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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | set g : 𝕜 → 𝕜 := fun x ↦ x + T with gdef | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
⊢ deriv f (x + T) = deriv f x | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
⊢ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
⊢ deriv f (x + T) = deriv f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | have diff_g : Differentiable 𝕜 g := by
apply differentiable_id.add_const | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
⊢ deriv f (x + T) = deriv f x | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
⊢ deriv ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | have : deriv (f ∘ g) x = ((deriv f) ∘ g) x := by
calc deriv (f ∘ g) x
_ = deriv g x • deriv f (g x) := deriv.scomp x (diff_f.differentiable (by norm_num)).differentiableAt diff_g.differentiableAt
_ = deriv f (g x) := by rw [gdef, deriv_add_const, deriv_id'']; simp | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ deriv f (x + T) = deriv f x | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ g) x = (deriv f ∘ g) x... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | rw [gdef] at this | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ g) x = (deriv f ∘ g) x... | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ fun x => x + T) x = (d... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | simp at this | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ fun x => x + T) x = (d... | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ fun x => x + T) x = de... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | convert this.symm | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ fun x => x + T) x = de... | case h.e'_3.h.e'_6
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ fun... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | ext y | case h.e'_3.h.e'_6
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ fun... | case h.e'_3.h.e'_6.h
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | simp | case h.e'_3.h.e'_6.h
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ f... | case h.e'_3.h.e'_6.h
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6.h
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fu... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | exact (periodic_f y).symm | case h.e'_3.h.e'_6.h
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
this : deriv (f ∘ f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6.h
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fu... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | apply differentiable_id.add_const | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
⊢ Differentiable 𝕜 g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
⊢ Differ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | calc deriv (f ∘ g) x
_ = deriv g x • deriv f (g x) := deriv.scomp x (diff_f.differentiable (by norm_num)).differentiableAt diff_g.differentiableAt
_ = deriv f (g x) := by rw [gdef, deriv_add_const, deriv_id'']; simp | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ deriv (f ∘ g) x = (deriv f ∘ g) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | norm_num | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ 1 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | rw [gdef, deriv_add_const, deriv_id''] | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ deriv g x • deriv f (g x) = deriv f (g... | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ (fun x => 1) x • deriv f ((fun x => x ... | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | simp | 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable 𝕜 g
⊢ (fun x => 1) x • deriv f ((fun x => x ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
F : Type
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
T : 𝕜
diff_f : ContDiff 𝕜 1 f
periodic_f : Function.Periodic f T
x : 𝕜
g : 𝕜 → 𝕜 := fun x => x + T
gdef : g = fun x => x + T
diff_g :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x := by
intro x _
rw [hasDerivAt_deriv_iff]
apply fdiff.differentiable (by norm_num) | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x := by
intro x _
rw [hasDerivAt_deriv_iff]
apply (contDiff_succ_iff_deriv.mp fdiff).2.differentiable (by norm_num) | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have fourierCoeffOn_eq {n : ℤ} (hn : n ≠ 0): (fourierCoeffOn Real.two_pi_pos f n) = - 1 / (n^2) * fourierCoeffOn Real.two_pi_pos (fun x ↦ deriv (deriv f) x) n := by
rw [fourierCoeffOn_of_hasDerivAt Real.two_pi_pos hn h, fourierCoeffOn_of_hasDerivAt Real.two_pi_pos hn h']
. have := periodicf 0
simp at this
s... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs (fourierCoeffOn Real.two_pi_pos f n) ≤ C / ↑n ^ 2 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 / ↑n ^ ... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
⊢ ∃ C, ∀ (n : ℤ), n ≠ 0 → Complex.abs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | obtain ⟨C, hC⟩ := fourierCoeffOn_bound (contDiff_one_iff_deriv.mp (contDiff_succ_iff_deriv.mp fdiff).2).2 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 / ↑n ^ ... | case intro
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n =... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | use C | case intro
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n =... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | intro n hn | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | rw [fourierCoeffOn_eq hn] | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp only [Nat.cast_one, Int.cast_pow, map_mul, map_div₀, map_neg_eq_map, map_one, map_pow,
Complex.abs_intCast, sq_abs, one_div] | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | rw [div_eq_mul_inv, mul_comm] | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | gcongr | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -1 ... | case h.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | exact hC n | case h.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n : ℤ},
n ≠ 0 →
fourierCoeffOn Real.two_pi_pos f n = -... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
∀ {n :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | intro x _ | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
⊢ ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ HasDerivAt f (deriv f x) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
⊢ ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | rw [hasDerivAt_deriv_iff] | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ HasDerivAt f (deriv f x) x | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ DifferentiableAt ℝ f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ HasDerivAt f (deriv f x) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | apply fdiff.differentiable (by norm_num) | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ DifferentiableAt ℝ f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ DifferentiableAt ℝ f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | norm_num | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ 1 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ 1 ≤ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | intro x _ | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
⊢ ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ HasDerivAt (deriv f) (deriv (deriv f) x) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
⊢ ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | rw [hasDerivAt_deriv_iff] | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ HasDerivAt (deriv f) (deriv (deriv f) x) x | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ DifferentiableAt ℝ (deriv f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ HasDerivAt (deriv f) (deriv (deriv f) x) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | apply (contDiff_succ_iff_deriv.mp fdiff).2.differentiable (by norm_num) | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ DifferentiableAt ℝ (deriv f) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ DifferentiableAt ℝ (deriv f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | norm_num | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ 1 ≤ ↑1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : ℝ
a✝ : x ∈ Set.uIcc 0 (2 * Real.pi)
⊢ 1 ≤ ↑1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | rw [fourierCoeffOn_of_hasDerivAt Real.two_pi_pos hn h, fourierCoeffOn_of_hasDerivAt Real.two_pi_pos hn h'] | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ fourierCoeffOn Real.two_pi_pos f n = -1 / ↑n ^ 2 * fourierCoeffOn Real.two_pi_p... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
((fourier (-n)) ↑0 * (f (2 * Real.... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ fourierCoeffOn Rea... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | . have := periodicf 0
simp at this
simp [this]
have periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi) := periodic_deriv (fdiff.of_le one_le_two) periodicf
have := periodic_deriv_f 0
simp at this
simp [this]
ring_nf
simp
left
rw [mul_inv_cancel, one_mul]
simp
exact Real.pi_pos.ne.symm | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
((fourier (-n)) ↑0 * (f (2 * Real.... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Rea... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑Real.pi... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | . apply Continuous.intervalIntegrable
exact (contDiff_one_iff_deriv.mp (contDiff_succ_iff_deriv.mp fdiff).2).2 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Rea... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | . apply Continuous.intervalIntegrable
exact (contDiff_succ_iff_deriv.mp fdiff).2.continuous | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have := periodicf 0 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
((fourier (-n)) ↑0 * (f (2 * Real.... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (0 + 2 * Real.pi) = f 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
(... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ 1 / (-2 * ↑Real.pi... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp at this | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (0 + 2 * Real.pi) = f 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
(... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.pi) = f 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
((fou... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (0 + 2 * Re... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp [this] | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.pi) = f 0
⊢ 1 / (-2 * ↑Real.pi * Complex.I * ↑n) *
((fou... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.pi) = f 0
⊢ -((-(2 * ↑Real.pi * Complex.I * ↑n))⁻¹ *
(... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.p... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi) := periodic_deriv (fdiff.of_le one_le_two) periodicf | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.pi) = f 0
⊢ -((-(2 * ↑Real.pi * Complex.I * ↑n))⁻¹ *
(... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * ... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.p... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have := periodic_deriv_f 0 | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * ... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this : f (2 * Real.p... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp at this | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp [this] | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | ring_nf | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | left | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 *... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv ... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | rw [mul_inv_cancel, one_mul] | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv ... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv ... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | exact Real.pi_pos.ne.symm | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
this✝ : f (2 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | apply Continuous.intervalIntegrable | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Rea... | case hu
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ Continuous fun x => deriv (deriv f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | exact (contDiff_one_iff_deriv.mp (contDiff_succ_iff_deriv.mp fdiff).2).2 | case hu
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ Continuous fun x => deriv (deriv f) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ Continuous... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | apply Continuous.intervalIntegrable | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | case hu
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ Continuous fun x => deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ IntervalIntegrable... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | exact (contDiff_succ_iff_deriv.mp fdiff).2.continuous | case hu
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ Continuous fun x => deriv f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
h : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : ∀ x ∈ Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : ℤ
hn : n ≠ 0
⊢ Continuous... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | have := hfa.nat_add_neg.tendsto_sum_nat | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N) ↑N, f n) atTop (𝓝 a) | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N) ↑N, f n) atTop (𝓝 a) | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N) ↑N, f n) atTop (𝓝 a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | have := (Filter.Tendsto.add_const (- (f 0))) this | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N) ↑N, f n) atTop (𝓝 a) | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun k => ∑ i ∈ range k, (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 (a + f 0 + -f 0))
⊢ Tendsto (fun N => ∑ n ∈ Icc... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N) ↑N, f n) atTop (�... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | simp at this | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun k => ∑ i ∈ range k, (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 (a + f 0 + -f 0))
⊢ Tendsto (fun N => ∑ n ∈ Icc... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun k => ∑ i ∈ range k, (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N)... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun k => ∑ i ∈ range k, (f ↑i + f (-↑i)) + -f... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [←tendsto_add_atTop_iff_nat 1] at this | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun k => ∑ i ∈ range k, (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.ofNat N)... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.of... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun k => ∑ i ∈ range k, (f ↑i + f (-↑i)) + -f... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | convert this using 1 | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ Tendsto (fun N => ∑ n ∈ Icc (-Int.of... | case h.e'_3
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ (fun N => ∑ n ∈ Icc (-In... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | ext N | case h.e'_3
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ (fun N => ∑ n ∈ Icc (-In... | case h.e'_3.h
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
⊢ ∑ n ∈ Icc (-Int.... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | induction' N with N ih | case h.e'_3.h
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
⊢ ∑ n ∈ Icc (-Int.... | case h.e'_3.h.zero
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ ∑ n ∈ Icc (-Int.o... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . simp | case h.e'_3.h.zero
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ ∑ n ∈ Icc (-Int.o... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Ic... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.zero
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . have : Icc (- Int.ofNat (Nat.succ N)) (Nat.succ N) = insert (↑(Nat.succ N)) (insert (-Int.ofNat (Nat.succ N)) (Icc (-Int.ofNat N) N)) := by
rw [←Ico_insert_right, ←Ioo_insert_left]
. congr
ext n
simp only [Int.ofNat_eq_coe, mem_Ioo, mem_Icc]
push_cast
rw [Int.lt_add_one_iff, neg_add, ←... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Ic... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | simp | case h.e'_3.h.zero
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
⊢ ∑ n ∈ Icc (-Int.o... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.zero
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | have : Icc (- Int.ofNat (Nat.succ N)) (Nat.succ N) = insert (↑(Nat.succ N)) (insert (-Int.ofNat (Nat.succ N)) (Icc (-Int.ofNat N) N)) := by
rw [←Ico_insert_right, ←Ioo_insert_left]
. congr
ext n
simp only [Int.ofNat_eq_coe, mem_Ioo, mem_Icc]
push_cast
rw [Int.lt_add_one_iff, neg_add, ←sub_eq_add_neg... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Ic... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [this, sum_insert, sum_insert, ih, ←add_assoc] | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | symm | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [sum_range_succ, add_comm, ←add_assoc, add_comm] | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | simp only [Nat.cast_add, Nat.cast_one, neg_add_rev, Int.reduceNeg, Nat.succ_eq_add_one,
Int.ofNat_eq_coe, add_right_inj] | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [add_comm] | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . simp | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . norm_num
linarith | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [←Ico_insert_right, ←Ioo_insert_left] | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . congr
ext n
simp only [Int.ofNat_eq_coe, mem_Ioo, mem_Icc]
push_cast
rw [Int.lt_add_one_iff, neg_add, ←sub_eq_add_neg, Int.sub_one_lt_iff] | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . norm_num
linarith | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . norm_num
linarith | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | congr | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | case e_a.e_a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-In... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | ext n | case e_a.e_a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-In... | case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | simp only [Int.ofNat_eq_coe, mem_Ioo, mem_Icc] | case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-... | case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | push_cast | case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-... | case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [Int.lt_add_one_iff, neg_add, ←sub_eq_add_neg, Int.sub_one_lt_iff] | case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.a
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | norm_num | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | linarith | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | norm_num | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | linarith | β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ Icc (-Int.ofNat N) ↑N... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | simp | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | norm_num | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | linarith | case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n + 1), (f ↑i + f (-↑i)) + -f 0) atTop (𝓝 a)
N : ℕ
ih : ∑ n ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
β : Type
inst✝² : AddCommGroup β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousAdd β
f : ℤ → β
a : β
hfa : HasSum f a
this✝¹ : Tendsto (fun n => ∑ i ∈ range n, (f ↑i + f (-↑i))) atTop (𝓝 (a + f 0))
this✝ : Tendsto (fun n => ∑ i ∈ range (n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have fact_two_pi_pos : Fact (0 < 2 * Real.pi) := by
rw [fact_iff]
exact Real.two_pi_pos | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
⊢ ∃ N₀, ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) ≤ ε | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
⊢ ∃ N₀, ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
⊢ ∃ N₀, ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | set g : C(AddCircle (2 * Real.pi), ℂ) := ⟨AddCircle.liftIco (2*Real.pi) 0 f, AddCircle.liftIco_continuous ((periodicf 0).symm) fdiff.continuous.continuousOn⟩ with g_def | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
⊢ ∃ N₀, ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) ≤ ε | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
⊢ ∃ N₀, ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have two_pi_pos' : 0 < 0 + 2 * Real.pi := by linarith [Real.two_pi_pos] | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have fourierCoeff_correspondence {i : ℤ} : fourierCoeff g i = fourierCoeffOn two_pi_pos' f i := fourierCoeff_liftIco_eq f i | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | simp at fourierCoeff_correspondence | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have function_sum : HasSum (fun (i : ℤ) => fourierCoeff g i • fourier i) g := by
apply hasSum_fourier_series_of_summable
obtain ⟨C, hC⟩ := fourierCoeffOn_ContDiff_two_bound periodicf fdiff
set maj : ℤ → ℝ := fun i ↦ 1 / (i ^ 2) * C with maj_def
have summable_maj : Summable maj := by
by_cases Ceq0 : C = 0
... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have := int_sum_nat function_sum | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | rw [ContinuousMap.tendsto_iff_tendstoUniformly, Metric.tendstoUniformly_iff] at this | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have := this ε εpos | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | rw [Filter.eventually_atTop] at this | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | obtain ⟨N₀, hN₀⟩ := this | f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_t... | case intro
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, c... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | use N₀ | case intro
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, c... | case h
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, conti... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℝ → ℂ
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff ℝ 2 f
ε : ℝ
εpos : ε > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), ℂ) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := ⋯ }
... |
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