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/Products.lean | product_pow' | [118, 1] | [120, 96] | rw [HasProd.tprod_eq h] | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ tprod f ^ p = ∏' (n : ℕ), f n ^ p | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ g ^ p = ∏' (n : ℕ), f n ^ p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rw [HasProd.tprod_eq _] | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ g ^ p = ∏' (n : ℕ), f n ^ p | f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ HasProd (fun n => f n ^ p) (g ^ p) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | exact product_pow p h | f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ HasProd (fun n => f n ^ p) (g ^ p) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | rw [HasProd] at h ⊢ | a g : ℂ
f : ℕ → ℂ
h : HasProd f g
⊢ HasProd (Stream'.cons a f) (a * g) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | have ha := Filter.Tendsto.comp (Continuous.tendsto (continuous_mul_left a) g) h | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | have s : ((fun z ↦ a * z) ∘ fun N : Finset ℕ ↦ N.prod f) =
(fun N : Finset ℕ ↦ N.prod (Stream'.cons a f)) ∘ push := by
apply funext; intro N; simp; exact push_prod | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | rw [s] at ha | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b... | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) ∘ push) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) at... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | exact tendsto_comp_push.mp ha | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) ∘ push) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) at... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | apply funext | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ ∀ (x : Finset ℕ), ((fun z => a * z) ∘ fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) ∘ push) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | intro N | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ ∀ (x : Finset ℕ), ((fun z => a * z) ∘ fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) ∘ push) x | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ ((fun z => a * z) ∘ fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) ∘ push) N |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | simp | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ ((fun z => a * z) ∘ fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) ∘ push) N | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (Stream'.cons a f) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | exact push_prod | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (Stream'.cons a f) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rcases h with ⟨g, h⟩ | a : ℂ
f : ℕ → ℂ
h : ProdExists f
⊢ tprod (Stream'.cons a f) = a * tprod f | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * tprod f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rw [HasProd.tprod_eq h] | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * tprod f | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * g |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rw [HasProd.tprod_eq _] | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * g | a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ HasProd (fun b => Stream'.cons a f b) (a * g) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | exact product_cons h | a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ HasProd (fun b => Stream'.cons a f b) (a * g) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | have c := @product_cons (f 0)⁻¹ _ _ h | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
⊢ HasProd (fun n => f (n + 1)) (g / f 0) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ HasProd (fun n => f (n + 1)) (g / f 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [HasProd] | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ HasProd (fun n => f (n + 1)) (g / f 0) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [inv_mul_eq_div, HasProd, ← tendsto_comp_push, ← tendsto_comp_push] at c | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | have s : ((fun N : Finset ℕ ↦ N.prod fun n ↦ (Stream'.cons (f 0)⁻¹ f) n) ∘ push) ∘ push =
fun N : Finset ℕ ↦ N.prod fun n ↦ f (n + 1) := by
clear c h g; apply funext; intro N; simp
nth_rw 2 [← Stream'.eta f]
simp only [←push_prod, Stream'.head, Stream'.tail, Stream'.get, ←mul_assoc, inv_mul_cancel f0,
one... | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) at... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [s] at c | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) at... | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | assumption | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | clear c h g | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
⊢ ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1) | f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | apply funext | f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ∀ (x : Finset ℕ), (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) x = x.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | intro N | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ∀ (x : Finset ℕ), (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) x = x.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) N = N.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | simp | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) N = N.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ ((push (push N)).prod fun n => Stream'.cons (f 0)⁻¹ f n) = N.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | nth_rw 2 [← Stream'.eta f] | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ ((push (push N)).prod fun n => Stream'.cons (f 0)⁻¹ f n) = N.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ ((push (push N)).prod fun n => Stream'.cons (f 0)⁻¹ (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | simp only [←push_prod, Stream'.head, Stream'.tail, Stream'.get, ←mul_assoc, inv_mul_cancel f0,
one_mul] | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ ((push (push N)).prod fun n => Stream'.cons (f 0)⁻¹ (Stream'.cons (Stream'.head f) (Stream'.tail f)) n) =
N.prod fun n => f (n + 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop' | [153, 1] | [155, 98] | rcases h with ⟨g, h⟩ | f : ℕ → ℂ
f0 : f 0 ≠ 0
h : ProdExists f
⊢ ∏' (n : ℕ), f (n + 1) = tprod f / f 0 | case intro
f : ℕ → ℂ
f0 : f 0 ≠ 0
g : ℂ
h : HasProd f g
⊢ ∏' (n : ℕ), f (n + 1) = tprod f / f 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop' | [153, 1] | [155, 98] | rw [HasProd.tprod_eq h] | case intro
f : ℕ → ℂ
f0 : f 0 ≠ 0
g : ℂ
h : HasProd f g
⊢ ∏' (n : ℕ), f (n + 1) = tprod f / f 0 | case intro
f : ℕ → ℂ
f0 : f 0 ≠ 0
g : ℂ
h : HasProd f g
⊢ ∏' (n : ℕ), f (n + 1) = g / f 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop' | [153, 1] | [155, 98] | rw [HasProd.tprod_eq _] | case intro
f : ℕ → ℂ
f0 : f 0 ≠ 0
g : ℂ
h : HasProd f g
⊢ ∏' (n : ℕ), f (n + 1) = g / f 0 | f : ℕ → ℂ
f0 : f 0 ≠ 0
g : ℂ
h : HasProd f g
⊢ HasProd (fun n => f (n + 1)) (g / f 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop' | [153, 1] | [155, 98] | exact product_drop f0 h | f : ℕ → ℂ
f0 : f 0 ≠ 0
g : ℂ
h : HasProd f g
⊢ HasProd (fun n => f (n + 1)) (g / f 0) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | rw [HasProd] | f : ℕ → ℂ
f0 : f 0 = 0
⊢ HasProd f 0 | f : ℕ → ℂ
f0 : f 0 = 0
⊢ Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | rw [Metric.tendsto_atTop] | f : ℕ → ℂ
f0 : f 0 = 0
⊢ Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 0) | f : ℕ → ℂ
f0 : f 0 = 0
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.prod fun b => f b) 0 < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | intro e ep | f : ℕ → ℂ
f0 : f 0 = 0
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.prod fun b => f b) 0 < ε | f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ n ≥ N, dist (n.prod fun b => f b) 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | use Finset.range 1 | f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ n ≥ N, dist (n.prod fun b => f b) 0 < e | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
⊢ ∀ n ≥ Finset.range 1, dist (n.prod fun b => f b) 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | intro N N1 | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
⊢ ∀ n ≥ Finset.range 1, dist (n.prod fun b => f b) 0 < e | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
N : Finset ℕ
N1 : N ≥ Finset.range 1
⊢ dist (N.prod fun b => f b) 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | simp at N1 | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
N : Finset ℕ
N1 : N ≥ Finset.range 1
⊢ dist (N.prod fun b => f b) 0 < e | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
N : Finset ℕ
N1 : 0 ∈ N
⊢ dist (N.prod fun b => f b) 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | rw [Finset.prod_eq_zero N1 f0] | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
N : Finset ℕ
N1 : 0 ∈ N
⊢ dist (N.prod fun b => f b) 0 < e | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
N : Finset ℕ
N1 : 0 ∈ N
⊢ dist 0 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_head_zero | [158, 1] | [161, 52] | simpa | case h
f : ℕ → ℂ
f0 : f 0 = 0
e : ℝ
ep : e > 0
N : Finset ℕ
N1 : 0 ∈ N
⊢ dist 0 0 < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_split | [164, 1] | [166, 38] | by_cases f0 : f 0 = 0 | f : ℕ → ℂ
h : ProdExists f
⊢ tprod f = f 0 * ∏' (n : ℕ), f (n + 1) | case pos
f : ℕ → ℂ
h : ProdExists f
f0 : f 0 = 0
⊢ tprod f = f 0 * ∏' (n : ℕ), f (n + 1)
case neg
f : ℕ → ℂ
h : ProdExists f
f0 : ¬f 0 = 0
⊢ tprod f = f 0 * ∏' (n : ℕ), f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_split | [164, 1] | [166, 38] | rw [product_drop' f0 h] | case neg
f : ℕ → ℂ
h : ProdExists f
f0 : ¬f 0 = 0
⊢ tprod f = f 0 * ∏' (n : ℕ), f (n + 1) | case neg
f : ℕ → ℂ
h : ProdExists f
f0 : ¬f 0 = 0
⊢ tprod f = f 0 * (tprod f / f 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_split | [164, 1] | [166, 38] | field_simp | case neg
f : ℕ → ℂ
h : ProdExists f
f0 : ¬f 0 = 0
⊢ tprod f = f 0 * (tprod f / f 0) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_split | [164, 1] | [166, 38] | rw [f0, (product_head_zero f0).tprod_eq] | case pos
f : ℕ → ℂ
h : ProdExists f
f0 : f 0 = 0
⊢ tprod f = f 0 * ∏' (n : ℕ), f (n + 1) | case pos
f : ℕ → ℂ
h : ProdExists f
f0 : f 0 = 0
⊢ 0 = 0 * ∏' (n : ℕ), f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_split | [164, 1] | [166, 38] | simp | case pos
f : ℕ → ℂ
h : ProdExists f
f0 : f 0 = 0
⊢ 0 = 0 * ∏' (n : ℕ), f (n + 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | have hd : DifferentiableOn ℂ f (closedBall z r) := by
intro x H; exact AnalyticAt.differentiableWithinAt (h x H) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | set p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
⊢ HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑r) z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z ↑r
⊢ HasFPowerSeriesOnBall f p z ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | exact DifferentiableOn.hasFPowerSeriesOnBall hd rp | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
hd : DifferentiableOn ℂ f (closedBall z ↑r)
p : FormalMultilinearSeries ℂ ℂ ℂ := cauchyPowerSeries f z ↑r
⊢ HasFPowerSeriesOnBall f p z ↑r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | intro x H | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
⊢ DifferentiableOn ℂ f (closedBall z ↑r) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
x : ℂ
H : x ∈ closedBall z ↑r
⊢ DifferentiableWithinAt ℂ f (closedBall z ↑r) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_on_cball_radius | [32, 1] | [38, 53] | exact AnalyticAt.differentiableWithinAt (h x H) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (closedBall z ↑r)
x : ℂ
H : x ∈ closedBall z ↑r
⊢ DifferentiableWithinAt ℂ f (closedBall z ↑r) x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | intro x hx | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
⊢ AnalyticOn ℂ f (closedBall z ↑s) | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ closedBall z ↑s
⊢ AnalyticAt ℂ f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | rw [closedBall] at hx | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ closedBall z ↑s
⊢ AnalyticAt ℂ f x | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ {y | dist y z ≤ ↑s}
⊢ AnalyticAt ℂ f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | simp at hx | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : x ∈ {y | dist y z ≤ ↑s}
⊢ AnalyticAt ℂ f x | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ AnalyticAt ℂ f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | have hb : x ∈ ball z r := by
rw [ball]; simp only [dist_lt_coe, Set.mem_setOf_eq]; exact lt_of_le_of_lt hx sr | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ AnalyticAt ℂ f x | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
hb : x ∈ ball z ↑r
⊢ AnalyticAt ℂ f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | exact h x hb | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
hb : x ∈ ball z ↑r
⊢ AnalyticAt ℂ f x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | rw [ball] | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ ball z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ {y | dist y z < ↑r} |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | simp only [dist_lt_coe, Set.mem_setOf_eq] | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ x ∈ {y | dist y z < ↑r} | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ nndist x z < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_small_cball | [45, 1] | [51, 15] | exact lt_of_le_of_lt hx sr | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
h : AnalyticOn ℂ f (ball z ↑r)
s : ℝ≥0
sr : s < r
x : ℂ
hx : nndist x z ≤ s
⊢ nndist x z < r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have h0 := analyticOn_small_cball h (r / 2) (NNReal.half_lt_self <| rp.ne') | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases analyticOn_cball_radius (half_pos rp) h0 with ⟨p, ph⟩ | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | set R := FormalMultilinearSeries.radius p | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | refine
⟨p, {
r_le := ?_
r_pos := ENNReal.coe_pos.mpr rp
hasSum := ?_ }⟩ | case intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∃ p, HasFPowerSeriesOnBall f p z ↑r | case intro.refine_1
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ↑r ≤ p.radius
case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | apply ENNReal.le_of_forall_pos_nnreal_lt | case intro.refine_1
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ↑r ≤ p.radius | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ (r_1 : ℝ≥0), 0 < r_1 → ↑r_1 < ↑r → ↑r_1 ≤ p.radius |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | intro t tp tr | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ (r_1 : ℝ≥0), 0 < r_1 → ↑r_1 < ↑r → ↑r_1 ≤ p.radius | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
⊢ ↑t ≤ p.radius |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have ht := analyticOn_small_cball h t (ENNReal.coe_lt_coe.mp tr) | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
⊢ ↑t ≤ p.radius | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
⊢ ↑t ≤ p.radius |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases analyticOn_cball_radius tp ht with ⟨p', hp'⟩ | case intro.refine_1.h
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
⊢ ↑t ≤ p.radius | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalM... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have pp : p = p' := HasFPowerSeriesAt.eq_formalMultilinearSeries ⟨↑(r / 2), ph⟩ ⟨t, hp'⟩ | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalM... | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalM... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [← pp] at hp' | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalM... | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalM... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | refine hp'.r_le | case intro.refine_1.h.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
t : ℝ≥0
tp : 0 < t
tr : ↑t < ↑r
ht : AnalyticOn ℂ f (closedBall z ↑t)
p' : FormalM... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | intro y yr | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
⊢ ∀ {y : ℂ}, y ∈ EMetric.ball 0 ↑r → HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : y ∈ EMetric.ball 0 ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [EMetric.ball, Set.mem_setOf] at yr | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : y ∈ EMetric.ball 0 ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases exists_between yr with ⟨t, t0, t1⟩ | case intro.refine_2
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
⊢ HasSum (fun n => (p n) fun x => y) (f (z + y)) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ HasSum ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have t1' : t.toNNReal < r := by
rw [← WithTop.coe_lt_coe]; exact lt_of_le_of_lt ENNReal.coe_toNNReal_le_self t1 | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ HasSum ... | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have ht := analyticOn_small_cball h t.toNNReal t1' | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.t... | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have tp : 0 < ENNReal.toNNReal t :=
ENNReal.toNNReal_pos (ne_of_gt <| pos_of_gt t0) (ne_top_of_lt t1) | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.t... | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rcases analyticOn_cball_radius tp ht with ⟨p', hp'⟩ | case intro.refine_2.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1' : t.t... | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | have pp : p = p' :=
HasFPowerSeriesAt.eq_formalMultilinearSeries ⟨↑(r / 2), ph⟩ ⟨t.toNNReal, hp'⟩ | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [← pp] at hp' | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | refine hp'.hasSum ?_ | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [EMetric.ball, Set.mem_setOf] | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | calc edist y 0
_ < t := t0
_ = ↑t.toNNReal := (ENNReal.coe_toNNReal <| ne_top_of_lt t1).symm | case intro.refine_2.intro.intro.intro
f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
t1'... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | rw [← WithTop.coe_lt_coe] | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ t.toNNReal < r | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ ↑t.toNNReal < ↑r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | analyticOn_ball_radius | [53, 1] | [87, 72] | exact lt_of_le_of_lt ENNReal.coe_toNNReal_le_self t1 | f : ℂ → ℂ
z : ℂ
r : ℝ≥0
rp : r > 0
h : AnalyticOn ℂ f (ball z ↑r)
h0 : AnalyticOn ℂ f (closedBall z ↑(r / 2))
p : FormalMultilinearSeries ℂ ℂ ℂ
ph : HasFPowerSeriesOnBall f p z ↑(r / 2)
R : ENNReal := p.radius
y : ℂ
yr : edist y 0 < ↑r
t : ENNReal
t0 : edist y 0 < t
t1 : t < ↑r
⊢ ↑t.toNNReal < ↑r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | set wr := abs w ^ n * r⁻¹ ^ n * d | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ wr |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [cauchyPowerSeries_apply f c r n w, smul_eq_mul, Complex.abs.map_mul] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs ((cauchyPowerSeries f c (↑r) n) fun x => w) ≤ wr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | generalize hg : (fun z ↦ (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (∮ (z : ℂ) in C(c, ↑r), (w / (z - c)) ^ n • (z - c)⁻¹ • f z) ≤ wr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have gs : ∀ z ∈ sphere c r, ‖g z‖ ≤ wr * r⁻¹ := by
intro z; simp only [mem_sphere_iff_norm, Complex.norm_eq_abs]; intro zr
simp only [← hg, zr, div_pow, Algebra.id.smul_eq_mul, AbsoluteValue.map_mul, map_div₀,
Complex.abs_pow, map_inv₀]
have zb : z ∈ closedBall c r := by
simp only [Metric.mem_closedBall, ... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ wr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have cn := circleIntegral.norm_integral_le_of_norm_le_const (NNReal.coe_nonneg r) gs | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
⊢ Complex.abs (2 * ↑π * I)⁻¹ * Complex.abs (circleIntegral g c ↑r) ≤ ... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
⊢ Comple... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | rw [Complex.norm_eq_abs] at cn | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : ‖∮ (z : ℂ) in C(c, ↑r), g z‖ ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
⊢ Comple... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [mul_inv_rev, Complex.inv_I, AbsoluteValue.map_neg, AbsoluteValue.map_mul,
Complex.abs_I, map_inv₀, Complex.abs_ofReal, Complex.abs_two, one_mul, div_pow,
Algebra.id.smul_eq_mul] at hg cn ⊢ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have p3 : |π| = π := abs_eq_self.mpr (by bound) | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | calc |π|⁻¹ * 2⁻¹ * abs (circleIntegral g c ↑r)
_ ≤ |π|⁻¹ * 2⁻¹ * (2 * π * r * (wr * r⁻¹)) := by bound
_ = π * |π|⁻¹ * (r * r⁻¹) * wr := by ring
_ = π * π⁻¹ * (r * r⁻¹) * wr := by rw [p3]
_ = 1 * (r * r⁻¹) * wr := by rw [mul_inv_cancel Real.pi_ne_zero]
_ = wr := by
rw [NNReal.coe_inv, mul_inv_cancel (NNRea... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
gs : ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹
cn : Complex.abs (∮ (z : ℂ) in C(c, ↑r), g z) ≤ 2 * π * ↑r * (wr * ↑r⁻¹)
hg : (fun z => w ^ n / (z - c) ^ n * ((z - c)⁻¹ * f ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | intro z | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
⊢ ∀ z ∈ sphere c ↑r, ‖g z‖ ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ z ∈ sphere c ↑r → ‖g z‖ ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | intro zr | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
⊢ Complex.abs (z - c) = ↑r → Complex.abs (g z) ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [← hg, zr, div_pow, Algebra.id.smul_eq_mul, AbsoluteValue.map_mul, map_div₀,
Complex.abs_pow, map_inv₀] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs (g z) ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have zb : z ∈ closedBall c r := by
simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist,
Complex.dist_eq, zr, le_refl] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.abs (f z)) ≤ wr * ↑r⁻¹ | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | have zs := h z zb | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
⊢ Complex.abs w ^ n / ↑r ^ n * ((↑r)⁻¹ * Complex.a... | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | calc abs w ^ n / ↑r ^ n * (r⁻¹ * abs (f z))
_ = abs w ^ n * (r⁻¹ ^ n : ℝ≥0) * (r⁻¹ * abs (f z)) := by
rw [div_eq_mul_inv, ← inv_pow, NNReal.coe_pow, NNReal.coe_inv]
_ ≤ abs w ^ n * r⁻¹ ^ n * (r⁻¹ * d) := by bound
_ = abs w ^ n * r⁻¹ ^ n * d * r⁻¹ := by ring
_ = wr * r⁻¹ := rfl | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
zb : z ∈ closedBall c ↑r
zs : Complex.abs (f z) ≤ ↑d
⊢ Complex.abs w ^ n / ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | cauchy_bound | [89, 1] | [121, 92] | simp only [Metric.mem_closedBall, dist_le_coe, ← NNReal.coe_le_coe, coe_nndist,
Complex.dist_eq, zr, le_refl] | f : ℂ → ℂ
c : ℂ
r d : ℝ≥0
w : ℂ
n : ℕ
rp : r > 0
h : ∀ w ∈ closedBall c ↑r, Complex.abs (f w) ≤ ↑d
wr : ℝ := Complex.abs w ^ n * ↑r⁻¹ ^ n * ↑d
g : ℂ → ℂ
hg : (fun z => (w / (z - c)) ^ n • (z - c)⁻¹ • f z) = g
z : ℂ
zr : Complex.abs (z - c) = ↑r
⊢ z ∈ closedBall c ↑r | no goals |
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