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/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | clear GS G h | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | set A := N ∪ M \ N | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
⊢ dist g (N.sum fun n => f n z) < e | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
⊢ dist g (N.su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have AM : M ⊆ A := subset_union_sdiff _ _ | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
⊢ dist g (N.su... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
⊢ d... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | simp at HM | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
HM : ∀ n ≥ M, dist (n.sum fun b => f b z) g < e / 4
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
⊢ d... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : ∀ (n : Finset ℕ), M ⊆ n → dist (n.sum fun b => f b... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | specialize HM A AM | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : ∀ (n : Finset ℕ), M ⊆ n → dist (n.sum fun b => f b... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist (A.sum fun b => f b z) g < e / 4
⊢ dist g (N.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [dist_comm] at HM | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist (A.sum fun b => f b z) g < e / 4
⊢ dist g (N.... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (N.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | calc dist g (N.sum fun n ↦ f n z)
_ ≤ dist g (A.sum fun n ↦ f n z) + dist (A.sum fun n ↦ f n z) (N.sum fun n ↦ f n z) := by bound
_ ≤ e / 4 + dist (A.sum fun n ↦ f n z) (N.sum fun n ↦ f n z) := by linarith
_ = e / 4 + dist ((N.sum fun n ↦ f n z) + (M \ N).sum fun n ↦ f n z)
(N.sum fun n ↦ f n z) := by rw [F... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (N.... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ e / 4 > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [← G] | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ HasSum (fun n => f n z) g | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ HasSum (fun n => f n z) (∑' (n : ℕ), ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | exact Summable.hasSum S | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ HasSum (fun n => f n z) (∑' (n : ℕ), ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ε
⊢ e /... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (N.sum fun n => ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ dist g (A.sum fun n => ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Finset.sum_union Finset.disjoint_sdiff] | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + dist (A.sum fun... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Complex.dist_eq] | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + dist ((N.sum fu... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring_nf | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + Complex.abs (((... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith [finset_complex_abs_sum_le (M \ N) fun n ↦ f n z] | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + Complex.abs ((M... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith [hm (M \ N) z (sdiff_late M Nm) zs] | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ (e / 4 + (M \ N).sum fu... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 4 + e / 4 = e / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
M : Finset ℕ
A : Finset ℕ := N ∪ M \ N
AM : M ⊆ A
HM : dist g (A.sum fun b => f b z) < e / 4
⊢ e / 2 < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ ∀ (n : ℕ), ∀ z ∈ s, f n z = 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
n : ℕ
z : ℂ
zs : z ∈ s
⊢ f n z = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | specialize hf n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
n : ℕ
z : ℂ
zs : z ∈ s
⊢ f n z = 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
⊢ f n z = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | have ca : c * a ^ n ≤ 0 := mul_nonpos_iff.mpr (Or.inr ⟨c0, by bound⟩) | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
⊢ f n z = 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
ca : c * a ^ n ≤ 0
⊢ f n z = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | exact Complex.abs.eq_zero.mp (le_antisymm (le_trans hf ca) (Complex.abs.nonneg _)) | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
ca : c * a ^ n ≤ 0
⊢ f n z = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | CNonpos.degenerate | [106, 1] | [110, 85] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
c0 : c ≤ 0
a0 : 0 ≤ a
n : ℕ
z : ℂ
zs : z ∈ s
hf : Complex.abs (f n z) ≤ c * a ^ n
⊢ 0 ≤ a ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | by_cases c0 : c ≤ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ HasUniformSum f (tsumOn f) s | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
⊢ HasUniformSum f (tsumOn f) s
case neg
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : ¬c ≤ 0
⊢ HasUniformSum f (tsumOn... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have fz := CNonpos.degenerate c0 a0 hf | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
⊢ HasUniformSum f (tsumOn f) s | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ HasUniformSum f (tsumOn f) s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [HasUniformSum, Metric.tendstoUniformlyOn_iff] | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ HasUniformSum f (tsumOn f) s | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro e ep | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | apply Filter.eventually_of_forall | case pos
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ (x : Finset ℕ), ∀ x_1 ∈ s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro n z zs | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
⊢ ∀ (x : Finset ℕ), ∀ x_1 ∈ s, dist (tsumOn f x_1) (x.sum fun n => f n x_1) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (n.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [tsumOn] | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (n.sum fun n => f n z) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (n.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp_rw [fz _ z zs] | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (n.sum fun n => f n z) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), 0) (n.sum fun n => 0) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp only [tsum_zero, Finset.sum_const_zero, dist_zero_left, Complex.norm_eq_abs,
AbsoluteValue.map_zero] | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), 0) (n.sum fun n => 0) < e | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | assumption | case pos.hp
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : c ≤ 0
fz : ∀ (n : ℕ), ∀ z ∈ s, f n z = 0
e : ℝ
ep : e > 0
n : Finset ℕ
z : ℂ
zs : z ∈ s
⊢ 0 < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | simp only [not_le] at c0 | case neg
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : ¬c ≤ 0
⊢ HasUniformSum f (tsumOn f) s | case neg
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ HasUniformSum f (tsumOn f) s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | apply uniformVanishing_to_tendsto_uniformly_on | case neg
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ HasUniformSum f (tsumOn f) s | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ UniformVanishing f s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro e ep | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
⊢ UniformVanishing f s | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | set t := (1 - ↑a) / ↑c * (e / 2) | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have tp : t > 0 := by bound | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rcases exists_pow_lt_of_lt_one tp a1 with ⟨n, nt⟩ | case neg.h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case neg.h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | use n | case neg.h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
⊢ ∃ n, ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
⊢ ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | intro N z NL zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
⊢ ∀ (N : Finset ℕ) (z : ℂ), Late N n → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
⊢ (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | have a1p : 1 - (a : ℝ) > 0 := by linarith | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
⊢ (N.sum fun n => Complex.abs (f n z)) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ (N.sum fun n => Complex.abs (f n z)) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | calc (N.sum fun n ↦ abs (f n z))
_ ≤ N.sum fun n ↦ c * a ^ n := Finset.sum_le_sum fun n _ ↦ hf n z zs
_ = c * N.sum fun n ↦ a ^ n := (Finset.mul_sum _ _ _).symm
_ ≤ c * (a ^ n * (1 - a)⁻¹) := by bound [late_geometric_bound NL a0 a1]
_ = a ^ n * (c * (1 - a)⁻¹) := by ring
_ ≤ t * (c * (1 - a)⁻¹) := by bound
... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ (N.sum fun n => Complex.abs (f n z)) < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
⊢ t > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
⊢ 1 - a > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound [late_geometric_bound NL a0 a1] | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ (c * N.sum fun n => a ^ n) ≤ c * (a ^ n * (1 - a)⁻¹) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ c * (a ^ n * (1 - a)⁻¹) = a ^ n * (c * (1 - a)⁻¹) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ a ^ n * (c * (1 - a)⁻¹) ≤ t * (c * (1 - a)⁻¹) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ (1 - a) / c * (e / 2) * (c * (1 - a)⁻¹) = (1 - a) * (1 -... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | rw [mul_inv_cancel a1p.ne', div_self c0.ne'] | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ (1 - a) * (1 - a)⁻¹ * (c / c) * (e / 2) = 1 * 1 * (e / 2... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ 1 * 1 * (e / 2) = e / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_uniformly_on | [113, 1] | [142, 27] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
c0 : 0 < c
e : ℝ
ep : e > 0
t : ℝ := (1 - a) / c * (e / 2)
tp : t > 0
n : ℕ
nt : a ^ n < t
N : Finset ℕ
z : ℂ
NL : Late N n
zs : z ∈ s
a1p : 1 - a > 0
⊢ e / 2 < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | set s : Set ℂ := {0} | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
⊢ Summable f | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
⊢ Summable f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | set g : ℕ → ℂ → ℂ := fun n _ ↦ f n | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
⊢ Summable f | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
⊢ Summable f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | have hg : ∀ n z, z ∈ s → abs (g n z) ≤ c * a ^ n := fun n z _ ↦ hf n | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
⊢ Summable f | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
⊢ Summable f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | have u := fast_series_converge_uniformly_on a0 a1 hg | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
⊢ Summable f | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
⊢ Summable f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | rw [HasUniformSum] at u | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : HasUniformSum (fun n z => g n z) (tsumOn fun n z => g n z) s
⊢ Summable f | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
⊢ Summable f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | rw [tendstoUniformlyOn_singleton_iff_tendsto] at u | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : TendstoUniformlyOn (fun N z => N.sum fun n => g n z) (tsumOn fun n z => g n z) atTop s
⊢ Summable f | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (𝓝 (tsumOn (fun n z => g n z) 0))
⊢ Summable f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | apply HasSum.summable | f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (𝓝 (tsumOn (fun n z => g n z) 0))
⊢ Summable f | case h
f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (𝓝 (tsumOn (fun n z => g n z) 0))
⊢ HasSum f ?a
case a
f : ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge_at | [145, 1] | [154, 36] | assumption | case h
f : ℕ → ℂ
c a : ℝ
a0 : 0 ≤ a
a1 : a < 1
hf : ∀ (n : ℕ), Complex.abs (f n) ≤ c * a ^ n
s : Set ℂ := {0}
g : ℕ → ℂ → ℂ := fun n x => f n
hg : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (g n z) ≤ c * a ^ n
u : Filter.Tendsto (fun n => n.sum fun n => g n 0) atTop (𝓝 (tsumOn (fun n z => g n z) 0))
⊢ HasSum f ?a
case a
f : ℕ →... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | use tsumOn f | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ ∃ g, AnalyticOn ℂ g s ∧ HasSumOn f g s | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ AnalyticOn ℂ (tsumOn f) s ∧ HasSumOn f (tsumOn f) s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | constructor | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ AnalyticOn ℂ (tsumOn f) s ∧ HasSumOn f (tsumOn f) s | case h.left
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ AnalyticOn ℂ (tsumOn f) s
case h.right
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | exact uniform_analytic_lim o (fun N ↦ N.analyticOn_sum fun _ _ ↦ h _)
(fast_series_converge_uniformly_on a0 a1 hf) | case h.left
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ AnalyticOn ℂ (tsumOn f) s | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | fast_series_converge | [157, 1] | [163, 87] | exact fun z zs ↦ Summable.hasSum (fast_series_converge_at a0 a1 fun n ↦ hf n z zs) | case h.right
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z) ≤ c * a ^ n
⊢ HasSumOn f (tsumOn f) s | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | rw [HasSum] at h ⊢ | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : HasSum f g
⊢ HasSum (Stream'.cons a f) (a + g) | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
⊢ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (𝓝 (a + g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | have ha := Filter.Tendsto.comp (Continuous.tendsto (continuous_add_left a) g) h | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
⊢ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (𝓝 (a + g)) | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
⊢ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (𝓝 (a + g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | have s : ((fun z ↦ a + z) ∘ fun N : Finset ℕ ↦ N.sum f) =
(fun N : Finset ℕ ↦ N.sum (Stream'.cons a f)) ∘ push := by
apply funext; intro N; simp; exact push_sum | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
⊢ Filter.Tendsto (fun s => s.sum fun b => Stream'.cons a f b) atTop (𝓝 (a + g)) | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
s : ((fun z => a + z) ∘ fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | rw [s] at ha | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
s : ((fun z => a + z) ∘ fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => ... | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) ∘ push) atTop (𝓝 (a + g))
s : ((fun z => a + z) ∘ fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | exact tendsto_comp_push.mp ha | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun N => N.sum (Stream'.cons a f)) ∘ push) atTop (𝓝 (a + g))
s : ((fun z => a + z) ∘ fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.su... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | apply funext | X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
⊢ ((fun z => a + z) ∘ fun N => N.sum f) = (fun N => N.sum (Stream'.cons a f)) ∘ push | case h
X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
⊢ ∀ (x : Finset ℕ), ((fun z => a + z) ∘ fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) ∘ push) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | intro N | case h
X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
⊢ ∀ (x : Finset ℕ), ((fun z => a + z) ∘ fun N => N.sum f) x = ((fun N => N.sum (Stream'.cons a f)) ∘ push) ... | case h
X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
N : Finset ℕ
⊢ ((fun z => a + z) ∘ fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) ∘ push) N |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | simp | case h
X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
N : Finset ℕ
⊢ ((fun z => a + z) ∘ fun N => N.sum f) N = ((fun N => N.sum (Stream'.cons a f)) ∘ push) N | case h
X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
N : Finset ℕ
⊢ a + N.sum f = (push N).sum (Stream'.cons a f) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons | [166, 1] | [174, 32] | exact push_sum | case h
X : Type
inst✝ : NormedAddCommGroup X
a g : X
f : ℕ → X
h : Filter.Tendsto (fun s => s.sum fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a + b) ∘ fun s => s.sum fun b => f b) atTop (𝓝 (a + g))
N : Finset ℕ
⊢ a + N.sum f = (push N).sum (Stream'.cons a f) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rcases h with ⟨g, h⟩ | a : ℂ
f : ℕ → ℂ
h : Summable f
⊢ tsum (Stream'.cons a f) = a + tsum f | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasSum f g
⊢ tsum (Stream'.cons a f) = a + tsum f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rw [HasSum.tsum_eq h] | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasSum f g
⊢ tsum (Stream'.cons a f) = a + tsum f | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasSum f g
⊢ tsum (Stream'.cons a f) = a + g |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | rw [HasSum.tsum_eq _] | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasSum f g
⊢ tsum (Stream'.cons a f) = a + g | a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasSum f g
⊢ HasSum (fun b => Stream'.cons a f b) (a + g) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_cons' | [176, 1] | [179, 87] | exact sum_cons h | a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasSum f g
⊢ HasSum (fun b => Stream'.cons a f b) (a + g) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | have c := sum_cons (a := -f 0) h | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
⊢ HasSum (fun n => f (n + 1)) (g - f 0) | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
⊢ HasSum (fun n => f (n + 1)) (g - f 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [HasSum] | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
⊢ HasSum (fun n => f (n + 1)) (g - f 0) | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
⊢ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (𝓝 (g - f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [neg_add_eq_sub, HasSum, ← tendsto_comp_push, ← tendsto_comp_push] at c | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : HasSum (Stream'.cons (-f 0) f) (-f 0 + g)
⊢ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (𝓝 (g - f 0)) | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) ∘ push) ∘ push) atTop (𝓝 (g - f 0))
⊢ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (𝓝 (g - f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | have s : ((fun N : Finset ℕ ↦ N.sum fun n ↦ (Stream'.cons (-f 0) f) n) ∘ push) ∘ push =
fun N : Finset ℕ ↦ N.sum fun n ↦ f (n + 1) := by
clear c h g; apply funext; intro N; simp
nth_rw 2 [← Stream'.eta f]
simp only [←push_sum, Stream'.head, Stream'.tail, Stream'.get]
abel | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) ∘ push) ∘ push) atTop (𝓝 (g - f 0))
⊢ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (𝓝 (g - f 0)) | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) ∘ push) ∘ push) atTop (𝓝 (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.sum fun ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | rw [s] at c | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) ∘ push) ∘ push) atTop (𝓝 (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.sum fun ... | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (𝓝 (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (𝓝 (g - f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | assumption | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (fun N => N.sum fun n => f (n + 1)) atTop (𝓝 (g - f 0))
s : ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.sum fun b => f (b + 1)) atTop (𝓝 (g - f... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | clear c h g | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
g : X
h : HasSum f g
c : Filter.Tendsto (((fun s => s.sum fun b => Stream'.cons (-f 0) f b) ∘ push) ∘ push) atTop (𝓝 (g - f 0))
⊢ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1) | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
⊢ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | apply funext | X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
⊢ ((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push = fun N => N.sum fun n => f (n + 1) | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
⊢ ∀ (x : Finset ℕ), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push) x = x.sum fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | intro N | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
⊢ ∀ (x : Finset ℕ), (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push) x = x.sum fun n => f (n + 1) | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push) N = N.sum fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | simp | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ (((fun N => N.sum fun n => Stream'.cons (-f 0) f n) ∘ push) ∘ push) N = N.sum fun n => f (n + 1) | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | nth_rw 2 [← Stream'.eta f] | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ ((push (push N)).sum fun n => Stream'.cons (-f 0) f n) = N.sum fun n => f (n + 1) | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | simp only [←push_sum, Stream'.head, Stream'.tail, Stream'.get] | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ ((push (push N)).sum fun n => Stream'.cons (-f 0) (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.sum fun n => f (n + 1) | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | sum_drop | [181, 1] | [193, 26] | abel | case h
X : Type
inst✝ : NormedAddCommGroup X
f : ℕ → X
N : Finset ℕ
⊢ -f 0 + (f 0 + N.sum fun x => f (x + 1)) = N.sum fun x => f (x + 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_mono | [21, 1] | [23, 73] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 1 < x
xy : x ≤ y
⊢ 0 < x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | have lx := neg_pos.mpr (Real.log_neg x0 x1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
⊢ (-y.log).log < (-x.log).log ↔ x < y | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
⊢ (-y.log).log < (-x.log).log ↔ x < y |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | have ly := neg_pos.mpr (Real.log_neg y0 y1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
⊢ (-y.log).log < (-x.log).log ↔ x < y | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
⊢ (-y.log).log < (-x.log).log ↔ x < y |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_neg_log_strict_anti | [25, 1] | [30, 76] | rw [Real.log_lt_log_iff ly lx, neg_lt_neg_iff, Real.log_lt_log_iff x0 y0] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 < x
y0 : 0 < y
x1 : x < 1
y1 : y < 1
lx : 0 < -x.log
ly : 0 < -y.log
⊢ (-y.log).log < (-x.log).log ↔ x < y | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | rw [Real.le_log_iff_exp_le (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ log 2 * x ≤ (1 + x).log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ (log 2 * x).exp ≤ 1 + x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | have x0' : 0 ≤ 1 - x := by linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ (log 2 * x).exp ≤ 1 + x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | have h := convexOn_exp.2 (mem_univ 0) (mem_univ (log 2)) x0' x0 (by abel) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : ((1 - x) • 0 + x • log 2).exp ≤ (1 - x) • exp 0 + x • (log 2).exp
⊢ (log 2 * x).exp ≤ 1 + x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | simp only [smul_eq_mul, mul_zero, zero_add, Real.exp_zero, mul_one, Real.exp_log zero_lt_two] at h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : ((1 - x) • 0 + x • log 2).exp ≤ (1 - x) • exp 0 + x • (log 2).exp
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 - x + x * 2
⊢ (log 2 * x).exp ≤ 1 + x |
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