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/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . rw [neg_add_rev, add_comm, Int.ofNat_eq_coe, Int.cast_neg, sub_eq_add_neg]
norm_cast | case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -(βN + 1 / 2) = β(-Int.ofNat N) - 1 / 2
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN | N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -(βN + 1 / 2) = β(-Int.ofNat N) - 1 / 2
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | all_goals simp | N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ext n | case e_a.e_s
N : β
x : β
h : cexp (I * βx) β 1
β’ Ico (-Int.ofNat N + 1) (βN + 1) = Ioc (-Int.ofNat N) βN | case e_a.e_s.a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ n β Ico (-Int.ofNat N + 1) (βN + 1) β n β Ioc (-Int.ofNat N) βN | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_s
N : β
x : β
h : cexp (I * βx) β 1
β’ Ico (-Int.ofNat N + 1) (βN + 1) = Ioc (-Int.ofNat N) βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [mem_Ico, mem_Ioc, Int.lt_iff_add_one_le, add_le_add_iff_right, β mem_Icc, Int.lt_iff_add_one_le, β mem_Icc] | case e_a.e_s.a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ n β Ico (-Int.ofNat N + 1) (βN + 1) β n β Ioc (-Int.ofNat N) βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_s.a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ n β Ico (-Int.ofNat N + 1) (βN + 1) β n β Ioc (-Int.ofNat N) βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ext n | case e_a.e_f
N : β
x : β
h : cexp (I * βx) β 1
β’ (fun x_1 => cexp ((β(x_1 + -1) + 1 / 2) * I * βx)) = fun x_1 => cexp ((βx_1 - 1 / 2) * I * βx) | case e_a.e_f.h
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ cexp ((β(n + -1) + 1 / 2) * I * βx) = cexp ((βn - 1 / 2) * I * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
N : β
x : β
h : cexp (I * βx) β 1
β’ (fun x_1 => cexp ((β(x_1 + -1) + 1 / 2) * I * βx)) = fun x_1 => cexp ((βx_1 - 1 / 2) * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | case e_a.e_f.h
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ cexp ((β(n + -1) + 1 / 2) * I * βx) = cexp ((βn - 1 / 2) * I * βx) | case e_a.e_f.h.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ β(n + -1) + 1 / 2 = βn - 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ cexp ((β(n + -1) + 1 / 2) * I * βx) = cexp ((βn - 1 / 2) * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | simp [add_assoc, sub_eq_add_neg] | case e_a.e_f.h.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ β(n + -1) + 1 / 2 = βn - 1 / 2 | case e_a.e_f.h.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ -1 + 2β»ΒΉ = -2β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ β(n + -1) + 1 / 2 = βn - 1 / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | norm_num | case e_a.e_f.h.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ -1 + 2β»ΒΉ = -2β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
n : β€
β’ -1 + 2β»ΒΉ = -2β»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [neg_add_rev, add_comm, Int.ofNat_eq_coe, Int.cast_neg, sub_eq_add_neg] | case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -(βN + 1 / 2) = β(-Int.ofNat N) - 1 / 2 | case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -βN + -(1 / 2) = -ββN + -(1 / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -(βN + 1 / 2) = β(-Int.ofNat N) - 1 / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | norm_cast | case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -βN + -(1 / 2) = -ββN + -(1 / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_z.e_a.e_a
N : β
x : β
h : cexp (I * βx) β 1
β’ -βN + -(1 / 2) = -ββN + -(1 / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | simp | N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
h : cexp (I * βx) β 1
β’ -Int.ofNat N β€ βN
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | contrapose! h | N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
β’ cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0 | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) = 0
β’ cexp (I * βx) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
β’ cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [sub_eq_zero] at h | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) = 0
β’ cexp (I * βx) = 1 | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) = 0
β’ cexp (I * βx) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | calc cexp (I * βx)
_ = cexp (1 / 2 * I * βx) * cexp (1 / 2 * I * βx) := by
rw [β exp_add]
congr
rw [mul_assoc, β mul_add]
ring
_ = cexp (1 / 2 * I * βx) * cexp (-1 / 2 * I * βx) := by
congr
_ = 1 := by
rw [β exp_add]
ring_nf
exact exp_zero | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [β exp_add] | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = cexp (1 / 2 * I * βx) * cexp (1 / 2 * I * βx) | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = cexp (1 / 2 * I * βx + 1 / 2 * I * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = cexp (1 / 2 * I * βx) * cexp (1 / 2 * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = cexp (1 / 2 * I * βx + 1 / 2 * I * βx) | case e_z
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ I * βx = 1 / 2 * I * βx + 1 / 2 * I * βx | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (I * βx) = cexp (1 / 2 * I * βx + 1 / 2 * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [mul_assoc, β mul_add] | case e_z
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ I * βx = 1 / 2 * I * βx + 1 / 2 * I * βx | case e_z
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ I * βx = 1 / 2 * (I * βx + I * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_z
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ I * βx = 1 / 2 * I * βx + 1 / 2 * I * βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring | case e_z
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ I * βx = 1 / 2 * (I * βx + I * βx) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_z
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ I * βx = 1 / 2 * (I * βx + I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx) * cexp (1 / 2 * I * βx) = cexp (1 / 2 * I * βx) * cexp (-1 / 2 * I * βx) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx) * cexp (1 / 2 * I * βx) = cexp (1 / 2 * I * βx) * cexp (-1 / 2 * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [β exp_add] | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx) * cexp (-1 / 2 * I * βx) = 1 | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx + -1 / 2 * I * βx) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx) * cexp (-1 / 2 * I * βx) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring_nf | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx + -1 / 2 * I * βx) = 1 | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp 0 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp (1 / 2 * I * βx + -1 / 2 * I * βx) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | exact exp_zero | N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h : cexp (1 / 2 * I * βx) = cexp (-1 / 2 * I * βx)
β’ cexp 0 = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [mul_div] | case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * (cexp (I * βN * βx) / (1 - cexp (-I * βx))) | case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) / (1 - cexp (-I * βx)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * (cexp (I * βN * βx) / (1 - cexp (-I * βx)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | apply eq_div_of_mul_eq | case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) / (1 - cexp (-I * βx)) | case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (-I * βx) β 0
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) * (1 - cexp (-I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) / (1 - cexp (-I * βx))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . contrapose! h
rwa [sub_eq_zero, neg_mul, exp_neg, eq_comm, inv_eq_one] at h | case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (-I * βx) β 0
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) * (1 - cexp (-I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) * (1 - cexp (-I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (-I * βx) β 0
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) * (1 - cexp (-I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring_nf | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) * (1 - cexp (-I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx) | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (βN * I * βx + I * βx * (1 / 2)) - cexp (βN * I * βx + I * βx * (1 / 2)) * cexp (-(I * βx)) =
cexp (I * βx * (1 / 2)) * cexp (βN * I * βx) - cexp (I * βx * (-1 / 2)) * cexp (βN * I * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp ((βN + 1 / 2) * I * βx) * (1 - cexp (-I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (I * βN * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [β exp_add, β exp_add, β exp_add] | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (βN * I * βx + I * βx * (1 / 2)) - cexp (βN * I * βx + I * βx * (1 / 2)) * cexp (-(I * βx)) =
cexp (I * βx * (1 / 2)) * cexp (βN * I * βx) - cexp (I * βx * (-1 / 2)) * cexp (βN * I * βx) | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (βN * I * βx + I * βx * (1 / 2)) - cexp (βN * I * βx + I * βx * (1 / 2) + -(I * βx)) =
cexp (I * βx * (1 / 2) + βN * I * βx) - cexp (I * βx * (-1 / 2) + βN * I * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (βN * I * βx + I * βx * (1 / 2)) - cexp (βN * I * βx + I * βx * (1 / 2)) * cexp (-(I * βx)) =
cexp (I * βx * (1 / 2)) * cexp (βN * I * βx) - cexp (I * βx * (-1 / 2)) * cexp (βN * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr 2 <;> ring | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (βN * I * βx + I * βx * (1 / 2)) - cexp (βN * I * βx + I * βx * (1 / 2) + -(I * βx)) =
cexp (I * βx * (1 / 2) + βN * I * βx) - cexp (I * βx * (-1 / 2) + βN * I * βx) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (βN * I * βx + I * βx * (1 / 2)) - cexp (βN * I * βx + I * βx * (1 / 2) + -(I * βx)) =
cexp (I * βx * (1 / 2) + βN * I * βx) - cexp (I * βx * (-1 / 2) + βN * I * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | contrapose! h | case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (-I * βx) β 0 | case e_a.hc
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
h : 1 - cexp (-I * βx) = 0
β’ cexp (I * βx) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (-I * βx) β 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rwa [sub_eq_zero, neg_mul, exp_neg, eq_comm, inv_eq_one] at h | case e_a.hc
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
h : 1 - cexp (-I * βx) = 0
β’ cexp (I * βx) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
h : 1 - cexp (-I * βx) = 0
β’ cexp (I * βx) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [mul_div] | case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * (cexp (-I * βN * βx) / (1 - cexp (I * βx))) | case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) / (1 - cexp (I * βx)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * (cexp (-I * βN * βx) / (1 - cexp (I * βx)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | apply eq_div_of_mul_eq | case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) / (1 - cexp (I * βx)) | case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (I * βx) β 0
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) * (1 - cexp (I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) / (1 - cexp (I * βx))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | . contrapose! h
rwa [sub_eq_zero, eq_comm] at h | case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (I * βx) β 0
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) * (1 - cexp (I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) * (1 - cexp (I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (I * βx) β 0
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) * (1 - cexp (I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring_nf | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) * (1 - cexp (I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx) | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN * I * βx) + I * βx * (-1 / 2)) + cexp (-(βN * I * βx) + I * βx * (-1 / 2)) * cexp (I * βx) =
cexp (I * βx * (1 / 2)) * cexp (-(βN * I * βx)) - cexp (I * βx * (-1 / 2)) * cexp (-(βN * I * βx)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN + 1 / 2) * I * βx) * (1 - cexp (I * βx)) =
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * cexp (-I * βN * βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [β exp_add, β exp_add, β exp_add, neg_add_eq_sub] | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN * I * βx) + I * βx * (-1 / 2)) + cexp (-(βN * I * βx) + I * βx * (-1 / 2)) * cexp (I * βx) =
cexp (I * βx * (1 / 2)) * cexp (-(βN * I * βx)) - cexp (I * βx * (-1 / 2)) * cexp (-(βN * I * βx)) | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (-(βN * I * βx) + I * βx * (-1 / 2) + I * βx) - cexp (-(βN * I * βx) + I * βx * (-1 / 2)) =
cexp (I * βx * (1 / 2) + -(βN * I * βx)) - cexp (I * βx * (-1 / 2) + -(βN * I * βx)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ -cexp (-(βN * I * βx) + I * βx * (-1 / 2)) + cexp (-(βN * I * βx) + I * βx * (-1 / 2)) * cexp (I * βx) =
cexp (I * βx * (1 / 2)) * cexp (-(βN * I * βx)) - cexp (I * βx * (-1 / 2)) * cexp (-(βN * I * βx))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr 2 <;> ring | case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (-(βN * I * βx) + I * βx * (-1 / 2) + I * βx) - cexp (-(βN * I * βx) + I * βx * (-1 / 2)) =
cexp (I * βx * (1 / 2) + -(βN * I * βx)) - cexp (I * βx * (-1 / 2) + -(βN * I * βx)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ cexp (-(βN * I * βx) + I * βx * (-1 / 2) + I * βx) - cexp (-(βN * I * βx) + I * βx * (-1 / 2)) =
cexp (I * βx * (1 / 2) + -(βN * I * βx)) - cexp (I * βx * (-1 / 2) + -(βN * I * βx))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | contrapose! h | case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (I * βx) β 0 | case e_a.hc
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
h : 1 - cexp (I * βx) = 0
β’ cexp (I * βx) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : β
x : β
h : cexp (I * βx) β 1
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
β’ 1 - cexp (I * βx) β 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rwa [sub_eq_zero, eq_comm] at h | case e_a.hc
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
h : 1 - cexp (I * βx) = 0
β’ cexp (I * βx) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : β
x : β
this :
(cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx)) * dirichletKernel N x =
cexp ((βN + 1 / 2) * I * βx) - cexp (-(βN + 1 / 2) * I * βx)
h' : cexp (1 / 2 * I * βx) - cexp (-1 / 2 * I * βx) β 0
h : 1 - cexp (I * βx) = 0
β’ cexp (I * βx) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_eq_zero | [103, 1] | [105, 7] | rw [dirichletKernel', neg_mul, exp_neg, h] | N : β
x : β
h : cexp (I * βx) = 1
β’ dirichletKernel' N x = 0 | N : β
x : β
h : cexp (I * βx) = 1
β’ cexp (I * βN * βx) / (1 - 1β»ΒΉ) + cexp (-I * βN * βx) / (1 - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
h : cexp (I * βx) = 1
β’ dirichletKernel' N x = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel'_eq_zero | [103, 1] | [105, 7] | simp | N : β
x : β
h : cexp (I * βx) = 1
β’ cexp (I * βN * βx) / (1 - 1β»ΒΉ) + cexp (-I * βN * βx) / (1 - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
h : cexp (I * βx) = 1
β’ cexp (I * βN * βx) / (1 - 1β»ΒΉ) + cexp (-I * βN * βx) / (1 - 1) = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | rw [MeasureTheory.ae_iff] | N : β
β’ βα΅ (x : β), dirichletKernel N x = dirichletKernel' N x | N : β
β’ MeasureTheory.volume {a | Β¬dirichletKernel N a = dirichletKernel' N a} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ βα΅ (x : β), dirichletKernel N x = dirichletKernel' N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | have : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n : β€, x = n * (2 * Real.pi)} := by
ext x
simp
constructor
. sorry
. sorry | N : β
β’ MeasureTheory.volume {a | Β¬dirichletKernel N a = dirichletKernel' N a} = 0 | N : β
this : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
β’ MeasureTheory.volume {a | Β¬dirichletKernel N a = dirichletKernel' N a} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ MeasureTheory.volume {a | Β¬dirichletKernel N a = dirichletKernel' N a} = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | rw [this] | N : β
this : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
β’ MeasureTheory.volume {a | Β¬dirichletKernel N a = dirichletKernel' N a} = 0 | N : β
this : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
β’ MeasureTheory.volume {x | β n, x = βn * (2 * Real.pi)} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
this : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
β’ MeasureTheory.volume {a | Β¬dirichletKernel N a = dirichletKernel' N a} = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | sorry | N : β
this : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
β’ MeasureTheory.volume {x | β n, x = βn * (2 * Real.pi)} = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
this : {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
β’ MeasureTheory.volume {x | β n, x = βn * (2 * Real.pi)} = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | ext x | N : β
β’ {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)} | case h
N : β
x : β
β’ x β {x | Β¬dirichletKernel N x = dirichletKernel' N x} β x β {x | β n, x = βn * (2 * Real.pi)} | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ {x | Β¬dirichletKernel N x = dirichletKernel' N x} = {x | β n, x = βn * (2 * Real.pi)}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | simp | case h
N : β
x : β
β’ x β {x | Β¬dirichletKernel N x = dirichletKernel' N x} β x β {x | β n, x = βn * (2 * Real.pi)} | case h
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β
x : β
β’ x β {x | Β¬dirichletKernel N x = dirichletKernel' N x} β x β {x | β n, x = βn * (2 * Real.pi)}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | constructor | case h
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi) | case h.mp
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi)
case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | . sorry | case h.mp
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi)
case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x | case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi)
case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | . sorry | case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | sorry | case h.mp
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : β
x : β
β’ Β¬dirichletKernel N x = dirichletKernel' N x β β n, x = βn * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq_ae | [108, 1] | [120, 8] | sorry | case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : β
x : β
β’ (β n, x = βn * (2 * Real.pi)) β Β¬dirichletKernel N x = dirichletKernel' N x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | rw [dirichletKernel] | N : β
x : β
β’ βdirichletKernel N xβ β€ 2 * βN + 1 | N : β
x : β
β’ ββ n β Icc (-Int.ofNat N) βN, (fourier n) βxβ β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ βdirichletKernel N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | calc ββ n β Icc (-Int.ofNat N) βN, (fourier n) βxβ
_ β€ β n β Icc (-Int.ofNat N) βN, β(fourier n) βxβ := by
apply norm_sum_le
_ β€ β n β Icc (-Int.ofNat N) βN, 1 := by
apply sum_le_sum
intro n _
have : Fact (0 < 2 * Real.pi) := by
rw [fact_iff]
exact Real.two_pi_pos
apply le_trans (ContinuousMap.norm_coe_le_norm (fourier n) x) (fourier_norm n).le
_ = 2 * N + 1 := by
rw [sum_const]
simp only [Int.ofNat_eq_coe, Int.card_Icc, sub_neg_eq_add, nsmul_eq_mul, mul_one]
norm_cast
rw [Int.toNat_ofNat]
ring | N : β
x : β
β’ ββ n β Icc (-Int.ofNat N) βN, (fourier n) βxβ β€ 2 * βN + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ ββ n β Icc (-Int.ofNat N) βN, (fourier n) βxβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | apply norm_sum_le | N : β
x : β
β’ ββ n β Icc (-Int.ofNat N) βN, (fourier n) βxβ β€ β n β Icc (-Int.ofNat N) βN, β(fourier n) βxβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ ββ n β Icc (-Int.ofNat N) βN, (fourier n) βxβ β€ β n β Icc (-Int.ofNat N) βN, β(fourier n) βxβ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | apply sum_le_sum | N : β
x : β
β’ β n β Icc (-Int.ofNat N) βN, β(fourier n) βxβ β€ β n β Icc (-Int.ofNat N) βN, 1 | case h
N : β
x : β
β’ β i β Icc (-Int.ofNat N) βN, β(fourier i) βxβ β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ β n β Icc (-Int.ofNat N) βN, β(fourier n) βxβ β€ β n β Icc (-Int.ofNat N) βN, 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | intro n _ | case h
N : β
x : β
β’ β i β Icc (-Int.ofNat N) βN, β(fourier i) βxβ β€ 1 | case h
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ β(fourier n) βxβ β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β
x : β
β’ β i β Icc (-Int.ofNat N) βN, β(fourier i) βxβ β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | have : Fact (0 < 2 * Real.pi) := by
rw [fact_iff]
exact Real.two_pi_pos | case h
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ β(fourier n) βxβ β€ 1 | case h
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
this : Fact (0 < 2 * Real.pi)
β’ β(fourier n) βxβ β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ β(fourier n) βxβ β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | apply le_trans (ContinuousMap.norm_coe_le_norm (fourier n) x) (fourier_norm n).le | case h
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
this : Fact (0 < 2 * Real.pi)
β’ β(fourier n) βxβ β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
this : Fact (0 < 2 * Real.pi)
β’ β(fourier n) βxβ β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | rw [fact_iff] | N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ Fact (0 < 2 * Real.pi) | N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ 0 < 2 * Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ Fact (0 < 2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | exact Real.two_pi_pos | N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ 0 < 2 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ 0 < 2 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | rw [sum_const] | N : β
x : β
β’ β n β Icc (-Int.ofNat N) βN, 1 = 2 * βN + 1 | N : β
x : β
β’ (Icc (-Int.ofNat N) βN).card β’ 1 = 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ β n β Icc (-Int.ofNat N) βN, 1 = 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | simp only [Int.ofNat_eq_coe, Int.card_Icc, sub_neg_eq_add, nsmul_eq_mul, mul_one] | N : β
x : β
β’ (Icc (-Int.ofNat N) βN).card β’ 1 = 2 * βN + 1 | N : β
x : β
β’ β(βN + 1 + βN).toNat = 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ (Icc (-Int.ofNat N) βN).card β’ 1 = 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | norm_cast | N : β
x : β
β’ β(βN + 1 + βN).toNat = 2 * βN + 1 | N : β
x : β
β’ (β(N + 1 + N)).toNat = 2 * N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ β(βN + 1 + βN).toNat = 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | rw [Int.toNat_ofNat] | N : β
x : β
β’ (β(N + 1 + N)).toNat = 2 * N + 1 | N : β
x : β
β’ N + 1 + N = 2 * N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ (β(N + 1 + N)).toNat = 2 * N + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | ring | N : β
x : β
β’ N + 1 + N = 2 * N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ N + 1 + N = 2 * N + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | by_cases h : cexp (I * x) β 1 | N : β
x : β
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
x : β
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | . rw [β dirichletKernel_eq]
apply norm_dirichletKernel_le
exact h | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | . push_neg at h
rw [dirichletKernel'_eq_zero h, norm_zero]
linarith | case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | rw [β dirichletKernel_eq] | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel N xβ β€ 2 * βN + 1
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ cexp (I * βx) β 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | apply norm_dirichletKernel_le | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel N xβ β€ 2 * βN + 1
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ cexp (I * βx) β 1 | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ cexp (I * βx) β 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ βdirichletKernel N xβ β€ 2 * βN + 1
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ cexp (I * βx) β 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | exact h | case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ cexp (I * βx) β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : β
x : β
h : cexp (I * βx) β 1
β’ cexp (I * βx) β 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | push_neg at h | case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | case neg
N : β
x : β
h : cexp (I * βx) = 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : β
x : β
h : Β¬cexp (I * βx) β 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | rw [dirichletKernel'_eq_zero h, norm_zero] | case neg
N : β
x : β
h : cexp (I * βx) = 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1 | case neg
N : β
x : β
h : cexp (I * βx) = 1
β’ 0 β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : β
x : β
h : cexp (I * βx) = 1
β’ βdirichletKernel' N xβ β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel'_le | [141, 1] | [148, 13] | linarith | case neg
N : β
x : β
h : cexp (I * βx) = 1
β’ 0 β€ 2 * βN + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : β
x : β
h : cexp (I * βx) = 1
β’ 0 β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | calc partialFourierSum f N x
_ = β n in Icc (-Int.ofNat N) βN, fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx := by
rw [partialFourierSum]
_ = β n in Icc (-Int.ofNat N) βN, (1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in (0 : β)..2 * Real.pi, (fourier (-n) βy β’ f y)) * (fourier n) βx) := by
congr 1
ext n
rw [fourierCoeffOn_eq_integral, smul_mul_assoc]
_ = (1 / (2 * Real.pi)) * β n in Icc (-Int.ofNat N) βN, ((β« (y : β) in (0 : β)..2 * Real.pi, (fourier (-n) βy β’ f y)) * (fourier n) βx) := by
rw [β smul_sum, real_smul, sub_zero]
norm_cast
_ = (1 / (2 * Real.pi)) * β n in Icc (-Int.ofNat N) βN, ((β« (y : β) in (0 : β)..2 * Real.pi, (fourier (-n) βy β’ f y) * (fourier n) βx)) := by
congr
ext n
symm
apply intervalIntegral.integral_mul_const
_ = (1 / (2 * Real.pi)) * β« (y : β) in (0 : β)..(2 * Real.pi), β n in Icc (-Int.ofNat N) βN, (fourier (-n)) y β’ f y * (fourier n) x := by
rw [β intervalIntegral.integral_finset_sum]
intro n _
apply IntervalIntegrable.mul_const
apply IntervalIntegrable.continuousOn_mul h fourier_uniformContinuous.continuous.continuousOn
_ = (1 / (2 * Real.pi)) * β« (y : β) in (0 : β)..(2 * Real.pi), f y * β n in Icc (-Int.ofNat N) βN, (fourier (-n)) y * (fourier n) x := by
congr
ext y
rw [mul_sum]
congr
ext n
rw [smul_eq_mul]
ring
_ = (1 / (2 * Real.pi)) * β« (y : β) in (0 : β)..(2 * Real.pi), f y * dirichletKernel N (x - y) := by
congr
ext y
rw [dirichletKernel]
congr
ext n
rw [fourier_coe_apply, fourier_coe_apply, fourier_coe_apply, βexp_add]
congr
field_simp
rw [mul_sub, sub_eq_neg_add] | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ partialFourierSum f N x = 1 / (2 * βReal.pi) * β« (y : β) in 0 ..2 * Real.pi, f y * dirichletKernel N (x - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ partialFourierSum f N x = 1 / (2 * βReal.pi) * β« (y : β) in 0 ..2 * Real.pi, f y * dirichletKernel N (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [partialFourierSum] | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ partialFourierSum f N x = β n β Icc (-Int.ofNat N) βN, fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ partialFourierSum f N x = β n β Icc (-Int.ofNat N) βN, fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr 1 | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β n β Icc (-Int.ofNat N) βN, fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx =
β n β Icc (-Int.ofNat N) βN,
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) | case e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx) = fun n =>
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β n β Icc (-Int.ofNat N) βN, fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx =
β n β Icc (-Int.ofNat N) βN,
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | ext n | case e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx) = fun n =>
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) | case e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx =
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx) = fun n =>
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [fourierCoeffOn_eq_integral, smul_mul_assoc] | case e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx =
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ fourierCoeffOn Real.two_pi_pos f n * (fourier n) βx =
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [β smul_sum, real_smul, sub_zero] | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β n β Icc (-Int.ofNat N) βN,
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β(1 / (2 * Real.pi)) *
β x_1 β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-x_1)) βy β’ f y) * (fourier x_1) βx =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β n β Icc (-Int.ofNat N) βN,
(1 / (2 * Real.pi - 0)) β’ ((β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | norm_cast | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β(1 / (2 * Real.pi)) *
β x_1 β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-x_1)) βy β’ f y) * (fourier x_1) βx =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β(1 / (2 * Real.pi)) *
β x_1 β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-x_1)) βy β’ f y) * (fourier x_1) βx =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β€ β β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx | case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun n => (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) = fun n =>
β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx =
1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | ext n | case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun n => (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) = fun n =>
β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx =
β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun n => (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx) = fun n =>
β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | symm | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx =
β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx =
(β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ (β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx =
β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | apply intervalIntegral.integral_mul_const | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx =
(β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
β’ β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx =
(β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y) * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [β intervalIntegral.integral_finset_sum] | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx =
1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β i β Icc (-Int.ofNat N) βN,
IntervalIntegrable (fun y => (fourier (-i)) βy β’ f y * (fourier i) βx) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β n β Icc (-Int.ofNat N) βN, β« (y : β) in 0 ..2 * Real.pi, (fourier (-n)) βy β’ f y * (fourier n) βx =
1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | intro n _ | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β i β Icc (-Int.ofNat N) βN,
IntervalIntegrable (fun y => (fourier (-i)) βy β’ f y * (fourier i) βx) MeasureTheory.volume 0 (2 * Real.pi) | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ IntervalIntegrable (fun y => (fourier (-n)) βy β’ f y * (fourier n) βx) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β i β Icc (-Int.ofNat N) βN,
IntervalIntegrable (fun y => (fourier (-i)) βy β’ f y * (fourier i) βx) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | apply IntervalIntegrable.mul_const | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ IntervalIntegrable (fun y => (fourier (-n)) βy β’ f y * (fourier n) βx) MeasureTheory.volume 0 (2 * Real.pi) | case hf
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ IntervalIntegrable (fun x => (fourier (-n)) βx β’ f x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ IntervalIntegrable (fun y => (fourier (-n)) βy β’ f y * (fourier n) βx) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | apply IntervalIntegrable.continuousOn_mul h fourier_uniformContinuous.continuous.continuousOn | case hf
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ IntervalIntegrable (fun x => (fourier (-n)) βx β’ f x) MeasureTheory.volume 0 (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : β€
aβ : n β Icc (-Int.ofNat N) βN
β’ IntervalIntegrable (fun x => (fourier (-n)) βx β’ f x) MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx | case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun y => β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx) = fun y =>
f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | ext y | case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun y => β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx) = fun y =>
f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun y => β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx) = fun y =>
f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [mul_sum] | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
β i β Icc (-Int.ofNat N) βN, f y * ((fourier (-i)) βy * (fourier i) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
β i β Icc (-Int.ofNat N) βN, f y * ((fourier (-i)) βy * (fourier i) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | case e_a.e_f.h.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ (fun n => (fourier (-n)) βy β’ f y * (fourier n) βx) = fun i => f y * ((fourier (-i)) βy * (fourier i) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy β’ f y * (fourier n) βx =
β i β Icc (-Int.ofNat N) βN, f y * ((fourier (-i)) βy * (fourier i) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | ext n | case e_a.e_f.h.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ (fun n => (fourier (-n)) βy β’ f y * (fourier n) βx) = fun i => f y * ((fourier (-i)) βy * (fourier i) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | case e_a.e_f.h.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
n : β€
β’ (fourier (-n)) βy β’ f y * (fourier n) βx = f y * ((fourier (-n)) βy * (fourier n) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ (fun n => (fourier (-n)) βy β’ f y * (fourier n) βx) = fun i => f y * ((fourier (-i)) βy * (fourier i) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [smul_eq_mul] | case e_a.e_f.h.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
n : β€
β’ (fourier (-n)) βy β’ f y * (fourier n) βx = f y * ((fourier (-n)) βy * (fourier n) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | case e_a.e_f.h.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
n : β€
β’ (fourier (-n)) βy * f y * (fourier n) βx = f y * ((fourier (-n)) βy * (fourier n) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
n : β€
β’ (fourier (-n)) βy β’ f y * (fourier n) βx = f y * ((fourier (-n)) βy * (fourier n) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | ring | case e_a.e_f.h.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
n : β€
β’ (fourier (-n)) βy * f y * (fourier n) βx = f y * ((fourier (-n)) βy * (fourier n) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
n : β€
β’ (fourier (-n)) βy * f y * (fourier n) βx = f y * ((fourier (-n)) βy * (fourier n) βx)
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ β β β€ β β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx =
1 / (2 * βReal.pi) * β« (y : β) in 0 ..2 * Real.pi, f y * dirichletKernel N (x - y) | case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun y => f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx) = fun y =>
f y * dirichletKernel N (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ 1 / (2 * βReal.pi) *
β« (y : β) in 0 ..2 * Real.pi, f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx =
1 / (2 * βReal.pi) * β« (y : β) in 0 ..2 * Real.pi, f y * dirichletKernel N (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | ext y | case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun y => f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx) = fun y =>
f y * dirichletKernel N (x - y) | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx = f y * dirichletKernel N (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
β’ (fun y => f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx) = fun y =>
f y * dirichletKernel N (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [dirichletKernel] | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx = f y * dirichletKernel N (x - y) | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx =
f y * β n β Icc (-Int.ofNat N) βN, (fourier n) β(x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx = f y * dirichletKernel N (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx =
f y * β n β Icc (-Int.ofNat N) βN, (fourier n) β(x - y) | case e_a.e_f.h.e_a.e_f
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ (fun n => (fourier (-n)) βy * (fourier n) βx) = fun n => (fourier n) β(x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : β β β
N : β
x : β
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : β
β’ f y * β n β Icc (-Int.ofNat N) βN, (fourier (-n)) βy * (fourier n) βx =
f y * β n β Icc (-Int.ofNat N) βN, (fourier n) β(x - y)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.