [eq_re_of_ofReal_le (h n)]
case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
let D : ℕ → ℝ := fun n ↦ (iteratedDeriv n f 0).re
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hD (n : ℕ) : iteratedDeriv n f 0 = D n := by refine Complex.ext rfl ?_ simp only [ofReal_im] exact (le_def.mp (h n)).2.symm
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
obtain ⟨F, hFd, hF⟩ := realValued_of_iteratedDeriv_real hf hD
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
rw [hF]
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
refine monotone_ofReal.comp_monotoneOn <| monotoneOn_of_deriv_nonneg (convex_Ici 0) hFd.continuous.continuousOn hFd.differentiableOn fun x hx ↦ ?_
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hD' (n : ℕ) : 0 ≤ iteratedDeriv n (deriv f) 0 := by rw [← iteratedDeriv_succ'] exact h (n + 1)
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hf' := (contDiff_succ_iff_deriv.mp <| hf.contDiff (n := 2)).2.differentiable rfl.le
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hx : (0 : ℂ) ≤ x := by norm_cast simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx exact hx.le
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have H := nonneg_of_iteratedDeriv_nonneg hf' hD' hx
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv f ↑x ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
rw [← deriv.comp_ofReal hf.differentiableAt] at H
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv f ↑x ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv (fun x => f ↑x) x ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
change 0 ≤ deriv (f ∘ ofReal') x at H
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv (fun x => f ↑x) x ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv (f ∘ ofReal') x ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
erw [hF, deriv.ofReal_comp] at H
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv (f ∘ ofReal') x ⊢ 0 ≤ deriv F x
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ ↑(deriv (fun y => F y) x) ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
norm_cast at H
case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ ↑(deriv (fun y => F y) x) ⊢ 0 ≤ deriv F x
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
refine Complex.ext rfl ?_
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re n : ℕ ⊢ iteratedDeriv n f 0 = ↑(D n)
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re n : ℕ ⊢ (iteratedDeriv n f 0).im = (↑(D n)).im
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
simp only [ofReal_im]
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re n : ℕ ⊢ (iteratedDeriv n f 0).im = (↑(D n)).im
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re n : ℕ ⊢ (iteratedDeriv n f 0).im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
exact (le_def.mp (h n)).2.symm
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re n : ℕ ⊢ (iteratedDeriv n f 0).im = 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
rw [← iteratedDeriv_succ']
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) n : ℕ ⊢ 0 ≤ iteratedDeriv n (deriv f) 0
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) n : ℕ ⊢ 0 ≤ iteratedDeriv (n + 1) f 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
exact h (n + 1)
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) n : ℕ ⊢ 0 ≤ iteratedDeriv (n + 1) f 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
norm_cast
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ ↑x
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ x
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 < x ⊢ 0 ≤ x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
exact hx.le
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 < x ⊢ 0 ≤ x
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
exact sub_nonneg.mp <| nonneg_of_iteratedDeriv_nonneg (hf.sub_const (f 0)) h' hz
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z h' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (fun x => f x - f 0) 0 ⊢ f 0 ≤ f z
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
cases n with | zero => simp only [iteratedDeriv_zero, sub_self, le_refl] | succ n => specialize h n.succ <| succ_ne_zero n rw [iteratedDeriv_succ'] at h ⊢ convert h using 2 ext w exact deriv_sub_const (f 0)
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z n : ℕ ⊢ 0 ≤ iteratedDeriv n (fun x => f x - f 0) 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
simp only [iteratedDeriv_zero, sub_self, le_refl]
case zero f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z ⊢ 0 ≤ iteratedDeriv 0 (fun x => f x - f 0) 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
specialize h n.succ <| succ_ne_zero n
case succ f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z n : ℕ ⊢ 0 ≤ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
case succ f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n.succ f 0 ⊢ 0 ≤ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
rw [iteratedDeriv_succ'] at h ⊢
case succ f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n.succ f 0 ⊢ 0 ≤ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
case succ f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ iteratedDeriv n (deriv fun x => f x - f 0) 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
convert h using 2
case succ f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ iteratedDeriv n (deriv fun x => f x - f 0) 0
case h.e'_4.h.e'_7 f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ (deriv fun x => f x - f 0) = deriv f
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
ext w
case h.e'_4.h.e'_7 f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ (deriv fun x => f x - f 0) = deriv f
case h.e'_4.h.e'_7.h f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n (deriv f) 0 w : ℂ ⊢ deriv (fun x => f x - f 0) w = deriv f w
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_nonneg
[255, 1]
[266, 83]
exact deriv_sub_const (f 0)
case h.e'_4.h.e'_7.h f : ℂ → ℂ hf : Differentiable ℂ f z : ℂ hz : 0 ≤ z n : ℕ h : 0 ≤ iteratedDeriv n (deriv f) 0 w : ℂ ⊢ deriv (fun x => f x - f 0) w = deriv f w
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_alternating
[271, 1]
[278, 66]
let F : ℂ → ℂ := fun z ↦ f (-z)
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 ⊢ f 0 ≤ f z
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) ⊢ f 0 ≤ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_alternating
[271, 1]
[278, 66]
convert at_zero_le_of_iteratedDeriv_nonneg (f := F) (hf.comp <| differentiable_neg) (fun n hn ↦ ?_) (neg_nonneg.mpr hz) using 1
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) ⊢ f 0 ≤ f z
case h.e'_3 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) ⊢ f 0 = F 0 case h.e'_4 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) ⊢ f z = F (-z) f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) n : ℕ hn : n ≠ 0 ⊢ 0 ≤ iteratedDeriv n F 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_alternating
[271, 1]
[278, 66]
simp only [F, neg_zero]
case h.e'_3 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) ⊢ f 0 = F 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_alternating
[271, 1]
[278, 66]
simp only [F, neg_neg]
case h.e'_4 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) ⊢ f z = F (-z)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.at_zero_le_of_iteratedDeriv_alternating
[271, 1]
[278, 66]
simpa only [F, iteratedDeriv_comp_neg, neg_zero] using h n hn
f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0 z : ℂ hz : z ≤ 0 F : ℂ → ℂ := fun z => f (-z) n : ℕ hn : n ≠ 0 ⊢ 0 ≤ iteratedDeriv n F 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
convert (Finset.sum_indicator_subset f Finset.mem_of_mem_filter).symm using 2 with _ _ m hm
R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n : ℕ ⊢ ∑ p ∈ n.primesBelow, f p = ∑ m ∈ Finset.range n, {p | p.Prime}.indicator f m
case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ {p | p.Prime}.indicator f m = (↑(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
simp only [Set.mem_setOf_eq, Finset.mem_range, Finset.coe_filter, not_and, Set.indicator_apply]
case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ {p | p.Prime}.indicator f m = (↑(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m
case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ (if m.Prime then f m else 0) = if m < n ∧ m.Prime then f m else 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
split_ifs with h₁ h₂ h₃
case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ (if m.Prime then f m else 0) = if m < n ∧ m.Prime then f m else 0
case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : m < n ∧ m.Prime ⊢ f m = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : ¬(m < n ∧ m.Prime) ⊢ f m = 0 case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : m < n ∧ m.Prime ⊢ 0 = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : ¬(m < n ∧ m.Prime) ⊢ 0 = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
rfl
case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : m < n ∧ m.Prime ⊢ f m = f m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
exact (h₂ ⟨Finset.mem_range.mp hm, h₁⟩).elim
case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : ¬(m < n ∧ m.Prime) ⊢ f m = 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
exact (h₁ h₃.2).elim
case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : m < n ∧ m.Prime ⊢ 0 = f m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
sum_primesBelow_eq_sum_range_indicator
[49, 1]
[58, 8]
rfl
case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : ¬(m < n ∧ m.Prime) ⊢ 0 = 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
tendsto_sum_primesBelow_tsum
[62, 1]
[69, 94]
rw [(show ∑' p : Nat.Primes, f p = ∑' p : {p : ℕ | p.Prime}, f p from rfl)]
R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : Nat.Primes), f ↑p))
R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : ↑{p | p.Prime}), f ↑p))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
tendsto_sum_primesBelow_tsum
[62, 1]
[69, 94]
simp_rw [tsum_subtype, sum_primesBelow_eq_sum_range_indicator]
R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : ↑{p | p.Prime}), f ↑p))
R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ m ∈ Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (∑' (x : ℕ), {p | p.Prime}.indicator f x))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
tendsto_sum_primesBelow_tsum
[62, 1]
[69, 94]
exact (Summable.hasSum_iff_tendsto_nat <| hsum.indicator _).mp <| (hsum.indicator _).hasSum
R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ m ∈ Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (∑' (x : ℕ), {p | p.Prime}.indicator f x))
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Complex.exp_tsum_primes
[71, 1]
[77, 81]
simpa only [← exp_sum] using Tendsto.cexp <| tendsto_sum_primesBelow_tsum hsum
f : ℕ → ℂ hsum : Summable f ⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (f p)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), f ↑p)))
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Summable.neg_clog_one_sub
[82, 1]
[91, 51]
let g (z : ℂ) : ℂ := -log (1 - z)
α : Type u_1 f : α → ℂ hsum : Summable f ⊢ Summable fun n => -(1 - f n).log
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ Summable fun n => -(1 - f n).log
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Summable.neg_clog_one_sub
[82, 1]
[91, 51]
have hg : DifferentiableAt ℂ g 0 := DifferentiableAt.neg <| ((differentiableAt_const 1).sub differentiableAt_id').clog <| by simp only [sub_zero, one_mem_slitPlane]
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ Summable fun n => -(1 - f n).log
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ Summable fun n => -(1 - f n).log
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Summable.neg_clog_one_sub
[82, 1]
[91, 51]
have : g =O[𝓝 0] id := by simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ Summable fun n => -(1 - f n).log
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 this : g =O[𝓝 0] id ⊢ Summable fun n => -(1 - f n).log
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Summable.neg_clog_one_sub
[82, 1]
[91, 51]
exact Asymptotics.IsBigO.comp_summable this hsum
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 this : g =O[𝓝 0] id ⊢ Summable fun n => -(1 - f n).log
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Summable.neg_clog_one_sub
[82, 1]
[91, 51]
simp only [sub_zero, one_mem_slitPlane]
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ 1 - 0 ∈ slitPlane
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
Summable.neg_clog_one_sub
[82, 1]
[91, 51]
simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg
α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ g =O[𝓝 0] id
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
have hs {p : ℕ} (hp : 1 < p) : ‖f p‖ < 1 := hsum.of_norm.norm_lt_one (f := f.toMonoidHom) hp
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
have H := Complex.exp_tsum_primes hsum.of_norm.neg_clog_one_sub
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
have help (n : ℕ) : n.primesBelow.prod (fun p ↦ cexp (-log (1 - f p))) = n.primesBelow.prod fun p ↦ (1 - f p)⁻¹ := by refine Finset.prod_congr rfl (fun p hp ↦ ?_) rw [exp_neg, exp_log ?_] rw [ne_eq, sub_eq_zero, ← ne_eq] exact fun h ↦ (norm_one (α := ℂ) ▸ h.symm ▸ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
simp_rw [help] at H
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ H : Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
exact tendsto_nhds_unique H <| eulerProduct_completely_multiplicative hsum
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ H : Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
refine Finset.prod_congr rfl (fun p hp ↦ ?_)
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n : ℕ ⊢ ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ cexp (-(1 - f p).log) = (1 - f p)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
rw [exp_neg, exp_log ?_]
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ cexp (-(1 - f p).log) = (1 - f p)⁻¹
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 - f p ≠ 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
rw [ne_eq, sub_eq_zero, ← ne_eq]
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 - f p ≠ 0
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 ≠ f p
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Logarithm.lean
EulerProduct.exp_sum_primes_log_eq_tsum
[96, 1]
[107, 77]
exact fun h ↦ (norm_one (α := ℂ) ▸ h.symm ▸ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false
f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 ≠ f p
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
LSeries.term_mul_aux
[21, 1]
[23, 90]
rw [mul_comm_div, div_div, ← mul_div_assoc, mul_comm (m : ℂ), natCast_mul_natCast_cpow]
a b : ℂ m n : ℕ s : ℂ ⊢ a / ↑m ^ s * (b / ↑n ^ s) = a * b / (↑m * ↑n) ^ s
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
LSeries.term_mul
[25, 1]
[30, 100]
rcases eq_or_ne (m * n) 0 with H | H
f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
case inl f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n = 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n case inr f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
LSeries.term_mul
[25, 1]
[30, 100]
rcases mul_eq_zero.mp H with rfl | rfl <;> simp only [term_zero, mul_zero, zero_mul]
case inl f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n = 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
LSeries.term_mul
[25, 1]
[30, 100]
obtain ⟨hm, hn⟩ := mul_ne_zero_iff.mp H
case inr f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
case inr.intro f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 hm : m ≠ 0 hn : n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
LSeries.term_mul
[25, 1]
[30, 100]
simp only [ne_eq, H, not_false_eq_true, term_of_ne_zero, Nat.cast_mul, hm, hn, h, term_mul_aux]
case inr.intro f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 hm : m ≠ 0 hn : n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
LSeries.term_at_one
[44, 1]
[45, 72]
rw [term_of_ne_zero one_ne_zero, h₁, Nat.cast_one, one_cpow, div_one]
f : ℕ → ℂ h₁ : f 1 = 1 s : ℂ ⊢ term f s 1 = 1
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
DirichletCharacter.toFun_on_nat_map_one
[86, 1]
[87, 32]
simp only [cast_one, map_one]
N : ℕ χ : DirichletCharacter ℂ N ⊢ (fun n => χ ↑n) 1 = 1
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
DirichletCharacter.toFun_on_nat_map_mul
[89, 1]
[91, 32]
simp only [cast_mul, map_mul]
N : ℕ χ : DirichletCharacter ℂ N m n : ℕ ⊢ (fun n => χ ↑n) (m * n) = (fun n => χ ↑n) m * (fun n => χ ↑n) n
no goals