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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.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/LSeriesUnique.lean	Complex.cpow_natCast_add_one_ne_zero	[8, 1]	[9, 67]	norm_cast at H	n : ℕ z : ℂ H : ↑n + 1 = 0 ∧ z ≠ 0 ⊢ False	n : ℕ z : ℂ H : False ∧ ¬z = 0 ⊢ False
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	Complex.cpow_natCast_add_one_ne_zero	[8, 1]	[9, 67]	exact H.1	n : ℕ z : ℂ H : False ∧ ¬z = 0 ⊢ False	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.abscissaOfAbsConv_binop_le	[13, 1]	[24, 59]	refine abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' fun x hx ↦ hF ?_ ?_	F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s f g : ℕ → ℂ ⊢ abscissaOfAbsConv (F f g) ≤ max (abscissaOfAbsConv f) (abscissaOfAbsConv g)	case refine_1 F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s f g : ℕ → ℂ x : ℝ hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x ⊢ LSeriesSummable f ↑x case refine_2 F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s f g : ℕ → ℂ x : ℝ hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x ⊢ LSeriesSummable g ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.abscissaOfAbsConv_binop_le	[13, 1]	[24, 59]	exact LSeriesSummable_of_abscissaOfAbsConv_lt_re <\| (ofReal_re x).symm ▸ (le_max_left ..).trans_lt hx	case refine_1 F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s f g : ℕ → ℂ x : ℝ hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x ⊢ LSeriesSummable f ↑x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.abscissaOfAbsConv_binop_le	[13, 1]	[24, 59]	exact LSeriesSummable_of_abscissaOfAbsConv_lt_re <\| (ofReal_re x).symm ▸ (le_max_right ..).trans_lt hx	case refine_2 F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s f g : ℕ → ℂ x : ℝ hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x ⊢ LSeriesSummable g ↑x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	have Hm : (0 : ℝ) ≤ m := m.cast_nonneg	m n : ℕ z : ℂ x : ℝ ⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m ⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	have Hn : (0 : ℝ) ≤ (n + 1 : ℝ)⁻¹ := by positivity	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m ⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	rw [← mul_div_assoc, mul_comm, div_eq_mul_inv z, mul_div_assoc]	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ z * ((↑n + 1) ^ ↑x / ↑m ^ ↑x) = z * ((↑m / (↑n + 1)) ^ ↑x)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	congr	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ z * ((↑n + 1) ^ ↑x / ↑m ^ ↑x) = z * ((↑m / (↑n + 1)) ^ ↑x)⁻¹	case e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1) ^ ↑x / ↑m ^ ↑x = ((↑m / (↑n + 1)) ^ ↑x)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	simp_rw [div_eq_mul_inv]	case e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1) ^ ↑x / ↑m ^ ↑x = ((↑m / (↑n + 1)) ^ ↑x)⁻¹	case e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1) ^ ↑x * (↑m ^ ↑x)⁻¹ = ((↑m * (↑n + 1)⁻¹) ^ ↑x)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	rw [show (n + 1 : ℂ)⁻¹ = (n + 1 : ℝ)⁻¹ by simp only [ofReal_inv, ofReal_add, ofReal_natCast, ofReal_one], show (n + 1 : ℂ) = (n + 1 : ℝ) by norm_cast, show (m : ℂ) = (m : ℝ) by norm_cast, mul_cpow_ofReal_nonneg Hm Hn, mul_inv, mul_comm]	case e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1) ^ ↑x * (↑m ^ ↑x)⁻¹ = ((↑m * (↑n + 1)⁻¹) ^ ↑x)⁻¹	case e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑↑m ^ ↑x)⁻¹ * ↑(↑n + 1) ^ ↑x = (↑↑m ^ ↑x)⁻¹ * (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	congr	case e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑↑m ^ ↑x)⁻¹ * ↑(↑n + 1) ^ ↑x = (↑↑m ^ ↑x)⁻¹ * (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹	case e_a.e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ ↑(↑n + 1) ^ ↑x = (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	rw [← cpow_neg, show (-x : ℂ) = (-1 : ℝ) * x by simp only [ofReal_neg, ofReal_one, neg_mul, one_mul], cpow_mul_ofReal_nonneg Hn, Real.rpow_neg_one, inv_inv]	case e_a.e_a m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ ↑(↑n + 1) ^ ↑x = (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	positivity	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m ⊢ 0 ≤ (↑n + 1)⁻¹	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	simp only [ofReal_inv, ofReal_add, ofReal_natCast, ofReal_one]	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ (↑n + 1)⁻¹ = ↑(↑n + 1)⁻¹	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	norm_cast	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ ↑n + 1 = ↑(↑n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	norm_cast	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ ↑m = ↑↑m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	foo	[38, 1]	[52, 78]	simp only [ofReal_neg, ofReal_one, neg_mul, one_mul]	m n : ℕ z : ℂ x : ℝ Hm : 0 ≤ ↑m Hn : 0 ≤ (↑n + 1)⁻¹ ⊢ -↑x = ↑(-1) * ↑x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.pow_mul_term_eq	[54, 1]	[58, 20]	simp only [term, add_eq_zero, one_ne_zero, and_false, ↓reduceIte, Nat.cast_add, Nat.cast_one, mul_div_assoc', ne_eq, cpow_natCast_add_one_ne_zero n _, not_false_eq_true, div_eq_iff, mul_comm (f _)]	f : ℕ → ℂ s : ℂ n : ℕ ⊢ (↑n + 1) ^ s * term f s (n + 1) = f (n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	obtain ⟨y, hay, hyt⟩ := exists_between ha	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))	case intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : EReal hay : abscissaOfAbsConv f < y hyt : y < ⊤ ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	lift y to ℝ using ⟨hyt.ne, ((OrderBot.bot_le _).trans_lt hay).ne'⟩	case intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : EReal hay : abscissaOfAbsConv f < y hyt : y < ⊤ ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	let F := fun (x : ℝ) ↦ {m \| n + 1 < m}.indicator (fun m ↦ f m / (m / (n + 1) : ℂ) ^ (x : ℂ))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have hF₀ (x : ℝ) {m : ℕ} (hm : m ≤ n + 1) : F x m = 0 := by simp only [Set.mem_setOf_eq, not_lt_of_le hm, not_false_eq_true, Set.indicator_of_not_mem, F]	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have hs {x : ℝ} (hx : x ≥ y) : Summable fun m ↦ (n + 1) ^ (x : ℂ) * term f x m := by refine (summable_mul_left_iff <\| cpow_natCast_add_one_ne_zero n _).mpr <\| LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_ simpa only [ofReal_re] using hay.trans_le <\| EReal.coe_le_coe_iff.mpr hx	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	conv => enter [3, 1]; rw [← add_zero (f _)]	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1) + 0))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	refine Tendsto.congr' (eventuallyEq_of_mem (s := {x \| y ≤ x}) (mem_atTop y) key).symm <\| tendsto_const_nhds.add ?_	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1) + 0))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m ⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rw [show (0 : ℂ) = tsum (fun _ : ℕ ↦ 0) from tsum_zero.symm]	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) ⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds 0)	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) ⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds (∑' (x : ℕ), 0))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	refine tendsto_tsum_of_dominated_convergence hys.norm hc <\| eventually_iff.mpr ?_	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) ⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds (∑' (x : ℕ), 0))	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) ⊢ {x \| ∀ (k : ℕ), ‖F x k‖ ≤ ‖F y k‖} ∈ atTop
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	filter_upwards [mem_atTop y] with y' hy' k	case intro.intro.intro f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) ⊢ {x \| ∀ (k : ℕ), ‖F x k‖ ≤ ‖F y k‖} ∈ atTop	case h f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ ⊢ ‖F y' k‖ ≤ ‖F y k‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rcases lt_or_le (n + 1) k with H \| H	case h f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ ⊢ ‖F y' k‖ ≤ ‖F y k‖	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ ‖F y' k‖ ≤ ‖F y k‖ case h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : k ≤ n + 1 ⊢ ‖F y' k‖ ≤ ‖F y k‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, not_lt_of_le hm, not_false_eq_true, Set.indicator_of_not_mem, F]	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x x : ℝ m : ℕ hm : m ≤ n + 1 ⊢ F x m = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rcases lt_trichotomy m (n + 1) with H \| rfl \| H	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : m < n + 1 ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m case inr.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ hm : n + 1 ≠ n + 1 ⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1) case inr.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : n + 1 < m ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, Nat.not_lt_of_gt H, not_false_eq_true, Set.indicator_of_not_mem, term, h m <\| Nat.lt_succ_iff.mp H, zero_div, ite_self, mul_zero, F]	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : m < n + 1 ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact (hm rfl).elim	case inr.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ hm : n + 1 ≠ n + 1 ⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, term, (n.zero_lt_succ.trans H).ne', ↓reduceIte, foo, F]	case inr.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : n + 1 < m ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	refine (summable_mul_left_iff <\| cpow_natCast_add_one_ne_zero n _).mpr <\| LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ abscissaOfAbsConv f < ↑(↑x).re
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simpa only [ofReal_re] using hay.trans_le <\| EReal.coe_le_coe_iff.mpr hx	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ abscissaOfAbsConv f < ↑(↑x).re	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	intro x hx	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m ⊢ ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rw [LSeries, ← tsum_mul_left, tsum_eq_add_tsum_ite (hs hx) (n + 1)]	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ ((↑n + 1) ^ ↑x * term f (↑x) (n + 1) + ∑' (n_1 : ℕ), if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = f (n + 1) + ∑' (m : ℕ), F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	congr	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ ((↑n + 1) ^ ↑x * term f (↑x) (n + 1) + ∑' (n_1 : ℕ), if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = f (n + 1) + ∑' (m : ℕ), F x m	case e_a f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * term f (↑x) (n + 1) = f (n + 1) case e_a.e_f f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (fun n_1 => if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = fun m => F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact pow_mul_term_eq f x n	case e_a f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * term f (↑x) (n + 1) = f (n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	ext m	case e_a.e_f f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (fun n_1 => if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = fun m => F x m	case e_a.e_f.h f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rcases eq_or_ne m (n + 1) with rfl \| hm	case e_a.e_f.h f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m	case e_a.e_f.h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (if n + 1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) (n + 1)) = F x (n + 1) case e_a.e_f.h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ hm : m ≠ n + 1 ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [↓reduceIte, hF₀ x le_rfl]	case e_a.e_f.h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (if n + 1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) (n + 1)) = F x (n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [hm, ↓reduceIte, ne_eq, not_false_eq_true, hF]	case e_a.e_f.h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ hm : m ≠ n + 1 ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	refine ((hs le_rfl).indicator {m \| n + 1 < m}).congr fun m ↦ ?_	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m ⊢ Summable (F y)	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ ⊢ {m \| n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	by_cases hm : n + 1 < m	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ ⊢ {m \| n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m	case pos f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : n + 1 < m ⊢ {m \| n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m case neg f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : ¬n + 1 < m ⊢ {m \| n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, hm, Set.indicator_of_mem, ne_eq, hm.ne', not_false_eq_true, hF]	case pos f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : n + 1 < m ⊢ {m \| n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, hm, not_false_eq_true, Set.indicator_of_not_mem, hF₀ _ (le_of_not_lt hm)]	case neg f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : ¬n + 1 < m ⊢ {m \| n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rcases lt_or_le (n + 1) k with H \| H	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k ⊢ Tendsto (fun x => F x k) atTop (nhds 0) case inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : k ≤ n + 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have H₀ : (0 : ℝ) ≤ k / (n + 1) := by positivity	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have H₀' : (0 : ℝ) ≤ (n + 1) / k := by positivity	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k ⊢ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have H₁ : (k / (n + 1) : ℂ) = (k / (n + 1) : ℝ) := by simp only [ofReal_div, ofReal_natCast, ofReal_add, ofReal_one]	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have H₂ : (n + 1) / k < (1 : ℝ) := (div_lt_one <\| by exact_mod_cast n.succ_pos.trans H).mpr <\| by exact_mod_cast H	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, F]	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k / (↑k / (↑n + 1)) ^ ↑x) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	conv => enter [1, x] rw [div_eq_mul_inv, H₁, ← ofReal_cpow H₀, ← ofReal_inv, ← Real.inv_rpow H₀, inv_div]	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k / (↑k / (↑n + 1)) ^ ↑x) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	conv => enter [3, 1]; rw [← mul_zero (f k)]	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds 0)	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds (f k * 0))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact (tendsto_rpow_atTop_of_base_lt_one _ (neg_one_lt_zero.trans_le H₀') H₂).ofReal.const_mul _	case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds (f k * 0))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	positivity	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k ⊢ 0 ≤ ↑k / (↑n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	positivity	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) ⊢ 0 ≤ (↑n + 1) / ↑k	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [ofReal_div, ofReal_natCast, ofReal_add, ofReal_one]	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k ⊢ ↑k / (↑n + 1) = ↑(↑k / (↑n + 1))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact_mod_cast n.succ_pos.trans H	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ 0 < ↑k	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact_mod_cast H	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ ↑n + 1 < ↑k	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [hF₀ _ H, tendsto_const_nhds_iff]	case inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : k ≤ n + 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, norm_div, Complex.norm_eq_abs, abs_cpow_real, map_div₀, abs_natCast, F]	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ ‖F y' k‖ ≤ ‖F y k‖	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	rw [← Nat.cast_one, ← Nat.cast_add, abs_natCast]	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	have hkn : 1 ≤ (k / (n + 1 :) : ℝ) := (one_le_div (by positivity)).mpr <\| by norm_cast; exact Nat.le_of_succ_le H	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k hkn : 1 ≤ ↑k / ↑(n + 1) ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact div_le_div_of_nonneg_left (Complex.abs.nonneg _) (rpow_pos_of_pos (zero_lt_one.trans_le hkn) _) <\| rpow_le_rpow_of_exponent_le hkn hy'	case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k hkn : 1 ≤ ↑k / ↑(n + 1) ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	positivity	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ 0 < ↑(n + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	norm_cast	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ ↑(n + 1) ≤ ↑k	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ n + 1 ≤ k
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	exact Nat.le_of_succ_le H	f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ n + 1 ≤ k	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_pow_mul_atTop	[61, 1]	[132, 44]	simp only [hF₀ _ H, norm_zero, le_refl]	case h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m \| n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : k ≤ n + 1 ⊢ ‖F y' k‖ ≤ ‖F y k‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	let F (n : ℕ) : ℂ := if n = 0 then 0 else f n	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	have hF₀ : F 0 = 0 := rfl	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := by simp only [hn, ↓reduceIte, F]	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	have ha' : abscissaOfAbsConv F < ⊤ := (abscissaOfAbsConv_congr hF).symm ▸ ha	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))	f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

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