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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/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	simp_rw [← LSeries_congr _ hF]	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))	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 (fun {n} => F n) ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	convert LSeries.tendsto_pow_mul_atTop (n := 0) (fun _ hm ↦ Nat.le_zero.mp hm ▸ hF₀) ha' using 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 (fun {n} => F n) ↑x) atTop (nhds (f 1))	case h.e'_3 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 < ⊤ ⊢ (fun x => LSeries (fun {n} => F n) ↑x) = fun x => (↑0 + 1) ^ ↑x * LSeries F ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	simp only [Nat.cast_zero, zero_add, one_cpow, one_mul]	case h.e'_3 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 < ⊤ ⊢ (fun x => LSeries (fun {n} => F n) ↑x) = fun x => (↑0 + 1) ^ ↑x * LSeries F ↑x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.tendsto_atTop	[135, 1]	[145, 57]	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 n : ℕ hn : n ≠ 0 ⊢ F n = f n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_of_abscissaOfAbsConv_eq_top	[147, 1]	[151, 57]	ext s	f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0	case h f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ s : ℂ ⊢ LSeries f s = 0 s
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_of_abscissaOfAbsConv_eq_top	[147, 1]	[151, 57]	exact LSeries.eq_zero_of_not_LSeriesSummable f s <\| mt LSeriesSummable.abscissaOfAbsConv_le <\| h ▸ fun H ↦ (H.trans_lt <\| EReal.coe_lt_top _).false	case h f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ s : ℂ ⊢ LSeries f s = 0 s	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	by_cases h : abscissaOfAbsConv f = ⊤ <;> simp only [h, or_true, or_false, iff_true]	f : ℕ → ℂ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ↔ (∀ (n : ℕ), n ≠ 0 → f n = 0) ∨ abscissaOfAbsConv f = ⊤	case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 case neg f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	refine eventually_of_forall ?_	case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0	case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ ∀ (x : ℝ), (fun x => LSeries f ↑x) x = 0 x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [LSeries_eq_zero_of_abscissaOfAbsConv_eq_top h, Pi.zero_apply, forall_const]	case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ ∀ (x : ℝ), (fun x => LSeries f ↑x) x = 0 x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	refine ⟨fun H ↦ ?_, fun H ↦ eventually_of_forall fun x ↦ ?_⟩	case neg f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0 case neg.refine_2 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∀ (n : ℕ), n ≠ 0 → f n = 0 x : ℝ ⊢ (fun x => LSeries f ↑x) x = 0 x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	let F (n : ℕ) : ℂ := if n = 0 then 0 else f n	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	have hF₀ : F 0 = 0 := rfl	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := by simp only [hn, ↓reduceIte, F]	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	suffices ∀ n, F n = 0 by peel hF with n hn h exact (this n ▸ h).symm	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 ⊢ ∀ (n : ℕ), F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	have ha : ¬ abscissaOfAbsConv F = ⊤ := abscissaOfAbsConv_congr hF ▸ h	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 ⊢ ∀ (n : ℕ), F n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ ⊢ ∀ (n : ℕ), F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	have h' (x : ℝ) : LSeries F x = LSeries f x := LSeries_congr x hF	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ ⊢ ∀ (n : ℕ), F n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x ⊢ ∀ (n : ℕ), F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	have H' (n : ℕ) : (fun x : ℝ ↦ (n ^ (x : ℂ)) * LSeries F x) =ᶠ[atTop] (fun _ ↦ 0) := by simp only [h'] rw [eventuallyEq_iff_exists_mem] at H ⊢ peel H with s hs refine ⟨hs.1, fun x hx ↦ ?_⟩ simp only [hs.2 hx, Pi.zero_apply, mul_zero]	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x ⊢ ∀ (n : ℕ), F n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ⊢ ∀ (n : ℕ), F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	intro n	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ⊢ ∀ (n : ℕ), F n = 0	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ⊢ F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	induction' n using Nat.strongInductionOn with n ih	case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ⊢ F n = 0	case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	suffices Tendsto (fun x : ℝ ↦ (n ^ (x : ℂ)) * LSeries F x) atTop (nhds (F n)) by replace this := this.congr' <\| H' n simp only [tendsto_const_nhds_iff] at this exact this.symm	case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ F n = 0	case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	cases n with \| zero => refine Tendsto.congr' (H' 0).symm ?_ simp only [zero_eq, hF₀, tendsto_const_nhds_iff] \| succ n => simp only [succ_eq_add_one, cast_add, cast_one] exact LSeries.tendsto_pow_mul_atTop (fun m hm ↦ ih m <\| lt_succ_of_le hm) <\| Ne.lt_top ha	case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [hn, ↓reduceIte, F]	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 n : ℕ hn : n ≠ 0 ⊢ F n = f n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	peel hF with n hn h	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 this : ∀ (n : ℕ), F n = 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0	case h.h f : ℕ → ℂ h✝ : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 this : ∀ (n : ℕ), F n = 0 n : ℕ hn : n ≠ 0 h : F n = f n ⊢ f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	exact (this n ▸ h).symm	case h.h f : ℕ → ℂ h✝ : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 this : ∀ (n : ℕ), F n = 0 n : ℕ hn : n ≠ 0 h : F n = f n ⊢ f n = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [h']	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) =ᶠ[atTop] fun x => 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	rw [eventuallyEq_iff_exists_mem] at H ⊢	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) =ᶠ[atTop] fun x => 0	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ ∃ s ∈ atTop, Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	peel H with s hs	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ ∃ s ∈ atTop, Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s	case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s ⊢ s ∈ atTop ∧ Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	refine ⟨hs.1, fun x hx ↦ ?_⟩	case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s ⊢ s ∈ atTop ∧ Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s	case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s x : ℝ hx : x ∈ s ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) x = (fun x => 0) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [hs.2 hx, Pi.zero_apply, mul_zero]	case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s x : ℝ hx : x ∈ s ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) x = (fun x => 0) x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	replace this := this.congr' <\| H' n	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n)) ⊢ F n = 0	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : Tendsto (fun x => 0) atTop (nhds (F n)) ⊢ F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [tendsto_const_nhds_iff] at this	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : Tendsto (fun x => 0) atTop (nhds (F n)) ⊢ F n = 0	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : 0 = F n ⊢ F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	exact this.symm	f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : 0 = F n ⊢ F n = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	refine Tendsto.congr' (H' 0).symm ?_	case neg.refine_1.ind.zero f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ih : ∀ m < 0, F m = 0 ⊢ Tendsto (fun x => ↑0 ^ ↑x * LSeries F ↑x) atTop (nhds (F 0))	case neg.refine_1.ind.zero f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ih : ∀ m < 0, F m = 0 ⊢ Tendsto (fun x => 0) atTop (nhds (F 0))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [zero_eq, hF₀, tendsto_const_nhds_iff]	case neg.refine_1.ind.zero f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ih : ∀ m < 0, F m = 0 ⊢ Tendsto (fun x => 0) atTop (nhds (F 0))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [succ_eq_add_one, cast_add, cast_one]	case neg.refine_1.ind.succ f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n + 1, F m = 0 ⊢ Tendsto (fun x => ↑(n + 1) ^ ↑x * LSeries F ↑x) atTop (nhds (F (n + 1)))	case neg.refine_1.ind.succ f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n + 1, F 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_eventually_eq_zero_iff'	[154, 1]	[191, 37]	exact LSeries.tendsto_pow_mul_atTop (fun m hm ↦ ih m <\| lt_succ_of_le hm) <\| Ne.lt_top ha	case neg.refine_1.ind.succ f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 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 = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n + 1, F m = 0 ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries F ↑x) atTop (nhds (F (n + 1)))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eventually_eq_zero_iff'	[154, 1]	[191, 37]	simp only [LSeries_congr x fun {n} ↦ H n, show (fun _ : ℕ ↦ (0 : ℂ)) = 0 from rfl, LSeries_zero, Pi.zero_apply]	case neg.refine_2 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∀ (n : ℕ), n ≠ 0 → f n = 0 x : ℝ ⊢ (fun x => LSeries f ↑x) x = 0 x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	by_cases h : abscissaOfAbsConv f = ⊤ <;> simp only [h, or_true, or_false, iff_true]	f : ℕ → ℂ hf : f 0 = 0 ⊢ LSeries f = 0 ↔ f = 0 ∨ abscissaOfAbsConv f = ⊤	case pos f : ℕ → ℂ hf : f 0 = 0 h : abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0 case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0 ↔ f = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	exact LSeries_eq_zero_of_abscissaOfAbsConv_eq_top h	case pos f : ℕ → ℂ hf : f 0 = 0 h : abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	refine ⟨fun H ↦ ?_, fun H ↦ H ▸ LSeries_zero⟩	case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0 ↔ f = 0	case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	convert (LSeries_eventually_eq_zero_iff'.mp ?_).resolve_right h	case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0	case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0 case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	refine ⟨fun H' _ _ ↦ by rw [H', Pi.zero_apply], fun H' ↦ ?_⟩	case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0	case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	ext ⟨- \| m⟩	case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f = 0	case a.h.zero f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f 0 = 0 0 case a.h.succ f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 n✝ : ℕ ⊢ f (n✝ + 1) = 0 (n✝ + 1)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	rw [H', Pi.zero_apply]	f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : f = 0 x✝¹ : ℕ x✝ : x✝¹ ≠ 0 ⊢ f x✝¹ = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	simp only [zero_eq, hf, Pi.zero_apply]	case a.h.zero f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f 0 = 0 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	simp only [ne_eq, succ_ne_zero, not_false_eq_true, H', Pi.zero_apply]	case a.h.succ f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 n✝ : ℕ ⊢ f (n✝ + 1) = 0 (n✝ + 1)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	simp only [H, Pi.zero_apply]	case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] 0	case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => 0) =ᶠ[Filter.atTop] 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_zero_iff	[194, 1]	[207, 36]	exact Filter.EventuallyEq.rfl	case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => 0) =ᶠ[Filter.atTop] 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	rw [EventuallyEq, eventually_atTop] at h ⊢	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x ⊢ (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : ∃ a, ∀ b ≥ a, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	obtain ⟨x₀, hx₀⟩ := h	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : ∃ a, ∀ b ≥ a, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b	case intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	obtain ⟨yf, hyf₁, hyf₂⟩ := exists_between hf	case intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b	case intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	obtain ⟨yg, hyg₁, hyg₂⟩ := exists_between hg	case intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b	case intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	lift yf to ℝ using ⟨hyf₂.ne, ((OrderBot.bot_le _).trans_lt hyf₁).ne'⟩	case intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b	case intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	lift yg to ℝ using ⟨hyg₂.ne, ((OrderBot.bot_le _).trans_lt hyg₁).ne'⟩	case intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	refine ⟨max x₀ (max yf yg), fun x hx ↦ ?_⟩	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ LSeries (f - g) ↑x = 0 x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	have Hf : LSeriesSummable f x := by refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <\| (ofReal_re x).symm ▸ hyf₁.trans_le ?_ refine (le_max_left _ (yg : EReal)).trans <\| (le_max_right (x₀ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ LSeries (f - g) ↑x = 0 x	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ LSeries (f - g) ↑x = 0 x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	have Hg : LSeriesSummable g x := by refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <\| (ofReal_re x).symm ▸ hyg₁.trans_le ?_ refine (le_max_right (yf : EReal) _).trans <\| (le_max_right (x₀ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ LSeries (f - g) ↑x = 0 x	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x Hg : LSeriesSummable g ↑x ⊢ LSeries (f - g) ↑x = 0 x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	rw [LSeries_sub Hf Hg, hx₀ x <\| (le_max_left ..).trans hx, sub_self, Pi.zero_apply]	case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x Hg : LSeriesSummable g ↑x ⊢ LSeries (f - g) ↑x = 0 x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <\| (ofReal_re x).symm ▸ hyf₁.trans_le ?_	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ LSeriesSummable f ↑x	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ ↑yf ≤ ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	refine (le_max_left _ (yg : EReal)).trans <\| (le_max_right (x₀ : EReal) _).trans ?_	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ ↑yf ≤ ↑x	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	refine LSeriesSummable_of_abscissaOfAbsConv_lt_re <\| (ofReal_re x).symm ▸ hyg₁.trans_le ?_	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ LSeriesSummable g ↑x	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ ↑yg ≤ ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	refine (le_max_right (yf : EReal) _).trans <\| (le_max_right (x₀ : EReal) _).trans ?_	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ ↑yg ≤ ↑x	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq	[210, 1]	[231, 86]	simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) Hf : LSeriesSummable f ↑x ⊢ max (↑x₀) (max ↑yf ↑yg) ≤ ↑x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.eq_of_LSeries_eventually_eq	[234, 1]	[245, 74]	have hsub : (fun x : ℝ ↦ LSeries (f - g) x) =ᶠ[atTop] (0 : ℝ → ℂ) := LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq hf hg h	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 ⊢ f n = g n	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ⊢ f n = g n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.eq_of_LSeries_eventually_eq	[234, 1]	[245, 74]	have ha : abscissaOfAbsConv (f - g) ≠ ⊤ := lt_top_iff_ne_top.mp <\| (abscissaOfAbsConv_sub_le f g).trans_lt <\| max_lt hf hg	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ⊢ f n = g n	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ha : abscissaOfAbsConv (f - g) ≠ ⊤ ⊢ f n = g n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries.eq_of_LSeries_eventually_eq	[234, 1]	[245, 74]	simpa only [Pi.sub_apply, sub_eq_zero] using (LSeries_eventually_eq_zero_iff'.mp hsub).resolve_right ha n hn	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x n : ℕ hn : n ≠ 0 hsub : (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0 ha : abscissaOfAbsConv (f - g) ≠ ⊤ ⊢ f n = g n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_iff_of_abscissaOfAbsConv_lt_top	[247, 1]	[254, 58]	refine eq_of_LSeries_eventually_eq hf hg ?_ hn	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ H : LSeries f = LSeries g n : ℕ hn : n ≠ 0 ⊢ f n = g n	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ H : LSeries f = LSeries g n : ℕ hn : n ≠ 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] fun x => LSeries g ↑x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/LSeriesUnique.lean	LSeries_eq_iff_of_abscissaOfAbsConv_lt_top	[247, 1]	[254, 58]	exact Filter.eventually_of_forall fun x ↦ congr_fun H x	f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ H : LSeries f = LSeries g n : ℕ hn : n ≠ 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] fun x => LSeries g ↑x	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

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