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Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
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

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