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/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
change 0 ≤ deriv (f ∘ ofReal') x at H
|
case intro.intro
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx✝ : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 ≤ ↑x
H : 0 ≤ deriv (fun x => f ↑x) x
⊢ 0 ≤ deriv F x
|
case intro.intro
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx✝ : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 ≤ ↑x
H : 0 ≤ deriv (f ∘ ofReal') x
⊢ 0 ≤ deriv F x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
erw [hF, deriv.ofReal_comp] at H
|
case intro.intro
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx✝ : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 ≤ ↑x
H : 0 ≤ deriv (f ∘ ofReal') x
⊢ 0 ≤ deriv F x
|
case intro.intro
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx✝ : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 ≤ ↑x
H : 0 ≤ ↑(deriv (fun y => F y) x)
⊢ 0 ≤ deriv F x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
norm_cast at H
|
case intro.intro
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx✝ : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 ≤ ↑x
H : 0 ≤ ↑(deriv (fun y => F y) x)
⊢ 0 ≤ deriv F x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
refine Complex.ext rfl ?_
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
n : ℕ
⊢ iteratedDeriv n f 0 = ↑(D n)
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
n : ℕ
⊢ (iteratedDeriv n f 0).im = (↑(D n)).im
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
simp only [ofReal_im]
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
n : ℕ
⊢ (iteratedDeriv n f 0).im = (↑(D n)).im
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
n : ℕ
⊢ (iteratedDeriv n f 0).im = 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
exact (le_def.mp (h n)).2.symm
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
n : ℕ
⊢ (iteratedDeriv n f 0).im = 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
rw [← iteratedDeriv_succ']
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx : x ∈ interior (Set.Ici 0)
n : ℕ
⊢ 0 ≤ iteratedDeriv n (deriv f) 0
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx : x ∈ interior (Set.Ici 0)
n : ℕ
⊢ 0 ≤ iteratedDeriv (n + 1) f 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
exact h (n + 1)
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx : x ∈ interior (Set.Ici 0)
n : ℕ
⊢ 0 ≤ iteratedDeriv (n + 1) f 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
norm_cast
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
⊢ 0 ≤ ↑x
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
⊢ 0 ≤ x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hx : x ∈ interior (Set.Ici 0)
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
⊢ 0 ≤ x
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 < x
⊢ 0 ≤ x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
exact hx.le
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0
D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re
hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n)
F : ℝ → ℝ
hFd : Differentiable ℝ F
hF : f ∘ ofReal' = ofReal' ∘ F
x : ℝ
hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0
hf' : Differentiable ℂ (deriv f)
hx : 0 < x
⊢ 0 ≤ x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
exact sub_nonneg.mp <| nonneg_of_iteratedDeriv_nonneg (hf.sub_const (f 0)) h' hz
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0
z : ℂ
hz : 0 ≤ z
h' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (fun x => f x - f 0) 0
⊢ f 0 ≤ f z
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
cases n with
| zero => simp only [iteratedDeriv_zero, sub_self, le_refl]
| succ n =>
specialize h n.succ <| succ_ne_zero n
rw [iteratedDeriv_succ'] at h ⊢
convert h using 2
ext w
exact deriv_sub_const (f 0)
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0
z : ℂ
hz : 0 ≤ z
n : ℕ
⊢ 0 ≤ iteratedDeriv n (fun x => f x - f 0) 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
simp only [iteratedDeriv_zero, sub_self, le_refl]
|
case zero
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0
z : ℂ
hz : 0 ≤ z
⊢ 0 ≤ iteratedDeriv 0 (fun x => f x - f 0) 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
specialize h n.succ <| succ_ne_zero n
|
case succ
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ iteratedDeriv n f 0
z : ℂ
hz : 0 ≤ z
n : ℕ
⊢ 0 ≤ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
|
case succ
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n.succ f 0
⊢ 0 ≤ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
rw [iteratedDeriv_succ'] at h ⊢
|
case succ
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n.succ f 0
⊢ 0 ≤ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
|
case succ
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (deriv f) 0
⊢ 0 ≤ iteratedDeriv n (deriv fun x => f x - f 0) 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
convert h using 2
|
case succ
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (deriv f) 0
⊢ 0 ≤ iteratedDeriv n (deriv fun x => f x - f 0) 0
|
case h.e'_4.h.e'_7
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (deriv f) 0
⊢ (deriv fun x => f x - f 0) = deriv f
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
ext w
|
case h.e'_4.h.e'_7
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (deriv f) 0
⊢ (deriv fun x => f x - f 0) = deriv f
|
case h.e'_4.h.e'_7.h
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (deriv f) 0
w : ℂ
⊢ deriv (fun x => f x - f 0) w = deriv f w
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
exact deriv_sub_const (f 0)
|
case h.e'_4.h.e'_7.h
f : ℂ → ℂ
hf : Differentiable ℂ f
z : ℂ
hz : 0 ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (deriv f) 0
w : ℂ
⊢ deriv (fun x => f x - f 0) w = deriv f w
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
let F : ℂ → ℂ := fun z ↦ f (-z)
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
⊢ f 0 ≤ f z
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
⊢ f 0 ≤ f z
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
convert at_zero_le_of_iteratedDeriv_nonneg (f := F) (hf.comp <| differentiable_neg)
(fun n hn ↦ ?_) (neg_nonneg.mpr hz) using 1
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
⊢ f 0 ≤ f z
|
case h.e'_3
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
⊢ f 0 = F 0
case h.e'_4
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
⊢ f z = F (-z)
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ iteratedDeriv n F 0
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
simp only [F, neg_zero]
|
case h.e'_3
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
⊢ f 0 = F 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
simp only [F, neg_neg]
|
case h.e'_4
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
⊢ f z = F (-z)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
simpa only [F, iteratedDeriv_comp_neg, neg_zero] using h n hn
|
f : ℂ → ℂ
hf : Differentiable ℂ f
h : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f 0
z : ℂ
hz : z ≤ 0
F : ℂ → ℂ := fun z => f (-z)
n : ℕ
hn : n ≠ 0
⊢ 0 ≤ iteratedDeriv n F 0
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
Complex.cpow_natCast_add_one_ne_zero
|
[8, 1]
|
[9, 67]
|
norm_cast at H
|
n : ℕ
z : ℂ
H : ↑n + 1 = 0 ∧ z ≠ 0
⊢ False
|
n : ℕ
z : ℂ
H : False ∧ ¬z = 0
⊢ False
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
Complex.cpow_natCast_add_one_ne_zero
|
[8, 1]
|
[9, 67]
|
exact H.1
|
n : ℕ
z : ℂ
H : False ∧ ¬z = 0
⊢ False
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.abscissaOfAbsConv_binop_le
|
[13, 1]
|
[24, 59]
|
refine abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' fun x hx ↦ hF ?_ ?_
|
F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ
hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s
f g : ℕ → ℂ
⊢ abscissaOfAbsConv (F f g) ≤ max (abscissaOfAbsConv f) (abscissaOfAbsConv g)
|
case refine_1
F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ
hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s
f g : ℕ → ℂ
x : ℝ
hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x
⊢ LSeriesSummable f ↑x
case refine_2
F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ
hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s
f g : ℕ → ℂ
x : ℝ
hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x
⊢ LSeriesSummable g ↑x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.abscissaOfAbsConv_binop_le
|
[13, 1]
|
[24, 59]
|
exact LSeriesSummable_of_abscissaOfAbsConv_lt_re <|
(ofReal_re x).symm ▸ (le_max_left ..).trans_lt hx
|
case refine_1
F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ
hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s
f g : ℕ → ℂ
x : ℝ
hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x
⊢ LSeriesSummable f ↑x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.abscissaOfAbsConv_binop_le
|
[13, 1]
|
[24, 59]
|
exact LSeriesSummable_of_abscissaOfAbsConv_lt_re <|
(ofReal_re x).symm ▸ (le_max_right ..).trans_lt hx
|
case refine_2
F : (ℕ → ℂ) → (ℕ → ℂ) → ℕ → ℂ
hF : ∀ {f g : ℕ → ℂ} {s : ℂ}, LSeriesSummable f s → LSeriesSummable g s → LSeriesSummable (F f g) s
f g : ℕ → ℂ
x : ℝ
hx : max (abscissaOfAbsConv f) (abscissaOfAbsConv g) < ↑x
⊢ LSeriesSummable g ↑x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
have Hm : (0 : ℝ) ≤ m := m.cast_nonneg
|
m n : ℕ
z : ℂ
x : ℝ
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
have Hn : (0 : ℝ) ≤ (n + 1 : ℝ)⁻¹ := by positivity
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
rw [← mul_div_assoc, mul_comm, div_eq_mul_inv z, mul_div_assoc]
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ z * ((↑n + 1) ^ ↑x / ↑m ^ ↑x) = z * ((↑m / (↑n + 1)) ^ ↑x)⁻¹
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
congr
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ z * ((↑n + 1) ^ ↑x / ↑m ^ ↑x) = z * ((↑m / (↑n + 1)) ^ ↑x)⁻¹
|
case e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x / ↑m ^ ↑x = ((↑m / (↑n + 1)) ^ ↑x)⁻¹
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
simp_rw [div_eq_mul_inv]
|
case e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x / ↑m ^ ↑x = ((↑m / (↑n + 1)) ^ ↑x)⁻¹
|
case e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x * (↑m ^ ↑x)⁻¹ = ((↑m * (↑n + 1)⁻¹) ^ ↑x)⁻¹
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
rw [show (n + 1 : ℂ)⁻¹ = (n + 1 : ℝ)⁻¹ by
simp only [ofReal_inv, ofReal_add, ofReal_natCast, ofReal_one],
show (n + 1 : ℂ) = (n + 1 : ℝ) by norm_cast, show (m : ℂ) = (m : ℝ) by norm_cast,
mul_cpow_ofReal_nonneg Hm Hn, mul_inv, mul_comm]
|
case e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x * (↑m ^ ↑x)⁻¹ = ((↑m * (↑n + 1)⁻¹) ^ ↑x)⁻¹
|
case e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑↑m ^ ↑x)⁻¹ * ↑(↑n + 1) ^ ↑x = (↑↑m ^ ↑x)⁻¹ * (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
congr
|
case e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑↑m ^ ↑x)⁻¹ * ↑(↑n + 1) ^ ↑x = (↑↑m ^ ↑x)⁻¹ * (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹
|
case e_a.e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ ↑(↑n + 1) ^ ↑x = (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
rw [← cpow_neg, show (-x : ℂ) = (-1 : ℝ) * x by simp only [ofReal_neg, ofReal_one,
neg_mul, one_mul], cpow_mul_ofReal_nonneg Hn, Real.rpow_neg_one, inv_inv]
|
case e_a.e_a
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ ↑(↑n + 1) ^ ↑x = (↑(↑n + 1)⁻¹ ^ ↑x)⁻¹
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
positivity
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
⊢ 0 ≤ (↑n + 1)⁻¹
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
simp only [ofReal_inv, ofReal_add, ofReal_natCast, ofReal_one]
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1)⁻¹ = ↑(↑n + 1)⁻¹
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
norm_cast
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ ↑n + 1 = ↑(↑n + 1)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
norm_cast
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ ↑m = ↑↑m
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
foo
|
[38, 1]
|
[52, 78]
|
simp only [ofReal_neg, ofReal_one,
neg_mul, one_mul]
|
m n : ℕ
z : ℂ
x : ℝ
Hm : 0 ≤ ↑m
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ -↑x = ↑(-1) * ↑x
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.pow_mul_term_eq
|
[54, 1]
|
[58, 20]
|
simp only [term, add_eq_zero, one_ne_zero, and_false, ↓reduceIte, Nat.cast_add, Nat.cast_one,
mul_div_assoc', ne_eq, cpow_natCast_add_one_ne_zero n _, not_false_eq_true, div_eq_iff,
mul_comm (f _)]
|
f : ℕ → ℂ
s : ℂ
n : ℕ
⊢ (↑n + 1) ^ s * term f s (n + 1) = f (n + 1)
|
no goals
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
obtain ⟨y, hay, hyt⟩ := exists_between ha
|
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
case intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : EReal
hay : abscissaOfAbsConv f < y
hyt : y < ⊤
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
lift y to ℝ using ⟨hyt.ne, ((OrderBot.bot_le _).trans_lt hay).ne'⟩
|
case intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : EReal
hay : abscissaOfAbsConv f < y
hyt : y < ⊤
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
let F := fun (x : ℝ) ↦ {m | n + 1 < m}.indicator (fun m ↦ f m / (m / (n + 1) : ℂ) ^ (x : ℂ))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
have hF₀ (x : ℝ) {m : ℕ} (hm : m ≤ n + 1) : F x m = 0 := by
simp only [Set.mem_setOf_eq, not_lt_of_le hm, not_false_eq_true, Set.indicator_of_not_mem, F]
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
have hs {x : ℝ} (hx : x ≥ y) : Summable fun m ↦ (n + 1) ^ (x : ℂ) * term f x m := by
refine (summable_mul_left_iff <| cpow_natCast_add_one_ne_zero n _).mpr <|
LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_
simpa only [ofReal_re] using hay.trans_le <| EReal.coe_le_coe_iff.mpr hx
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
conv => enter [3, 1]; rw [← add_zero (f _)]
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1) + 0))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
refine Tendsto.congr'
(eventuallyEq_of_mem (s := {x | y ≤ x}) (mem_atTop y) key).symm <| tendsto_const_nhds.add ?_
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
⊢ Tendsto (fun x => (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1) + 0))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds 0)
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
rw [show (0 : ℂ) = tsum (fun _ : ℕ ↦ 0) from tsum_zero.symm]
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds 0)
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds (∑' (x : ℕ), 0))
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
refine tendsto_tsum_of_dominated_convergence hys.norm hc <| eventually_iff.mpr ?_
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
⊢ Tendsto (fun x => ∑' (m : ℕ), F x m) atTop (nhds (∑' (x : ℕ), 0))
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
⊢ {x | ∀ (k : ℕ), ‖F x k‖ ≤ ‖F y k‖} ∈ atTop
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
filter_upwards [mem_atTop y] with y' hy' k
|
case intro.intro.intro
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
⊢ {x | ∀ (k : ℕ), ‖F x k‖ ≤ ‖F y k‖} ∈ atTop
|
case h
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
y' : ℝ
hy' : y ≤ y'
k : ℕ
⊢ ‖F y' k‖ ≤ ‖F y k‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
rcases lt_or_le (n + 1) k with H | H
|
case h
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
y' : ℝ
hy' : y ≤ y'
k : ℕ
⊢ ‖F y' k‖ ≤ ‖F y k‖
|
case h.inl
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
y' : ℝ
hy' : y ≤ y'
k : ℕ
H : n + 1 < k
⊢ ‖F y' k‖ ≤ ‖F y k‖
case h.inr
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑x) m
hs : ∀ {x : ℝ}, x ≥ y → Summable fun m => (↑n + 1) ^ ↑x * term f (↑x) m
key : ∀ x ≥ y, (↑n + 1) ^ ↑x * LSeries f ↑x = f (n + 1) + ∑' (m : ℕ), F x m
hys : Summable (F y)
hc : ∀ (k : ℕ), Tendsto (fun x => F x k) atTop (nhds 0)
y' : ℝ
hy' : y ≤ y'
k : ℕ
H : k ≤ n + 1
⊢ ‖F y' k‖ ≤ ‖F y k‖
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/LSeriesUnique.lean
|
LSeries.tendsto_pow_mul_atTop
|
[61, 1]
|
[132, 44]
|
simp only [Set.mem_setOf_eq, not_lt_of_le hm, not_false_eq_true, Set.indicator_of_not_mem, F]
|
f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
x : ℝ
m : ℕ
hm : m ≤ n + 1
⊢ F x m = 0
|
no goals
|
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))
|
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