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 : π β π := fun x β¦ x + T with gdef | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
β’ deriv f (x + T) = deriv f x | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
β’ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
β’ 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 π g := by
apply differentiable_id.add_const | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
β’ deriv f (x + T) = deriv f x | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
β’ 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 : deriv (f β g) x = ((deriv f) β g) x := by
calc deriv (f β g) x
_ = deriv g x β’ 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 | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ deriv f (x + T) = deriv f x | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β g) x = (deriv f β g) x
β’ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ 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] | rw [gdef] at this | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β g) x = (deriv f β g) x
β’ deriv f (x + T) = deriv f x | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = (deriv f β fun x => x + T) x
β’ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β g) x = (deriv f β g) x
β’ 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] | simp at this | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = (deriv f β fun x => x + T) x
β’ deriv f (x + T) = deriv f x | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
β’ deriv f (x + T) = deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = (deriv f β fun x => x + T) x
β’ 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] | convert this.symm | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
β’ deriv f (x + T) = deriv f x | case h.e'_3.h.e'_6
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
β’ f = f β fun x => x + T | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
β’ 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] | ext y | case h.e'_3.h.e'_6
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
β’ f = f β fun x => x + T | case h.e'_3.h.e'_6.h
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
y : π
β’ f y = (f β fun x => x + T) y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
β’ f = f β fun x => x + T
TACTIC:
|
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
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
y : π
β’ f y = (f β fun x => x + T) y | case h.e'_3.h.e'_6.h
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
y : π
β’ f y = f (y + T) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6.h
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
y : π
β’ f y = (f β fun x => x + T) y
TACTIC:
|
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
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
y : π
β’ f y = f (y + T) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_6.h
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
this : deriv (f β fun x => x + T) x = deriv f (x + T)
y : π
β’ f y = f (y + T)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | apply differentiable_id.add_const | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
β’ Differentiable π g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
β’ Differentiable π g
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | calc deriv (f β g) x
_ = deriv g x β’ 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 | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ deriv (f β g) x = (deriv f β g) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ deriv (f β g) x = (deriv f β g) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | norm_num | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ 1 β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ 1 β€ 1
TACTIC:
|
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''] | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ deriv g x β’ deriv f (g x) = deriv f (g x) | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ (fun x => 1) x β’ deriv f ((fun x => x + T) x) = deriv f ((fun x => x + T) x) | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ deriv g x β’ deriv f (g x) = deriv f (g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | periodic_deriv | [196, 1] | [211, 28] | simp | π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ (fun x => 1) x β’ deriv f ((fun x => x + T) x) = deriv f ((fun x => x + T) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ² : NontriviallyNormedField π
F : Type
instβΒΉ : NormedAddCommGroup F
instβ : NormedSpace π F
f : π β F
T : π
diff_f : ContDiff π 1 f
periodic_f : Function.Periodic f T
x : π
g : π β π := fun x => x + T
gdef : g = fun x => x + T
diff_g : Differentiable π g
β’ (fun x => 1) x β’ deriv f ((fun x => x + T) x) = deriv f ((fun x => x + T) x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have h : β x β 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have h' : β x β 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 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 : β€} (hn : n β 0): (fourierCoeffOn Real.two_pi_pos f n) = - 1 / (n^2) * fourierCoeffOn Real.two_pi_pos (fun x β¦ 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
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
. apply Continuous.intervalIntegrable
exact (contDiff_one_iff_deriv.mp (contDiff_succ_iff_deriv.mp fdiff).2).2
. apply Continuous.intervalIntegrable
exact (contDiff_succ_iff_deriv.mp fdiff).2.continuous | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | obtain β¨C, hCβ© := fourierCoeffOn_bound (contDiff_one_iff_deriv.mp (contDiff_succ_iff_deriv.mp fdiff).2).2 | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | case intro
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
β’ β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
β’ β C, β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
β’ β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
β’ β (n : β€), n β 0 β Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos f n) β€ C / βn ^ 2
TACTIC:
|
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_divβ, map_neg_eq_map, map_one, map_pow,
Complex.abs_intCast, sq_abs, one_div] | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C / βn ^ 2 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ (βn ^ 2)β»ΒΉ * Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C / βn ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C / βn ^ 2
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ (βn ^ 2)β»ΒΉ * Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C / βn ^ 2 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) * (βn ^ 2)β»ΒΉ β€ C * (βn ^ 2)β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ (βn ^ 2)β»ΒΉ * Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C / βn ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | gcongr | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) * (βn ^ 2)β»ΒΉ β€ C * (βn ^ 2)β»ΒΉ | case h.h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) * (βn ^ 2)β»ΒΉ β€ C * (βn ^ 2)β»ΒΉ
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
fourierCoeffOn_eq :
β {n : β€},
n β 0 β
fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
C : β
hC : β (n : β€), Complex.abs (fourierCoeffOn Real.two_pi_pos (deriv (deriv f)) n) β€ C
n : β€
hn : n β 0
β’ Complex.abs (fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n) β€ C
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | intro x _ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
β’ β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ HasDerivAt f (deriv f x) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
β’ β x β 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ HasDerivAt f (deriv f x) x | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ DifferentiableAt β f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ DifferentiableAt β f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ DifferentiableAt β 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ 1 β€ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ 1 β€ 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
β’ β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ HasDerivAt (deriv f) (deriv (deriv f) x) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
β’ β x β 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ HasDerivAt (deriv f) (deriv (deriv f) x) x | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ DifferentiableAt β (deriv f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ DifferentiableAt β (deriv f) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ DifferentiableAt β (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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ 1 β€ β1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
x : β
aβ : x β Set.uIcc 0 (2 * Real.pi)
β’ 1 β€ β1
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ fourierCoeffOn Real.two_pi_pos f n = -1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ 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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | have := periodicf 0 | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (0 + 2 * Real.pi) = f 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp at this | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (0 + 2 * Real.pi) = f 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (0 + 2 * Real.pi) = f 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp [this] | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
β’ 1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (f (2 * Real.pi) - f 0) -
(β(2 * Real.pi) - β0) *
(1 / (-2 * βReal.pi * Complex.I * βn) *
((fourier (-n)) β0 * (deriv f (2 * Real.pi) - deriv f 0) -
(β(2 * Real.pi) - β0) * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (0 + 2 * Real.pi) = deriv f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
this : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp at this | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (0 + 2 * Real.pi) = deriv f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (0 + 2 * Real.pi) = deriv f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp [this] | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ (-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ -((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(β(AddCircle.toCircle 0) * (deriv f (2 * Real.pi) - deriv f 0) -
2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n)))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | ring_nf | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ (-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi)β»ΒΉ ^ 2 * Complex.Iβ»ΒΉ ^ 2 * (βn)β»ΒΉ ^ 2 * fourierCoeffOn β― (fun x => deriv (deriv f) x) n =
-((βn)β»ΒΉ ^ 2 * fourierCoeffOn β― (fun x => deriv (deriv f) x) n) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ (-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi *
((-(2 * βReal.pi * Complex.I * βn))β»ΒΉ *
(2 * βReal.pi * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n))) =
-1 / βn ^ 2 * fourierCoeffOn Real.two_pi_pos (fun x => deriv (deriv f) x) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi)β»ΒΉ ^ 2 * Complex.Iβ»ΒΉ ^ 2 * (βn)β»ΒΉ ^ 2 * fourierCoeffOn β― (fun x => deriv (deriv f) x) n =
-((βn)β»ΒΉ ^ 2 * fourierCoeffOn β― (fun x => deriv (deriv f) x) n) | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi ^ 2)β»ΒΉ * (βn ^ 2)β»ΒΉ = (βn ^ 2)β»ΒΉ β¨ fourierCoeffOn β― (fun x => deriv (deriv f) x) n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi)β»ΒΉ ^ 2 * Complex.Iβ»ΒΉ ^ 2 * (βn)β»ΒΉ ^ 2 * fourierCoeffOn β― (fun x => deriv (deriv f) x) n =
-((βn)β»ΒΉ ^ 2 * fourierCoeffOn β― (fun x => deriv (deriv f) x) n)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | left | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi ^ 2)β»ΒΉ * (βn ^ 2)β»ΒΉ = (βn ^ 2)β»ΒΉ β¨ fourierCoeffOn β― (fun x => deriv (deriv f) x) n = 0 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi ^ 2)β»ΒΉ * (βn ^ 2)β»ΒΉ = (βn ^ 2)β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi ^ 2)β»ΒΉ * (βn ^ 2)β»ΒΉ = (βn ^ 2)β»ΒΉ β¨ fourierCoeffOn β― (fun x => deriv (deriv f) x) n = 0
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi ^ 2)β»ΒΉ * (βn ^ 2)β»ΒΉ = (βn ^ 2)β»ΒΉ | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 * (βReal.pi ^ 2)β»ΒΉ * (βn ^ 2)β»ΒΉ = (βn ^ 2)β»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | simp | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 β 0 | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ Β¬Real.pi = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ βReal.pi ^ 2 β 0
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ Β¬Real.pi = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
thisβ : f (2 * Real.pi) = f 0
periodic_deriv_f : Function.Periodic (deriv f) (2 * Real.pi)
this : deriv f (2 * Real.pi) = deriv f 0
β’ Β¬Real.pi = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | apply Continuous.intervalIntegrable | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi) | case hu
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ Continuous fun x => deriv (deriv f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv (deriv f) x) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ Continuous fun x => deriv (deriv f) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ Continuous fun x => deriv (deriv f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierCoeffOn_ContDiff_two_bound | [215, 1] | [255, 13] | apply Continuous.intervalIntegrable | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi) | case hu
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ Continuous fun x => deriv f x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ IntervalIntegrable (fun x => deriv f x) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ Continuous fun x => deriv f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
h : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt f (deriv f x) x
h' : β x β Set.uIcc 0 (2 * Real.pi), HasDerivAt (deriv f) (deriv (deriv f) x) x
n : β€
hn : n β 0
β’ Continuous fun x => deriv f x
TACTIC:
|
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 | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
this : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π 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 | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
this : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun k => β i β range k, (f βi + f (-βi)) + -f 0) atTop (π (a + f 0 + -f 0))
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
this : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | simp at this | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun k => β i β range k, (f βi + f (-βi)) + -f 0) atTop (π (a + f 0 + -f 0))
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun k => β i β range k, (f βi + f (-βi)) + -f 0) atTop (π a)
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun k => β i β range k, (f βi + f (-βi)) + -f 0) atTop (π (a + f 0 + -f 0))
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [βtendsto_add_atTop_iff_nat 1] at this | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun k => β i β range k, (f βi + f (-βi)) + -f 0) atTop (π a)
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun k => β i β range k, (f βi + f (-βi)) + -f 0) atTop (π a)
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | convert this using 1 | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a) | case h.e'_3
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ (fun N => β n β Icc (-Int.ofNat N) βN, f n) = fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, f n) atTop (π a)
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ (fun N => β n β Icc (-Int.ofNat N) βN, f n) = fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0 | case h.e'_3.h
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
β’ β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ (fun N => β n β Icc (-Int.ofNat N) βN, f n) = fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
β’ β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0 | case h.e'_3.h.zero
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ β n β Icc (-Int.ofNat 0) β0, f n = β i β range (0 + 1), (f βi + f (-βi)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
β’ β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ β n β Icc (-Int.ofNat 0) β0, f n = β i β range (0 + 1), (f βi + f (-βi)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.zero
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ β n β Icc (-Int.ofNat 0) β0, f n = β i β range (0 + 1), (f βi + f (-βi)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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 (β(Nat.succ N)) (insert (-Int.ofNat (Nat.succ N)) (Icc (-Int.ofNat N) N)) := by
rw [βIco_insert_right, βIoo_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, βsub_eq_add_neg, Int.sub_one_lt_iff]
. norm_num
linarith
. norm_num
linarith
rw [this, sum_insert, sum_insert, ih, βadd_assoc]
symm
rw [sum_range_succ, add_comm, βadd_assoc, add_comm]
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]
rw [add_comm]
. simp
. norm_num
linarith | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ β n β Icc (-Int.ofNat 0) β0, f n = β i β range (0 + 1), (f βi + f (-βi)) + -f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.zero
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
β’ β n β Icc (-Int.ofNat 0) β0, f n = β i β range (0 + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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 (β(Nat.succ N)) (insert (-Int.ofNat (Nat.succ N)) (Icc (-Int.ofNat N) N)) := by
rw [βIco_insert_right, βIoo_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, βsub_eq_add_neg, Int.sub_one_lt_iff]
. norm_num
linarith
. norm_num
linarith | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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, βadd_assoc] | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0) =
β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ β n β Icc (-Int.ofNat (N + 1)) β(N + 1), f n = β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0) =
β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 =
f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0)
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0) =
β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
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, βadd_assoc, add_comm] | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 =
f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0)
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ f β(N + 1) + f (-β(N + 1)) + (-f 0 + β x β range (N + 1), (f βx + f (-βx))) =
f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0)
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ β i β range (N + 1 + 1), (f βi + f (-βi)) + -f 0 =
f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0)
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ f β(N + 1) + f (-β(N + 1)) + (-f 0 + β x β range (N + 1), (f βx + f (-βx))) =
f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0)
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -f 0 + β x β range (N + 1), (f βx + f (-βx)) = β x β range (N + 1), (f βx + f (-βx)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ f β(N + 1) + f (-β(N + 1)) + (-f 0 + β x β range (N + 1), (f βx + f (-βx))) =
f βN.succ + f (-Int.ofNat N.succ) + (β i β range (N + 1), (f βi + f (-βi)) + -f 0)
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -f 0 + β x β range (N + 1), (f βx + f (-βx)) = β x β range (N + 1), (f βx + f (-βx)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -f 0 + β x β range (N + 1), (f βx + f (-βx)) = β x β range (N + 1), (f βx + f (-βx)) + -f 0
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | rw [βIco_insert_right, βIoo_insert_left] | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)) | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ insert (βN.succ) (insert (-Int.ofNat N.succ) (Ioo (-Int.ofNat N.succ) βN.succ)) =
insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
TACTIC:
|
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, βsub_eq_add_neg, Int.sub_one_lt_iff] | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ insert (βN.succ) (insert (-Int.ofNat N.succ) (Ioo (-Int.ofNat N.succ) βN.succ)) =
insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ insert (βN.succ) (insert (-Int.ofNat N.succ) (Ioo (-Int.ofNat N.succ) βN.succ)) =
insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . norm_num
linarith | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | . norm_num
linarith | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | congr | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ insert (βN.succ) (insert (-Int.ofNat N.succ) (Ioo (-Int.ofNat N.succ) βN.succ)) =
insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)) | case e_a.e_a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ Ioo (-Int.ofNat N.succ) βN.succ = Icc (-Int.ofNat N) βN | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ insert (βN.succ) (insert (-Int.ofNat N.succ) (Ioo (-Int.ofNat N.succ) βN.succ)) =
insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ Ioo (-Int.ofNat N.succ) βN.succ = Icc (-Int.ofNat N) βN | case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ n β Ioo (-Int.ofNat N.succ) βN.succ β n β Icc (-Int.ofNat N) βN | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ Ioo (-Int.ofNat N.succ) βN.succ = Icc (-Int.ofNat N) βN
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ n β Ioo (-Int.ofNat N.succ) βN.succ β n β Icc (-Int.ofNat N) βN | case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ -βN.succ < n β§ n < βN.succ β -βN β€ n β§ n β€ βN | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ n β Ioo (-Int.ofNat N.succ) βN.succ β n β Icc (-Int.ofNat N) βN
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ -βN.succ < n β§ n < βN.succ β -βN β€ n β§ n β€ βN | case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ -(βN + 1) < n β§ n < βN + 1 β -βN β€ n β§ n β€ βN | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ -βN.succ < n β§ n < βN.succ β -βN β€ n β§ n β€ βN
TACTIC:
|
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, βsub_eq_add_neg, Int.sub_one_lt_iff] | case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ -(βN + 1) < n β§ n < βN + 1 β -βN β€ n β§ n β€ βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.a
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
n : β€
β’ -(βN + 1) < n β§ n < βN + 1 β -βN β€ n β§ n β€ βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | norm_num | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -1 < βN + 1 + βN | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ < βN.succ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | linarith | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -1 < βN + 1 + βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -1 < βN + 1 + βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | norm_num | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -1 β€ βN + 1 + βN | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -Int.ofNat N.succ β€ βN.succ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | int_sum_nat | [260, 1] | [290, 15] | linarith | Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -1 β€ βN + 1 + βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
this : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
β’ -1 β€ βN + 1 + βN
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ -Int.ofNat N.succ β Icc (-Int.ofNat N) βN
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN) | case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ Β¬βN + 1 = -1 + -βN | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ βN.succ β insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN)
TACTIC:
|
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
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ Β¬βN + 1 = -1 + -βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.succ
Ξ² : Type
instβΒ² : AddCommGroup Ξ²
instβΒΉ : TopologicalSpace Ξ²
instβ : ContinuousAdd Ξ²
f : β€ β Ξ²
a : Ξ²
hfa : HasSum f a
thisβΒΉ : Tendsto (fun n => β i β range n, (f βi + f (-βi))) atTop (π (a + f 0))
thisβ : Tendsto (fun n => β i β range (n + 1), (f βi + f (-βi)) + -f 0) atTop (π a)
N : β
ih : β n β Icc (-Int.ofNat N) βN, f n = β i β range (N + 1), (f βi + f (-βi)) + -f 0
this : Icc (-Int.ofNat N.succ) βN.succ = insert (βN.succ) (insert (-Int.ofNat N.succ) (Icc (-Int.ofNat N) βN))
β’ Β¬βN + 1 = -1 + -βN
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
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), β) := β¨AddCircle.liftIco (2*Real.pi) 0 f, AddCircle.liftIco_continuous ((periodicf 0).symm) fdiff.continuous.continuousOnβ© with g_def | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have fourierCoeff_correspondence {i : β€} : fourierCoeff g i = fourierCoeffOn two_pi_pos' f i := fourierCoeff_liftIco_eq f i | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn two_pi_pos' f i
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | simp at fourierCoeff_correspondence | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn two_pi_pos' f i
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn two_pi_pos' f i
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have function_sum : HasSum (fun (i : β€) => fourierCoeff g i β’ fourier i) g := by
apply hasSum_fourier_series_of_summable
obtain β¨C, hCβ© := fourierCoeffOn_ContDiff_two_bound periodicf fdiff
set maj : β€ β β := fun i β¦ 1 / (i ^ 2) * C with maj_def
have summable_maj : Summable maj := by
by_cases Ceq0 : C = 0
. rw [maj_def, Ceq0]
simp only [one_div, mul_zero]
exact summable_zero
. rw [β summable_div_const_iff Ceq0]
convert Real.summable_one_div_int_pow.mpr one_lt_two using 1
rw [maj_def]
ext i
simp only [one_div]
rw [mul_div_cancel_rightβ]
exact Ceq0
rw [summable_congr @fourierCoeff_correspondence, βsummable_norm_iff]
apply summable_of_le_on_nonzero _ _ summable_maj
. intro i
simp
. intro i ine0
rw [maj_def, Complex.norm_eq_abs]
field_simp
exact hC i ine0 | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this : Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, fourierCoeff (βg) n β’ fourier n) atTop (π g)
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this : Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, fourierCoeff (βg) n β’ fourier n) atTop (π g)
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this : Tendsto (fun N => β n β Icc (-Int.ofNat N) βN, fourierCoeff (βg) n β’ fourier n) atTop (π g)
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | have := this Ξ΅ Ξ΅pos | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
thisβ :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
this :
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
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 : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
thisβ :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
this :
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
thisβ :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
this :
β a,
β b β₯ a,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
thisβ :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
this :
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | obtain β¨Nβ, hNββ© := this | f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
thisβ :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
this :
β a,
β b β₯ a,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | case intro
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
Nβ : β
hNβ :
β b β₯ Nβ,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
thisβ :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
this :
β a,
β b β₯ a,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Approximation.lean | fourierConv_ofTwiceDifferentiable | [297, 1] | [362, 9] | use Nβ | case intro
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
Nβ : β
hNβ :
β b β₯ Nβ,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | case h
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
Nβ : β
hNβ :
β b β₯ Nβ,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
periodicf : Function.Periodic f (2 * Real.pi)
fdiff : ContDiff β 2 f
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
fact_two_pi_pos : Fact (0 < 2 * Real.pi)
g : C(AddCircle (2 * Real.pi), β) := { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
g_def : g = { toFun := AddCircle.liftIco (2 * Real.pi) 0 f, continuous_toFun := β― }
two_pi_pos' : 0 < 0 + 2 * Real.pi
fourierCoeff_correspondence : β {i : β€}, fourierCoeff (βg) i = fourierCoeffOn β― f i
function_sum : HasSum (fun i => fourierCoeff (βg) i β’ fourier i) g
this :
β Ξ΅ > 0,
βαΆ (n : β) in atTop,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat n) βn, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
Nβ : β
hNβ :
β b β₯ Nβ,
β (x : AddCircle (2 * Real.pi)), dist (g x) ((β n β Icc (-Int.ofNat b) βb, fourierCoeff (βg) n β’ fourier n) x) < Ξ΅
β’ β Nβ, β N > Nβ, β x β Set.Icc 0 (2 * Real.pi), Complex.abs (f x - partialFourierSum f N x) β€ Ξ΅
TACTIC:
|
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