url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_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
|
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