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https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
rcases lt_trichotomy m (n + 1) with H | rfl | H
f : 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 x : ℝ m : β„• hm : m β‰  n + 1 ⊒ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
case 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 x : ℝ m : β„• hm : m β‰  n + 1 H : m < n + 1 ⊒ F x m = (↑n + 1) ^ ↑x * term f (↑x) m case inr.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 x : ℝ hm : n + 1 β‰  n + 1 ⊒ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1) case inr.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 x : ℝ m : β„• hm : m β‰  n + 1 H : n + 1 < m ⊒ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [Set.mem_setOf_eq, Nat.not_lt_of_gt H, not_false_eq_true, Set.indicator_of_not_mem, term, h m <| Nat.lt_succ_iff.mp H, zero_div, ite_self, mul_zero, F]
case inl f : 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 x : ℝ m : β„• hm : m β‰  n + 1 H : m < n + 1 ⊒ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact (hm rfl).elim
case inr.inl f : 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 x : ℝ hm : n + 1 β‰  n + 1 ⊒ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, term, (n.zero_lt_succ.trans H).ne', ↓reduceIte, foo, F]
case inr.inr f : 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 x : ℝ m : β„• hm : m β‰  n + 1 H : n + 1 < m ⊒ F x m = (↑n + 1) ^ ↑x * term f (↑x) m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
refine (summable_mul_left_iff <| cpow_natCast_add_one_ne_zero n _).mpr <| LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_
f : 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 x : ℝ hx : x β‰₯ y ⊒ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
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 x : ℝ hx : x β‰₯ y ⊒ abscissaOfAbsConv f < ↑(↑x).re
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simpa only [ofReal_re] using hay.trans_le <| EReal.coe_le_coe_iff.mpr hx
f : 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 x : ℝ hx : x β‰₯ y ⊒ abscissaOfAbsConv f < ↑(↑x).re
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
intro x hx
f : 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 ⊒ βˆ€ x β‰₯ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m
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 x : ℝ hx : x β‰₯ y ⊒ (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
rw [LSeries, ← tsum_mul_left, tsum_eq_add_tsum_ite (hs hx) (n + 1)]
f : 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 x : ℝ hx : x β‰₯ y ⊒ (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + βˆ‘' (m : β„•), F x m
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 x : ℝ hx : x β‰₯ y ⊒ ((↑n + 1) ^ ↑x * term f (↑x) (n + 1) + βˆ‘' (n_1 : β„•), if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = f (n + 1) + βˆ‘' (m : β„•), F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
congr
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 x : ℝ hx : x β‰₯ y ⊒ ((↑n + 1) ^ ↑x * term f (↑x) (n + 1) + βˆ‘' (n_1 : β„•), if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = f (n + 1) + βˆ‘' (m : β„•), F x m
case e_a 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 x : ℝ hx : x β‰₯ y ⊒ (↑n + 1) ^ ↑x * term f (↑x) (n + 1) = f (n + 1) case e_a.e_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 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 x : ℝ hx : x β‰₯ y ⊒ (fun n_1 => if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = fun m => F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact pow_mul_term_eq f x n
case e_a f : 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 x : ℝ hx : x β‰₯ y ⊒ (↑n + 1) ^ ↑x * term f (↑x) (n + 1) = f (n + 1)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
ext m
case e_a.e_f f : 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 x : ℝ hx : x β‰₯ y ⊒ (fun n_1 => if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = fun m => F x m
case e_a.e_f.h f : 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 x : ℝ hx : x β‰₯ y m : β„• ⊒ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
rcases eq_or_ne m (n + 1) with rfl | hm
case e_a.e_f.h f : 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 x : ℝ hx : x β‰₯ y m : β„• ⊒ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m
case e_a.e_f.h.inl f : 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 x : ℝ hx : x β‰₯ y ⊒ (if n + 1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) (n + 1)) = F x (n + 1) case e_a.e_f.h.inr f : 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 x : ℝ hx : x β‰₯ y m : β„• hm : m β‰  n + 1 ⊒ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [↓reduceIte, hFβ‚€ x le_rfl]
case e_a.e_f.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 x : ℝ hx : x β‰₯ y ⊒ (if n + 1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) (n + 1)) = F x (n + 1)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [hm, ↓reduceIte, ne_eq, not_false_eq_true, hF]
case e_a.e_f.h.inr f : 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 x : ℝ hx : x β‰₯ y m : β„• hm : m β‰  n + 1 ⊒ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
refine ((hs le_rfl).indicator {m | n + 1 < m}).congr fun m ↦ ?_
f : 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 ⊒ Summable (F y)
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 m : β„• ⊒ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
by_cases hm : n + 1 < m
f : 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 m : β„• ⊒ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
case pos 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 m : β„• hm : n + 1 < m ⊒ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m case neg f : 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 m : β„• hm : Β¬n + 1 < m ⊒ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [Set.mem_setOf_eq, hm, Set.indicator_of_mem, ne_eq, hm.ne', not_false_eq_true, hF]
case pos f : 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 m : β„• hm : n + 1 < m ⊒ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [Set.mem_setOf_eq, hm, not_false_eq_true, Set.indicator_of_not_mem, hFβ‚€ _ (le_of_not_lt hm)]
case neg 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 m : β„• hm : Β¬n + 1 < m ⊒ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
rcases lt_or_le (n + 1) k with H | H
f : 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) k : β„• ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k ⊒ Tendsto (fun x => F x k) atTop (nhds 0) case 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) k : β„• H : k ≀ n + 1 ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
have Hβ‚€ : (0 : ℝ) ≀ k / (n + 1) := by positivity
case 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) k : β„• H : n + 1 < k ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
have Hβ‚€' : (0 : ℝ) ≀ (n + 1) / k := by positivity
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
have H₁ : (k / (n + 1) : β„‚) = (k / (n + 1) : ℝ) := by simp only [ofReal_div, ofReal_natCast, ofReal_add, ofReal_one]
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
have Hβ‚‚ : (n + 1) / k < (1 : ℝ) := (div_lt_one <| by exact_mod_cast n.succ_pos.trans H).mpr <| by exact_mod_cast H
case inl f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, F]
case inl f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => f k / (↑k / (↑n + 1)) ^ ↑x) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
conv => enter [1, x] rw [div_eq_mul_inv, H₁, ← ofReal_cpow Hβ‚€, ← ofReal_inv, ← Real.inv_rpow Hβ‚€, inv_div]
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => f k / (↑k / (↑n + 1)) ^ ↑x) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
conv => enter [3, 1]; rw [← mul_zero (f k)]
case inl f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds 0)
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds (f k * 0))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact (tendsto_rpow_atTop_of_base_lt_one _ (neg_one_lt_zero.trans_le Hβ‚€') Hβ‚‚).ofReal.const_mul _
case 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) Hβ‚‚ : (↑n + 1) / ↑k < 1 ⊒ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds (f k * 0))
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
positivity
f : 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) k : β„• H : n + 1 < k ⊒ 0 ≀ ↑k / (↑n + 1)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
positivity
f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) ⊒ 0 ≀ (↑n + 1) / ↑k
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [ofReal_div, ofReal_natCast, ofReal_add, ofReal_one]
f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k ⊒ ↑k / (↑n + 1) = ↑(↑k / (↑n + 1))
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact_mod_cast n.succ_pos.trans H
f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊒ 0 < ↑k
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact_mod_cast H
f : 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) k : β„• H : n + 1 < k Hβ‚€ : 0 ≀ ↑k / (↑n + 1) Hβ‚€' : 0 ≀ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊒ ↑n + 1 < ↑k
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [hFβ‚€ _ H, tendsto_const_nhds_iff]
case 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) k : β„• H : k ≀ n + 1 ⊒ Tendsto (fun x => F x k) atTop (nhds 0)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, norm_div, Complex.norm_eq_abs, abs_cpow_real, map_divβ‚€, abs_natCast, F]
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.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 ⊒ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y' ≀ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
rw [← Nat.cast_one, ← Nat.cast_add, abs_natCast]
case h.inl f : 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 ⊒ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y' ≀ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y
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 ⊒ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≀ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
have hkn : 1 ≀ (k / (n + 1 :) : ℝ) := (one_le_div (by positivity)).mpr <| by norm_cast; exact Nat.le_of_succ_le H
case h.inl f : 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 ⊒ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≀ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y
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 hkn : 1 ≀ ↑k / ↑(n + 1) ⊒ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≀ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact div_le_div_of_nonneg_left (Complex.abs.nonneg _) (rpow_pos_of_pos (zero_lt_one.trans_le hkn) _) <| rpow_le_rpow_of_exponent_le hkn hy'
case h.inl f : 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 hkn : 1 ≀ ↑k / ↑(n + 1) ⊒ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≀ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
positivity
f : 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 ⊒ 0 < ↑(n + 1)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
norm_cast
f : 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 ⊒ ↑(n + 1) ≀ ↑k
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 ⊒ n + 1 ≀ k
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
exact Nat.le_of_succ_le H
f : 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 ⊒ n + 1 ≀ k
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_pow_mul_atTop
[61, 1]
[132, 44]
simp only [hFβ‚€ _ H, norm_zero, le_refl]
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€–
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_atTop
[135, 1]
[145, 57]
let F (n : β„•) : β„‚ := if n = 0 then 0 else f n
f : β„• β†’ β„‚ 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 ⊒ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_atTop
[135, 1]
[145, 57]
have hFβ‚€ : F 0 = 0 := rfl
f : β„• β†’ β„‚ ha : abscissaOfAbsConv f < ⊀ F : β„• β†’ β„‚ := fun n => if n = 0 then 0 else f n ⊒ 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 ⊒ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_atTop
[135, 1]
[145, 57]
have hF {n : β„•} (hn : n β‰  0) : F n = f n := by 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 ⊒ 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 ⊒ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/LSeriesUnique.lean
LSeries.tendsto_atTop
[135, 1]
[145, 57]
have ha' : abscissaOfAbsConv F < ⊀ := (abscissaOfAbsConv_congr hF).symm β–Έ ha
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 ⊒ 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 f ↑x) atTop (nhds (f 1))
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