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https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_atTop
[135, 1]
[145, 57]
simp_rw [← LSeries_congr _ hF]
f : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha' : abscissaOfAbsConv F < ⊀ ⊒ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
f : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ 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 β–Έ hFβ‚€) ha' using 1
f : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ 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 : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ 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 : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ 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 : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 n : β„• hn : n β‰  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 : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ ⊒ LSeries f = 0
case h f : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ s : β„‚ ⊒ 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 β–Έ fun H ↦ (H.trans_lt <| EReal.coe_lt_top _).false
case h f : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ s : β„‚ ⊒ 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 : β„• β†’ β„‚ ⊒ (fun x => LSeries f ↑x) =αΆ [atTop] 0 ↔ (βˆ€ (n : β„•), n β‰  0 β†’ f n = 0) ∨ abscissaOfAbsConv f = ⊀
case pos f : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ ⊒ (fun x => LSeries f ↑x) =αΆ [atTop] 0 case neg f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ ⊒ (fun x => LSeries f ↑x) =αΆ [atTop] 0 ↔ βˆ€ (n : β„•), n β‰  0 β†’ 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 : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ ⊒ (fun x => LSeries f ↑x) =αΆ [atTop] 0
case pos f : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ ⊒ βˆ€ (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 : β„• β†’ β„‚ h : abscissaOfAbsConv f = ⊀ ⊒ βˆ€ (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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ ⊒ (fun x => LSeries f ↑x) =αΆ [atTop] 0 ↔ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0 case neg.refine_2 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆ€ (n : β„•), n β‰  0 β†’ 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 : β„•) : β„‚ := if n = 0 then 0 else f n
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries_eventually_eq_zero_iff'
[154, 1]
[191, 37]
have hFβ‚€ : F 0 = 0 := rfl
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 ⊒ βˆ€ (n : β„•), n β‰  0 β†’ 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 : β„•} (hn : n β‰  0) : F n = f n := by simp only [hn, ↓reduceIte, F]
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries_eventually_eq_zero_iff'
[154, 1]
[191, 37]
suffices βˆ€ n, F n = 0 by peel hF with n hn h exact (this n β–Έ h).symm
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ⊒ βˆ€ (n : β„•), F n = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries_eventually_eq_zero_iff'
[154, 1]
[191, 37]
have ha : Β¬ abscissaOfAbsConv F = ⊀ := abscissaOfAbsConv_congr hF β–Έ h
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ⊒ βˆ€ (n : β„•), F n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ ⊒ βˆ€ (n : β„•), 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ ⊒ βˆ€ (n : β„•), F n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x ⊒ βˆ€ (n : β„•), 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 : β„•) : (fun x : ℝ ↦ (n ^ (x : β„‚)) * LSeries F x) =αΆ [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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x ⊒ βˆ€ (n : β„•), F n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 ⊒ βˆ€ (n : β„•), 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 ⊒ βˆ€ (n : β„•), F n = 0
case neg.refine_1 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ⊒ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ⊒ F n = 0
case neg.refine_1.ind f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„‚)) * 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ m < n, F m = 0 ⊒ F n = 0
case neg.refine_1.ind f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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, hFβ‚€, 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 n : β„• hn : n β‰  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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n this : βˆ€ (n : β„•), F n = 0 ⊒ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case h.h f : β„• β†’ β„‚ h✝ : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n this : βˆ€ (n : β„•), F n = 0 n : β„• hn : n β‰  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 β–Έ h).symm
case h.h f : β„• β†’ β„‚ h✝ : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n this : βˆ€ (n : β„•), F n = 0 n : β„• hn : n β‰  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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• ⊒ (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0
f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• ⊒ (fun x => ↑n ^ ↑x * LSeries f ↑x) =αΆ [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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• ⊒ (fun x => ↑n ^ ↑x * LSeries f ↑x) =αΆ [atTop] fun x => 0
f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆƒ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• ⊒ βˆƒ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆƒ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• ⊒ βˆƒ s ∈ atTop, Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s
case h f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆƒ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆƒ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆƒ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆƒ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : β„• 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ m < n, F m = 0 this : Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n)) ⊒ F n = 0
f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ m < n, F m = 0 this : Tendsto (fun x => 0) atTop (nhds (F n)) ⊒ F n = 0
f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 ih : βˆ€ m < 0, F m = 0 ⊒ Tendsto (fun x => ↑0 ^ ↑x * LSeries F ↑x) atTop (nhds (F 0))
case neg.refine_1.ind.zero f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 ih : βˆ€ 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, hFβ‚€, tendsto_const_nhds_iff]
case neg.refine_1.ind.zero f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : (fun x => LSeries f ↑x) =αΆ [atTop] 0 F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n hFβ‚€ : F 0 = 0 hF : βˆ€ {n : β„•}, n β‰  0 β†’ F n = f n ha : Β¬abscissaOfAbsConv F = ⊀ h' : βˆ€ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : βˆ€ (n : β„•), (fun x => ↑n ^ ↑x * LSeries F ↑x) =αΆ [atTop] fun x => 0 n : β„• ih : βˆ€ 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 _ : β„• ↦ (0 : β„‚)) = 0 from rfl, LSeries_zero, Pi.zero_apply]
case neg.refine_2 f : β„• β†’ β„‚ h : Β¬abscissaOfAbsConv f = ⊀ H : βˆ€ (n : β„•), n β‰  0 β†’ 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 : β„• β†’ β„‚ hf : f 0 = 0 ⊒ LSeries f = 0 ↔ f = 0 ∨ abscissaOfAbsConv f = ⊀
case pos f : β„• β†’ β„‚ hf : f 0 = 0 h : abscissaOfAbsConv f = ⊀ ⊒ LSeries f = 0 case neg f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬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 : β„• β†’ β„‚ 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 β–Έ LSeries_zero⟩
case neg f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ ⊒ LSeries f = 0 ↔ f = 0
case neg f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ f = 0
case a f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ f = 0 ↔ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0 case neg f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ (fun x => LSeries f ↑x) =αΆ [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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ f = 0 ↔ βˆ€ (n : β„•), n β‰  0 β†’ f n = 0
case a f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : βˆ€ (n : β„•), n β‰  0 β†’ 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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : βˆ€ (n : β„•), n β‰  0 β†’ f n = 0 ⊒ f = 0
case a.h.zero f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : βˆ€ (n : β„•), n β‰  0 β†’ f n = 0 ⊒ f 0 = 0 0 case a.h.succ f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : βˆ€ (n : β„•), n β‰  0 β†’ f n = 0 n✝ : β„• ⊒ 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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : f = 0 x✝¹ : β„• x✝ : x✝¹ β‰  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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : βˆ€ (n : β„•), n β‰  0 β†’ 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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 H' : βˆ€ (n : β„•), n β‰  0 β†’ f n = 0 n✝ : β„• ⊒ 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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ (fun x => LSeries f ↑x) =αΆ [Filter.atTop] 0
case neg f : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ (fun x => 0) =αΆ [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 : β„• β†’ β„‚ hf : f 0 = 0 h : Β¬abscissaOfAbsConv f = ⊀ H : LSeries f = 0 ⊒ (fun x => 0) =αΆ [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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x ⊒ (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : βˆƒ a, βˆ€ b β‰₯ a, LSeries f ↑b = LSeries g ↑b ⊒ βˆƒ a, βˆ€ b β‰₯ 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 ⟨xβ‚€, hxβ‚€βŸ© := h
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : βˆƒ a, βˆ€ b β‰₯ a, LSeries f ↑b = LSeries g ↑b ⊒ βˆƒ a, βˆ€ b β‰₯ a, LSeries (f - g) ↑b = 0 b
case intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b ⊒ βˆƒ a, βˆ€ b β‰₯ 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, hyfβ‚‚βŸ© := exists_between hf
case intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b ⊒ βˆƒ a, βˆ€ b β‰₯ a, LSeries (f - g) ↑b = 0 b
case intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyfβ‚‚ : yf < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ 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, hygβ‚‚βŸ© := exists_between hg
case intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyfβ‚‚ : yf < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ a, LSeries (f - g) ↑b = 0 b
case intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyfβ‚‚ : yf < ⊀ yg : EReal hyg₁ : abscissaOfAbsConv g < yg hygβ‚‚ : yg < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ 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 ⟨hyfβ‚‚.ne, ((OrderBot.bot_le _).trans_lt hyf₁).ne'⟩
case intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyfβ‚‚ : yf < ⊀ yg : EReal hyg₁ : abscissaOfAbsConv g < yg hygβ‚‚ : yg < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ a, LSeries (f - g) ↑b = 0 b
case intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yg : EReal hyg₁ : abscissaOfAbsConv g < yg hygβ‚‚ : yg < ⊀ yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ 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 ⟨hygβ‚‚.ne, ((OrderBot.bot_le _).trans_lt hyg₁).ne'⟩
case intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yg : EReal hyg₁ : abscissaOfAbsConv g < yg hygβ‚‚ : yg < ⊀ yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ a, LSeries (f - g) ↑b = 0 b
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ 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 xβ‚€ (max yf yg), fun x hx ↦ ?_⟩
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ ⊒ βˆƒ a, βˆ€ b β‰₯ a, LSeries (f - g) ↑b = 0 b
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 β–Έ hyf₁.trans_le ?_ refine (le_max_left _ (yg : EReal)).trans <| (le_max_right (xβ‚€ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ LSeries (f - g) ↑x = 0 x
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 β–Έ hyg₁.trans_le ?_ refine (le_max_right (yf : EReal) _).trans <| (le_max_right (xβ‚€ : EReal) _).trans ?_ simpa only [max_le_iff, EReal.coe_le_coe_iff] using hx
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ LSeries (f - g) ↑x = 0 x
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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, hxβ‚€ x <| (le_max_left ..).trans hx, sub_self, Pi.zero_apply]
case intro.intro.intro.intro.intro.intro.intro f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 β–Έ hyf₁.trans_le ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ LSeriesSummable f ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 (xβ‚€ : EReal) _).trans ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ ↑yf ≀ ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ max (↑xβ‚€) (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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) ⊒ max (↑xβ‚€) (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 β–Έ hyg₁.trans_le ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ LSeriesSummable g ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (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 (xβ‚€ : EReal) _).trans ?_
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ ↑yg ≀ ↑x
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ max (↑xβ‚€) (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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ xβ‚€ : ℝ hxβ‚€ : βˆ€ b β‰₯ xβ‚€, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyfβ‚‚ : ↑yf < ⊀ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hygβ‚‚ : ↑yg < ⊀ x : ℝ hx : x β‰₯ max xβ‚€ (max yf yg) Hf : LSeriesSummable f ↑x ⊒ max (↑xβ‚€) (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) =αΆ [atTop] (0 : ℝ β†’ β„‚) := LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq hf hg h
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 ⊒ f n = g n
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [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) β‰  ⊀ := lt_top_iff_ne_top.mp <| (abscissaOfAbsConv_sub_le f g).trans_lt <| max_lt hf hg
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0 ⊒ f n = g n
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0 ha : abscissaOfAbsConv (f - g) β‰  ⊀ ⊒ 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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ h : (fun x => LSeries f ↑x) =αΆ [atTop] fun x => LSeries g ↑x n : β„• hn : n β‰  0 hsub : (fun x => LSeries (f - g) ↑x) =αΆ [atTop] 0 ha : abscissaOfAbsConv (f - g) β‰  ⊀ ⊒ 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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ H : LSeries f = LSeries g n : β„• hn : n β‰  0 ⊒ f n = g n
f g : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ H : LSeries f = LSeries g n : β„• hn : n β‰  0 ⊒ (fun x => LSeries f ↑x) =αΆ [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 : β„• β†’ β„‚ hf : abscissaOfAbsConv f < ⊀ hg : abscissaOfAbsConv g < ⊀ H : LSeries f = LSeries g n : β„• hn : n β‰  0 ⊒ (fun x => LSeries f ↑x) =αΆ [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 : β„• z : β„‚ H : ↑n + 1 = 0 ∧ z β‰  0 ⊒ False
n : β„• z : β„‚ H : False ∧ Β¬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 : β„• z : β„‚ H : False ∧ Β¬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 : (β„• β†’ β„‚) β†’ (β„• β†’ β„‚) β†’ β„• β†’ β„‚ hF : βˆ€ {f g : β„• β†’ β„‚} {s : β„‚}, LSeriesSummable f s β†’ LSeriesSummable g s β†’ LSeriesSummable (F f g) s f g : β„• β†’ β„‚ ⊒ abscissaOfAbsConv (F f g) ≀ max (abscissaOfAbsConv f) (abscissaOfAbsConv g)
case refine_1 F : (β„• β†’ β„‚) β†’ (β„• β†’ β„‚) β†’ β„• β†’ β„‚ hF : βˆ€ {f g : β„• β†’ β„‚} {s : β„‚}, LSeriesSummable f s β†’ LSeriesSummable g s β†’ LSeriesSummable (F f g) s f g : β„• β†’ β„‚ x : ℝ hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x ⊒ LSeriesSummable f ↑x case refine_2 F : (β„• β†’ β„‚) β†’ (β„• β†’ β„‚) β†’ β„• β†’ β„‚ hF : βˆ€ {f g : β„• β†’ β„‚} {s : β„‚}, LSeriesSummable f s β†’ LSeriesSummable g s β†’ LSeriesSummable (F f g) s f g : β„• β†’ β„‚ 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 β–Έ (le_max_left ..).trans_lt hx
case refine_1 F : (β„• β†’ β„‚) β†’ (β„• β†’ β„‚) β†’ β„• β†’ β„‚ hF : βˆ€ {f g : β„• β†’ β„‚} {s : β„‚}, LSeriesSummable f s β†’ LSeriesSummable g s β†’ LSeriesSummable (F f g) s f g : β„• β†’ β„‚ 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 β–Έ (le_max_right ..).trans_lt hx
case refine_2 F : (β„• β†’ β„‚) β†’ (β„• β†’ β„‚) β†’ β„• β†’ β„‚ hF : βˆ€ {f g : β„• β†’ β„‚} {s : β„‚}, LSeriesSummable f s β†’ LSeriesSummable g s β†’ LSeriesSummable (F f g) s f g : β„• β†’ β„‚ 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 : β„• z : β„‚ x : ℝ ⊒ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
m n : β„• z : β„‚ 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 : β„• z : β„‚ x : ℝ Hm : 0 ≀ ↑m ⊒ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
m n : β„• z : β„‚ 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 : β„• z : β„‚ x : ℝ Hm : 0 ≀ ↑m Hn : 0 ≀ (↑n + 1)⁻¹ ⊒ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
m n : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„‚)⁻¹ = (n + 1 : ℝ)⁻¹ by simp only [ofReal_inv, ofReal_add, ofReal_natCast, ofReal_one], show (n + 1 : β„‚) = (n + 1 : ℝ) by norm_cast, show (m : β„‚) = (m : ℝ) by norm_cast, mul_cpow_ofReal_nonneg Hm Hn, mul_inv, mul_comm]
case e_a m n : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„‚) = (-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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• z : β„‚ 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 : β„• β†’ β„‚ s : β„‚ n : β„• ⊒ (↑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 : β„• β†’ β„‚ n : β„• h : βˆ€ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ 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) : β„‚) ^ (x : β„‚))
case intro.intro.intro f : β„• β†’ β„‚ n : β„• h : βˆ€ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := 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 hFβ‚€ (x : ℝ) {m : β„•} (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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ 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 β‰₯ y) : Summable fun m ↦ (n + 1) ^ (x : β„‚) * 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m ⊒ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
case intro.intro.intro f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m ⊒ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1) + 0))
case intro.intro.intro f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m ⊒ Tendsto (fun x => βˆ‘' (m : β„•), 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 : β„‚) = tsum (fun _ : β„• ↦ 0) from tsum_zero.symm]
case intro.intro.intro f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) ⊒ Tendsto (fun x => βˆ‘' (m : β„•), F x m) atTop (nhds 0)
case intro.intro.intro f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) ⊒ Tendsto (fun x => βˆ‘' (m : β„•), F x m) atTop (nhds (βˆ‘' (x : β„•), 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) ⊒ Tendsto (fun x => βˆ‘' (m : β„•), F x m) atTop (nhds (βˆ‘' (x : β„•), 0))
case intro.intro.intro f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) ⊒ {x | βˆ€ (k : β„•), β€–F x kβ€– ≀ β€–F y kβ€–} ∈ 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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) ⊒ {x | βˆ€ (k : β„•), β€–F x kβ€– ≀ β€–F y kβ€–} ∈ atTop
case h f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≀ y' k : β„• ⊒ β€–F y' kβ€– ≀ β€–F y kβ€–
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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≀ y' k : β„• ⊒ β€–F y' kβ€– ≀ β€–F y kβ€–
case h.inl f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≀ y' k : β„• H : n + 1 < k ⊒ β€–F y' kβ€– ≀ β€–F y kβ€– case h.inr f : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hFβ‚€ : βˆ€ (x : ℝ) {m : β„•}, m ≀ n + 1 β†’ F x m = 0 hF : βˆ€ (x : ℝ) {m : β„•}, m β‰  n + 1 β†’ F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : βˆ€ {x : ℝ}, x β‰₯ y β†’ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m hys : Summable (F y) hc : βˆ€ (k : β„•), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≀ y' k : β„• H : k ≀ n + 1 ⊒ β€–F y' kβ€– ≀ β€–F y kβ€–
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 : β„• β†’ β„‚ n : β„• h : βˆ€ m ≀ n, f m = 0 ha : abscissaOfAbsConv f < ⊀ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊀ F : ℝ β†’ β„• β†’ β„‚ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x x : ℝ m : β„• hm : m ≀ n + 1 ⊒ F x m = 0
no goals