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like 21

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

rcases lt_trichotomy m (n + 1) with H | rfl | H

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : m < n + 1 ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m case inr.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ hm : n + 1 ≠ n + 1 ⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1) case inr.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : n + 1 < m ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [Set.mem_setOf_eq, Nat.not_lt_of_gt H, not_false_eq_true, Set.indicator_of_not_mem, term, h m <| Nat.lt_succ_iff.mp H, zero_div, ite_self, mul_zero, F]

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : m < n + 1 ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact (hm rfl).elim

case inr.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ hm : n + 1 ≠ n + 1 ⊢ F x (n + 1) = (↑n + 1) ^ ↑x * term f (↑x) (n + 1)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, term, (n.zero_lt_succ.trans H).ne', ↓reduceIte, foo, F]

case inr.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 x : ℝ m : ℕ hm : m ≠ n + 1 H : n + 1 < m ⊢ F x m = (↑n + 1) ^ ↑x * term f (↑x) m

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

refine (summable_mul_left_iff <| cpow_natCast_add_one_ne_zero n _).mpr <| LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ abscissaOfAbsConv f < ↑(↑x).re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simpa only [ofReal_re] using hay.trans_le <| EReal.coe_le_coe_iff.mpr hx

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ abscissaOfAbsConv f < ↑(↑x).re

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

intro x hx

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m ⊢ ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

rw [LSeries, ← tsum_mul_left, tsum_eq_add_tsum_ite (hs hx) (n + 1)]

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ ((↑n + 1) ^ ↑x * term f (↑x) (n + 1) + ∑' (n_1 : ℕ), if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = f (n + 1) + ∑' (m : ℕ), F x m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

congr

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ ((↑n + 1) ^ ↑x * term f (↑x) (n + 1) + ∑' (n_1 : ℕ), if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = f (n + 1) + ∑' (m : ℕ), F x m

case e_a f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * term f (↑x) (n + 1) = f (n + 1) case e_a.e_f f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (fun n_1 => if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = fun m => F x m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact pow_mul_term_eq f x n

case e_a f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (↑n + 1) ^ ↑x * term f (↑x) (n + 1) = f (n + 1)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

ext m

case e_a.e_f f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (fun n_1 => if n_1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) n_1) = fun m => F x m

case e_a.e_f.h f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

rcases eq_or_ne m (n + 1) with rfl | hm

case e_a.e_f.h f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m

case e_a.e_f.h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (if n + 1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) (n + 1)) = F x (n + 1) case e_a.e_f.h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ hm : m ≠ n + 1 ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [↓reduceIte, hF₀ x le_rfl]

case e_a.e_f.h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y ⊢ (if n + 1 = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) (n + 1)) = F x (n + 1)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [hm, ↓reduceIte, ne_eq, not_false_eq_true, hF]

case e_a.e_f.h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m x : ℝ hx : x ≥ y m : ℕ hm : m ≠ n + 1 ⊢ (if m = n + 1 then 0 else (↑n + 1) ^ ↑x * term f (↑x) m) = F x m

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

refine ((hs le_rfl).indicator {m | n + 1 < m}).congr fun m ↦ ?_

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m ⊢ Summable (F y)

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ ⊢ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

by_cases hm : n + 1 < m

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ ⊢ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m

case pos f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : n + 1 < m ⊢ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m case neg f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : ¬n + 1 < m ⊢ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [Set.mem_setOf_eq, hm, Set.indicator_of_mem, ne_eq, hm.ne', not_false_eq_true, hF]

case pos f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : n + 1 < m ⊢ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [Set.mem_setOf_eq, hm, not_false_eq_true, Set.indicator_of_not_mem, hF₀ _ (le_of_not_lt hm)]

case neg f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m m : ℕ hm : ¬n + 1 < m ⊢ {m | n + 1 < m}.indicator (fun m => (↑n + 1) ^ ↑y * term f (↑y) m) m = F y m

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

rcases lt_or_le (n + 1) k with H | H

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k ⊢ Tendsto (fun x => F x k) atTop (nhds 0) case inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : k ≤ n + 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

have H₀ : (0 : ℝ) ≤ k / (n + 1) := by positivity

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

have H₀' : (0 : ℝ) ≤ (n + 1) / k := by positivity

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

have H₁ : (k / (n + 1) : ℂ) = (k / (n + 1) : ℝ) := by simp only [ofReal_div, ofReal_natCast, ofReal_add, ofReal_one]

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

have H₂ : (n + 1) / k < (1 : ℝ) := (div_lt_one <| by exact_mod_cast n.succ_pos.trans H).mpr <| by exact_mod_cast H

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, F]

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k / (↑k / (↑n + 1)) ^ ↑x) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

conv => enter [1, x] rw [div_eq_mul_inv, H₁, ← ofReal_cpow H₀, ← ofReal_inv, ← Real.inv_rpow H₀, inv_div]

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k / (↑k / (↑n + 1)) ^ ↑x) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

conv => enter [3, 1]; rw [← mul_zero (f k)]

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds 0)

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds (f k * 0))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact (tendsto_rpow_atTop_of_base_lt_one _ (neg_one_lt_zero.trans_le H₀') H₂).ofReal.const_mul _

case inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) H₂ : (↑n + 1) / ↑k < 1 ⊢ Tendsto (fun x => f k * ↑(((↑n + 1) / ↑k) ^ x)) atTop (nhds (f k * 0))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

positivity

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k ⊢ 0 ≤ ↑k / (↑n + 1)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

positivity

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) ⊢ 0 ≤ (↑n + 1) / ↑k

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [ofReal_div, ofReal_natCast, ofReal_add, ofReal_one]

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k ⊢ ↑k / (↑n + 1) = ↑(↑k / (↑n + 1))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact_mod_cast n.succ_pos.trans H

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ 0 < ↑k

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact_mod_cast H

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : n + 1 < k H₀ : 0 ≤ ↑k / (↑n + 1) H₀' : 0 ≤ (↑n + 1) / ↑k H₁ : ↑k / (↑n + 1) = ↑(↑k / (↑n + 1)) ⊢ ↑n + 1 < ↑k

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [hF₀ _ H, tendsto_const_nhds_iff]

case inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) k : ℕ H : k ≤ n + 1 ⊢ Tendsto (fun x => F x k) atTop (nhds 0)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [Set.mem_setOf_eq, H, Set.indicator_of_mem, norm_div, Complex.norm_eq_abs, abs_cpow_real, map_div₀, abs_natCast, F]

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ ‖F y' k‖ ≤ ‖F y k‖

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

rw [← Nat.cast_one, ← Nat.cast_add, abs_natCast]

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / Complex.abs (↑n + 1)) ^ y

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

have hkn : 1 ≤ (k / (n + 1 :) : ℝ) := (one_le_div (by positivity)).mpr <| by norm_cast; exact Nat.le_of_succ_le H

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k hkn : 1 ≤ ↑k / ↑(n + 1) ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact div_le_div_of_nonneg_left (Complex.abs.nonneg _) (rpow_pos_of_pos (zero_lt_one.trans_le hkn) _) <| rpow_le_rpow_of_exponent_le hkn hy'

case h.inl f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k hkn : 1 ≤ ↑k / ↑(n + 1) ⊢ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y' ≤ Complex.abs (f k) / (↑k / ↑(n + 1)) ^ y

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

positivity

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ 0 < ↑(n + 1)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

norm_cast

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ ↑(n + 1) ≤ ↑k

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ n + 1 ≤ k

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

exact Nat.le_of_succ_le H

f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : n + 1 < k ⊢ n + 1 ≤ k

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_pow_mul_atTop

[61, 1]

[132, 44]

simp only [hF₀ _ H, norm_zero, le_refl]

case h.inr f : ℕ → ℂ n : ℕ h : ∀ m ≤ n, f m = 0 ha : abscissaOfAbsConv f < ⊤ y : ℝ hay : abscissaOfAbsConv f < ↑y hyt : ↑y < ⊤ F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0 hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m hys : Summable (F y) hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0) y' : ℝ hy' : y ≤ y' k : ℕ H : k ≤ n + 1 ⊢ ‖F y' k‖ ≤ ‖F y k‖

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

let F (n : ℕ) : ℂ := if n = 0 then 0 else f n

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

have hF₀ : F 0 = 0 := rfl

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := by simp only [hn, ↓reduceIte, F]

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

have ha' : abscissaOfAbsConv F < ⊤ := (abscissaOfAbsConv_congr hF).symm ▸ ha

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

simp_rw [← LSeries_congr _ hF]

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ Tendsto (fun x => LSeries f ↑x) atTop (nhds (f 1))

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ Tendsto (fun x => LSeries (fun {n} => F n) ↑x) atTop (nhds (f 1))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

convert LSeries.tendsto_pow_mul_atTop (n := 0) (fun _ hm ↦ Nat.le_zero.mp hm ▸ hF₀) ha' using 1

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ Tendsto (fun x => LSeries (fun {n} => F n) ↑x) atTop (nhds (f 1))

case h.e'_3 f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ (fun x => LSeries (fun {n} => F n) ↑x) = fun x => (↑0 + 1) ^ ↑x * LSeries F ↑x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

simp only [Nat.cast_zero, zero_add, one_cpow, one_mul]

case h.e'_3 f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha' : abscissaOfAbsConv F < ⊤ ⊢ (fun x => LSeries (fun {n} => F n) ↑x) = fun x => (↑0 + 1) ^ ↑x * LSeries F ↑x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries.tendsto_atTop

[135, 1]

[145, 57]

simp only [hn, ↓reduceIte, F]

f : ℕ → ℂ ha : abscissaOfAbsConv f < ⊤ F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 n : ℕ hn : n ≠ 0 ⊢ F n = f n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_of_abscissaOfAbsConv_eq_top

[147, 1]

[151, 57]

ext s

f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0

case h f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ s : ℂ ⊢ LSeries f s = 0 s

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_of_abscissaOfAbsConv_eq_top

[147, 1]

[151, 57]

exact LSeries.eq_zero_of_not_LSeriesSummable f s <| mt LSeriesSummable.abscissaOfAbsConv_le <| h ▸ fun H ↦ (H.trans_lt <| EReal.coe_lt_top _).false

case h f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ s : ℂ ⊢ LSeries f s = 0 s

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

by_cases h : abscissaOfAbsConv f = ⊤ <;> simp only [h, or_true, or_false, iff_true]

f : ℕ → ℂ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ↔ (∀ (n : ℕ), n ≠ 0 → f n = 0) ∨ abscissaOfAbsConv f = ⊤

case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 case neg f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

refine eventually_of_forall ?_

case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0

case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ ∀ (x : ℝ), (fun x => LSeries f ↑x) x = 0 x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [LSeries_eq_zero_of_abscissaOfAbsConv_eq_top h, Pi.zero_apply, forall_const]

case pos f : ℕ → ℂ h : abscissaOfAbsConv f = ⊤ ⊢ ∀ (x : ℝ), (fun x => LSeries f ↑x) x = 0 x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

refine ⟨fun H ↦ ?_, fun H ↦ eventually_of_forall fun x ↦ ?_⟩

case neg f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ ⊢ (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0 case neg.refine_2 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∀ (n : ℕ), n ≠ 0 → f n = 0 x : ℝ ⊢ (fun x => LSeries f ↑x) x = 0 x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

let F (n : ℕ) : ℂ := if n = 0 then 0 else f n

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

have hF₀ : F 0 = 0 := rfl

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

have hF {n : ℕ} (hn : n ≠ 0) : F n = f n := by simp only [hn, ↓reduceIte, F]

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

suffices ∀ n, F n = 0 by peel hF with n hn h exact (this n ▸ h).symm

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ ∀ (n : ℕ), F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

have ha : ¬ abscissaOfAbsConv F = ⊤ := abscissaOfAbsConv_congr hF ▸ h

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ⊢ ∀ (n : ℕ), F n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ ⊢ ∀ (n : ℕ), F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

have h' (x : ℝ) : LSeries F x = LSeries f x := LSeries_congr x hF

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ ⊢ ∀ (n : ℕ), F n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x ⊢ ∀ (n : ℕ), F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

have H' (n : ℕ) : (fun x : ℝ ↦ (n ^ (x : ℂ)) * LSeries F x) =ᶠ[atTop] (fun _ ↦ 0) := by simp only [h'] rw [eventuallyEq_iff_exists_mem] at H ⊢ peel H with s hs refine ⟨hs.1, fun x hx ↦ ?_⟩ simp only [hs.2 hx, Pi.zero_apply, mul_zero]

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x ⊢ ∀ (n : ℕ), F n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ⊢ ∀ (n : ℕ), F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

intro n

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ⊢ ∀ (n : ℕ), F n = 0

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ⊢ F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

induction' n using Nat.strongInductionOn with n ih

case neg.refine_1 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ⊢ F n = 0

case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

suffices Tendsto (fun x : ℝ ↦ (n ^ (x : ℂ)) * LSeries F x) atTop (nhds (F n)) by replace this := this.congr' <| H' n simp only [tendsto_const_nhds_iff] at this exact this.symm

case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ F n = 0

case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

cases n with | zero => refine Tendsto.congr' (H' 0).symm ?_ simp only [zero_eq, hF₀, tendsto_const_nhds_iff] | succ n => simp only [succ_eq_add_one, cast_add, cast_one] exact LSeries.tendsto_pow_mul_atTop (fun m hm ↦ ih m <| lt_succ_of_le hm) <| Ne.lt_top ha

case neg.refine_1.ind f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 ⊢ Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [hn, ↓reduceIte, F]

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 n : ℕ hn : n ≠ 0 ⊢ F n = f n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

peel hF with n hn h

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n this : ∀ (n : ℕ), F n = 0 ⊢ ∀ (n : ℕ), n ≠ 0 → f n = 0

case h.h f : ℕ → ℂ h✝ : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n this : ∀ (n : ℕ), F n = 0 n : ℕ hn : n ≠ 0 h : F n = f n ⊢ f n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

exact (this n ▸ h).symm

case h.h f : ℕ → ℂ h✝ : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n this : ∀ (n : ℕ), F n = 0 n : ℕ hn : n ≠ 0 h : F n = f n ⊢ f n = 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [h']

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) =ᶠ[atTop] fun x => 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

rw [eventuallyEq_iff_exists_mem] at H ⊢

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) =ᶠ[atTop] fun x => 0

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ ∃ s ∈ atTop, Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

peel H with s hs

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ ⊢ ∃ s ∈ atTop, Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s

case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s ⊢ s ∈ atTop ∧ Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

refine ⟨hs.1, fun x hx ↦ ?_⟩

case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s ⊢ s ∈ atTop ∧ Set.EqOn (fun x => ↑n ^ ↑x * LSeries f ↑x) (fun x => 0) s

case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s x : ℝ hx : x ∈ s ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) x = (fun x => 0) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [hs.2 hx, Pi.zero_apply, mul_zero]

case h f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∃ s ∈ atTop, Set.EqOn (fun x => LSeries f ↑x) 0 s F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x n : ℕ s : Set ℝ hs : s ∈ atTop ∧ Set.EqOn (fun x => LSeries f ↑x) 0 s x : ℝ hx : x ∈ s ⊢ (fun x => ↑n ^ ↑x * LSeries f ↑x) x = (fun x => 0) x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

replace this := this.congr' <| H' n

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : Tendsto (fun x => ↑n ^ ↑x * LSeries F ↑x) atTop (nhds (F n)) ⊢ F n = 0

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : Tendsto (fun x => 0) atTop (nhds (F n)) ⊢ F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [tendsto_const_nhds_iff] at this

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : Tendsto (fun x => 0) atTop (nhds (F n)) ⊢ F n = 0

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : 0 = F n ⊢ F n = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

exact this.symm

f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n, F m = 0 this : 0 = F n ⊢ F n = 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

refine Tendsto.congr' (H' 0).symm ?_

case neg.refine_1.ind.zero f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ih : ∀ m < 0, F m = 0 ⊢ Tendsto (fun x => ↑0 ^ ↑x * LSeries F ↑x) atTop (nhds (F 0))

case neg.refine_1.ind.zero f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ih : ∀ m < 0, F m = 0 ⊢ Tendsto (fun x => 0) atTop (nhds (F 0))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [zero_eq, hF₀, tendsto_const_nhds_iff]

case neg.refine_1.ind.zero f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 ih : ∀ m < 0, F m = 0 ⊢ Tendsto (fun x => 0) atTop (nhds (F 0))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [succ_eq_add_one, cast_add, cast_one]

case neg.refine_1.ind.succ f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n + 1, F m = 0 ⊢ Tendsto (fun x => ↑(n + 1) ^ ↑x * LSeries F ↑x) atTop (nhds (F (n + 1)))

case neg.refine_1.ind.succ f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n + 1, F m = 0 ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries F ↑x) atTop (nhds (F (n + 1)))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

exact LSeries.tendsto_pow_mul_atTop (fun m hm ↦ ih m <| lt_succ_of_le hm) <| Ne.lt_top ha

case neg.refine_1.ind.succ f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0 F : ℕ → ℂ := fun n => if n = 0 then 0 else f n hF₀ : F 0 = 0 hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n ha : ¬abscissaOfAbsConv F = ⊤ h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0 n : ℕ ih : ∀ m < n + 1, F m = 0 ⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries F ↑x) atTop (nhds (F (n + 1)))

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eventually_eq_zero_iff'

[154, 1]

[191, 37]

simp only [LSeries_congr x fun {n} ↦ H n, show (fun _ : ℕ ↦ (0 : ℂ)) = 0 from rfl, LSeries_zero, Pi.zero_apply]

case neg.refine_2 f : ℕ → ℂ h : ¬abscissaOfAbsConv f = ⊤ H : ∀ (n : ℕ), n ≠ 0 → f n = 0 x : ℝ ⊢ (fun x => LSeries f ↑x) x = 0 x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

by_cases h : abscissaOfAbsConv f = ⊤ <;> simp only [h, or_true, or_false, iff_true]

f : ℕ → ℂ hf : f 0 = 0 ⊢ LSeries f = 0 ↔ f = 0 ∨ abscissaOfAbsConv f = ⊤

case pos f : ℕ → ℂ hf : f 0 = 0 h : abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0 case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0 ↔ f = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

exact LSeries_eq_zero_of_abscissaOfAbsConv_eq_top h

case pos f : ℕ → ℂ hf : f 0 = 0 h : abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

refine ⟨fun H ↦ ?_, fun H ↦ H ▸ LSeries_zero⟩

case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ ⊢ LSeries f = 0 ↔ f = 0

case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

convert (LSeries_eventually_eq_zero_iff'.mp ?_).resolve_right h

case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0

case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0 case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

refine ⟨fun H' _ _ ↦ by rw [H', Pi.zero_apply], fun H' ↦ ?_⟩

case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ f = 0 ↔ ∀ (n : ℕ), n ≠ 0 → f n = 0

case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

ext ⟨- | m⟩

case a f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f = 0

case a.h.zero f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f 0 = 0 0 case a.h.succ f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 n✝ : ℕ ⊢ f (n✝ + 1) = 0 (n✝ + 1)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

rw [H', Pi.zero_apply]

f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : f = 0 x✝¹ : ℕ x✝ : x✝¹ ≠ 0 ⊢ f x✝¹ = 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

simp only [zero_eq, hf, Pi.zero_apply]

case a.h.zero f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 ⊢ f 0 = 0 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

simp only [ne_eq, succ_ne_zero, not_false_eq_true, H', Pi.zero_apply]

case a.h.succ f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 H' : ∀ (n : ℕ), n ≠ 0 → f n = 0 n✝ : ℕ ⊢ f (n✝ + 1) = 0 (n✝ + 1)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

simp only [H, Pi.zero_apply]

case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => LSeries f ↑x) =ᶠ[Filter.atTop] 0

case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => 0) =ᶠ[Filter.atTop] 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_eq_zero_iff

[194, 1]

[207, 36]

exact Filter.EventuallyEq.rfl

case neg f : ℕ → ℂ hf : f 0 = 0 h : ¬abscissaOfAbsConv f = ⊤ H : LSeries f = 0 ⊢ (fun x => 0) =ᶠ[Filter.atTop] 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

rw [EventuallyEq, eventually_atTop] at h ⊢

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : (fun x => LSeries f ↑x) =ᶠ[atTop] fun x => LSeries g ↑x ⊢ (fun x => LSeries (f - g) ↑x) =ᶠ[atTop] 0

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : ∃ a, ∀ b ≥ a, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

obtain ⟨x₀, hx₀⟩ := h

f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ h : ∃ a, ∀ b ≥ a, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

case intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

obtain ⟨yf, hyf₁, hyf₂⟩ := exists_between hf

case intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

case intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

obtain ⟨yg, hyg₁, hyg₂⟩ := exists_between hg

case intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

case intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

lift yf to ℝ using ⟨hyf₂.ne, ((OrderBot.bot_le _).trans_lt hyf₁).ne'⟩

case intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : EReal hyf₁ : abscissaOfAbsConv f < yf hyf₂ : yf < ⊤ yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

case intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

lift yg to ℝ using ⟨hyg₂.ne, ((OrderBot.bot_le _).trans_lt hyg₁).ne'⟩

case intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yg : EReal hyg₁ : abscissaOfAbsConv g < yg hyg₂ : yg < ⊤ yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/LSeriesUnique.lean

LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq

[210, 1]

[231, 86]

refine ⟨max x₀ (max yf yg), fun x hx ↦ ?_⟩

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ ⊢ ∃ a, ∀ b ≥ a, LSeries (f - g) ↑b = 0 b

case intro.intro.intro.intro.intro.intro.intro f g : ℕ → ℂ hf : abscissaOfAbsConv f < ⊤ hg : abscissaOfAbsConv g < ⊤ x₀ : ℝ hx₀ : ∀ b ≥ x₀, LSeries f ↑b = LSeries g ↑b yf : ℝ hyf₁ : abscissaOfAbsConv f < ↑yf hyf₂ : ↑yf < ⊤ yg : ℝ hyg₁ : abscissaOfAbsConv g < ↑yg hyg₂ : ↑yg < ⊤ x : ℝ hx : x ≥ max x₀ (max yf yg) ⊢ LSeries (f - g) ↑x = 0 x