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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | all_goals simp | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ Ico (-Int.ofNat N + 1) (↑N + 1) = Ioc (-Int.ofNat N) ↑N | case e_a.e_s.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ n ∈ Ico (-Int.ofNat N + 1) (↑N + 1) ↔ n ∈ Ioc (-Int.ofNat N) ↑N | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_s
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ Ico (-Int.ofNat N + 1) (↑N + 1) = Ioc (-Int.ofNat N) ↑N
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, ← mem_Icc, Int.lt_iff_add_one_le, ← mem_Icc] | case e_a.e_s.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ n ∈ Ico (-Int.ofNat N + 1) (↑N + 1) ↔ n ∈ Ioc (-Int.ofNat N) ↑N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_s.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ n ∈ Ico (-Int.ofNat N + 1) (↑N + 1) ↔ n ∈ Ioc (-Int.ofNat N) ↑N
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x) | case e_a.e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ cexp ((↑(n + -1) + 1 / 2) * I * ↑x) = cexp ((↑n - 1 / 2) * I * ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ (fun x_1 => cexp ((↑(x_1 + -1) + 1 / 2) * I * ↑x)) = fun x_1 => cexp ((↑x_1 - 1 / 2) * I * ↑x)
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ cexp ((↑(n + -1) + 1 / 2) * I * ↑x) = cexp ((↑n - 1 / 2) * I * ↑x) | case e_a.e_f.h.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑(n + -1) + 1 / 2 = ↑n - 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ cexp ((↑(n + -1) + 1 / 2) * I * ↑x) = cexp ((↑n - 1 / 2) * I * ↑x)
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑(n + -1) + 1 / 2 = ↑n - 1 / 2 | case e_a.e_f.h.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ -1 + 2⁻¹ = -2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ ↑(n + -1) + 1 / 2 = ↑n - 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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ -1 + 2⁻¹ = -2⁻¹ | 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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
n : ℤ
⊢ -1 + 2⁻¹ = -2⁻¹
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-Int.ofNat N) - 1 / 2 | case e_a.e_a.e_z.e_a.e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -↑N + -(1 / 2) = -↑↑N + -(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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -(↑N + 1 / 2) = ↑(-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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -↑N + -(1 / 2) = -↑↑N + -(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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -↑N + -(1 / 2) = -↑↑N + -(1 / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | simp | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ -Int.ofNat N ≤ ↑N
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | contrapose! h | N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0 | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) = 0
⊢ cexp (I * ↑x) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 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 : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) = 0
⊢ cexp (I * ↑x) = 1 | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) = 0
⊢ cexp (I * ↑x) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | calc cexp (I * ↑x)
_ = cexp (1 / 2 * I * ↑x) * cexp (1 / 2 * I * ↑x) := by
rw [← exp_add]
congr
rw [mul_assoc, ← mul_add]
ring
_ = cexp (1 / 2 * I * ↑x) * cexp (-1 / 2 * I * ↑x) := by
congr
_ = 1 := by
rw [← exp_add]
ring_nf
exact exp_zero | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← exp_add] | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = cexp (1 / 2 * I * ↑x) * cexp (1 / 2 * I * ↑x) | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = cexp (1 / 2 * I * ↑x + 1 / 2 * I * ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = cexp (1 / 2 * I * ↑x) * cexp (1... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = cexp (1 / 2 * I * ↑x + 1 / 2 * I * ↑x) | case e_z
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ I * ↑x = 1 / 2 * I * ↑x + 1 / 2 * I * ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (I * ↑x) = cexp (1 / 2 * I * ↑x + 1 / 2 * ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [mul_assoc, ← mul_add] | case e_z
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ I * ↑x = 1 / 2 * I * ↑x + 1 / 2 * I * ↑x | case e_z
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ I * ↑x = 1 / 2 * (I * ↑x + I * ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_z
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ I * ↑x = 1 / 2 * I * ↑x + 1 / 2 * I * ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring | case e_z
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ I * ↑x = 1 / 2 * (I * ↑x + I * ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_z
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ I * ↑x = 1 / 2 * (I * ↑x + I * ↑x)
TAC... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | congr | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x) * cexp (1 / 2 * I * ↑x) = cexp (1 / 2 * I * ↑x) * cexp (-1 / 2 * I * ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x) * cexp (1 / 2 * I * ↑x) =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← exp_add] | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x) * cexp (-1 / 2 * I * ↑x) = 1 | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x + -1 / 2 * I * ↑x) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x) * cexp (-1 / 2 * I * ↑x) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | ring_nf | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x + -1 / 2 * I * ↑x) = 1 | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp 0 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp (1 / 2 * I * ↑x + -1 / 2 * I * ↑x) = 1
TAC... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | exact exp_zero | N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ cexp 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h : cexp (1 / 2 * I * ↑x) = cexp (-1 / 2 * I * ↑x)
⊢ 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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N +... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2... | case e_a.hc
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp (-I * ↑x) ≠ 0
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N +... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp (-I * ↑x) ≠ 0
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
... | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) * (1 - cexp (-I * ↑x)) =
(cexp (1 / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N + 1 / 2) * I * ↑x) * (1 - cexp (-I * ↑x)) =
(cexp (1 / 2... | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (↑N * I * ↑x + I * ↑x * (1 / 2)) - cexp (↑N * I * ↑x + I * ↑x * (... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp ((↑N... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← exp_add, ← exp_add, ← exp_add] | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (↑N * I * ↑x + I * ↑x * (1 / 2)) - cexp (↑N * I * ↑x + I * ↑x * (... | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (↑N * I * ↑x + I * ↑x * (1 / 2)) - cexp (↑N * I * ↑x + I * ↑x * (... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (↑N ... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (↑N * I * ↑x + I * ↑x * (1 / 2)) - cexp (↑N * I * ↑x + I * ↑x * (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (↑N ... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp (-I * ↑x) ≠ 0 | case e_a.hc
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
h : 1 - cexp (-I * ↑x) = 0
⊢ cexp (I * ↑x) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp... |
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 : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
h : 1 - cexp (-I * ↑x) = 0
⊢ cexp (I * ↑x) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
h : 1 - cexp (-I * ↑x) = 0
⊢ cex... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 /... | case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 /... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) =
(cexp (1 / 2 * I * ↑x) - cexp (-1 /... | case e_a.hc
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp (I * ↑x) ≠ 0
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
t... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp (I * ↑x) ≠ 0
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
t... | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) * (1 - cexp (I * ↑x)) =
(cexp (1 / ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N + 1 / 2) * I * ↑x) * (1 - cexp (I * ↑x)) =
(cexp (1 / ... | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N * I * ↑x) + I * ↑x * (-1 / 2)) + cexp (-(↑N * I * ↑x) + I ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | dirichletKernel_eq | [31, 1] | [101, 21] | rw [← exp_add, ← exp_add, ← exp_add, neg_add_eq_sub] | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(↑N * I * ↑x) + I * ↑x * (-1 / 2)) + cexp (-(↑N * I * ↑x) + I ... | case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (-(↑N * I * ↑x) + I * ↑x * (-1 / 2) + I * ↑x) - cexp (-(↑N * I * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ -cexp (-(... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (-(↑N * I * ↑x) + I * ↑x * (-1 / 2) + I * ↑x) - cexp (-(↑N * I * ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.a
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ cexp (-(↑... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp (I * ↑x) ≠ 0 | case e_a.hc
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
h : 1 - cexp (I * ↑x) = 0
⊢ cexp (I * ↑x) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
⊢ 1 - cexp... |
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 : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
h : 1 - cexp (I * ↑x) = 0
⊢ cexp (I * ↑x) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hc
N : ℕ
x : ℝ
this :
(cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x)) * dirichletKernel N x =
cexp ((↑N + 1 / 2) * I * ↑x) - cexp (-(↑N + 1 / 2) * I * ↑x)
h' : cexp (1 / 2 * I * ↑x) - cexp (-1 / 2 * I * ↑x) ≠ 0
h : 1 - cexp (I * ↑x) = 0
⊢ cexp... |
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 : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ dirichletKernel' N x = 0 | N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ cexp (I * ↑N * ↑x) / (1 - 1⁻¹) + cexp (-I * ↑N * ↑x) / (1 - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ 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 : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ cexp (I * ↑N * ↑x) / (1 - 1⁻¹) + cexp (-I * ↑N * ↑x) / (1 - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ cexp (I * ↑N * ↑x) / (1 - 1⁻¹) + cexp (-I * ↑N * ↑x) / (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 : ℕ
⊢ ∀ᵐ (x : ℝ), dirichletKernel N x = dirichletKernel' N x | N : ℕ
⊢ MeasureTheory.volume {a | ¬dirichletKernel N a = dirichletKernel' N a} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
⊢ ∀ᵐ (x : ℝ), 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 | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n : ℤ, x = n * (2 * Real.pi)} := by
ext x
simp
constructor
. sorry
. sorry | N : ℕ
⊢ MeasureTheory.volume {a | ¬dirichletKernel N a = dirichletKernel' N a} = 0 | N : ℕ
this : {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)}
⊢ MeasureTheory.volume {a | ¬dirichletKernel N a = dirichletKernel' N a} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
⊢ MeasureTheory.volume {a | ¬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 : ℕ
this : {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)}
⊢ MeasureTheory.volume {a | ¬dirichletKernel N a = dirichletKernel' N a} = 0 | N : ℕ
this : {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)}
⊢ MeasureTheory.volume {x | ∃ n, x = ↑n * (2 * Real.pi)} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
this : {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)}
⊢ MeasureTheory.volume {a | ¬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 : ℕ
this : {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)}
⊢ MeasureTheory.volume {x | ∃ n, x = ↑n * (2 * Real.pi)} = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
this : {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)}
⊢ MeasureTheory.volume {x | ∃ n, x = ↑n * (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 : ℕ
⊢ {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (2 * Real.pi)} | case h
N : ℕ
x : ℝ
⊢ x ∈ {x | ¬dirichletKernel N x = dirichletKernel' N x} ↔ x ∈ {x | ∃ n, x = ↑n * (2 * Real.pi)} | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
⊢ {x | ¬dirichletKernel N x = dirichletKernel' N x} = {x | ∃ n, x = ↑n * (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 : ℕ
x : ℝ
⊢ x ∈ {x | ¬dirichletKernel N x = dirichletKernel' N x} ↔ x ∈ {x | ∃ n, x = ↑n * (2 * Real.pi)} | case h
N : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x ↔ ∃ n, x = ↑n * (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : ℕ
x : ℝ
⊢ x ∈ {x | ¬dirichletKernel N x = dirichletKernel' N x} ↔ x ∈ {x | ∃ n, x = ↑n * (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 : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x ↔ ∃ n, x = ↑n * (2 * Real.pi) | case h.mp
N : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x → ∃ n, x = ↑n * (2 * Real.pi)
case h.mpr
N : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬dirichletKernel N x = dirichletKernel' N x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x ↔ ∃ n, x = ↑n * (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 : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x → ∃ n, x = ↑n * (2 * Real.pi)
case h.mpr
N : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬dirichletKernel N x = dirichletKernel' N x | case h.mpr
N : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬dirichletKernel N x = dirichletKernel' N x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x → ∃ n, x = ↑n * (2 * Real.pi)
case h.mpr
N : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬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 : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬dirichletKernel N x = dirichletKernel' N x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬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 : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x → ∃ n, x = ↑n * (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : ℕ
x : ℝ
⊢ ¬dirichletKernel N x = dirichletKernel' N x → ∃ n, x = ↑n * (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 : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬dirichletKernel N x = dirichletKernel' N x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : ℕ
x : ℝ
⊢ (∃ n, x = ↑n * (2 * Real.pi)) → ¬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 : ℕ
x : ℝ
⊢ ‖dirichletKernel N x‖ ≤ 2 * ↑N + 1 | N : ℕ
x : ℝ
⊢ ‖∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑x‖ ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ‖dirichletKernel N x‖ ≤ 2 * ↑N + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | norm_dirichletKernel_le | [122, 1] | [139, 11] | calc ‖∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑x‖
_ ≤ ∑ n ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier n) ↑x‖ := by
apply norm_sum_le
_ ≤ ∑ n ∈ Icc (-Int.ofNat N) ↑N, 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 (Con... | N : ℕ
x : ℝ
⊢ ‖∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑x‖ ≤ 2 * ↑N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ‖∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑x‖ ≤ 2 * ↑N + 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 : ℕ
x : ℝ
⊢ ‖∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑x‖ ≤ ∑ n ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier n) ↑x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ‖∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑x‖ ≤ ∑ n ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier n) ↑x‖
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 : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier n) ↑x‖ ≤ ∑ n ∈ Icc (-Int.ofNat N) ↑N, 1 | case h
N : ℕ
x : ℝ
⊢ ∀ i ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier i) ↑x‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier n) ↑x‖ ≤ ∑ n ∈ Icc (-Int.ofNat N) ↑N, 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 : ℕ
x : ℝ
⊢ ∀ i ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier i) ↑x‖ ≤ 1 | case h
N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ ‖(fourier n) ↑x‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : ℕ
x : ℝ
⊢ ∀ i ∈ Icc (-Int.ofNat N) ↑N, ‖(fourier i) ↑x‖ ≤ 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 : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ ‖(fourier n) ↑x‖ ≤ 1 | case h
N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
this : Fact (0 < 2 * Real.pi)
⊢ ‖(fourier n) ↑x‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ ‖(fourier n) ↑x‖ ≤ 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 : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
this : Fact (0 < 2 * Real.pi)
⊢ ‖(fourier n) ↑x‖ ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
this : Fact (0 < 2 * Real.pi)
⊢ ‖(fourier n) ↑x‖ ≤ 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 : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ Fact (0 < 2 * Real.pi) | N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ 0 < 2 * Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ 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 : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ 0 < 2 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ 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 : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, 1 = 2 * ↑N + 1 | N : ℕ
x : ℝ
⊢ (Icc (-Int.ofNat N) ↑N).card • 1 = 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, 1 = 2 * ↑N + 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 : ℕ
x : ℝ
⊢ (Icc (-Int.ofNat N) ↑N).card • 1 = 2 * ↑N + 1 | N : ℕ
x : ℝ
⊢ ↑(↑N + 1 + ↑N).toNat = 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ (Icc (-Int.ofNat N) ↑N).card • 1 = 2 * ↑N + 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 : ℕ
x : ℝ
⊢ ↑(↑N + 1 + ↑N).toNat = 2 * ↑N + 1 | N : ℕ
x : ℝ
⊢ (↑(N + 1 + N)).toNat = 2 * N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ↑(↑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] | rw [Int.toNat_ofNat] | N : ℕ
x : ℝ
⊢ (↑(N + 1 + N)).toNat = 2 * N + 1 | N : ℕ
x : ℝ
⊢ N + 1 + N = 2 * N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ (↑(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 : ℕ
x : ℝ
⊢ N + 1 + N = 2 * N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ 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) ≠ 1 | N : ℕ
x : ℝ
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1
case neg
N : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x : ℝ
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 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]
apply norm_dirichletKernel_le
exact h | case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1
case neg
N : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | case neg
N : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1
case neg
N : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 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 : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 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] | case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel N x‖ ≤ 2 * ↑N + 1
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp (I * ↑x) ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel N x‖ ≤ 2 * ↑N + 1
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp (I * ↑x) ≠ 1 | case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp (I * ↑x) ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel N x‖ ≤ 2 * ↑N + 1
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp (I * ↑x) ≠ 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 : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp (I * ↑x) ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : ℕ
x : ℝ
h : cexp (I * ↑x) ≠ 1
⊢ cexp (I * ↑x) ≠ 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 : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | case neg
N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : ℕ
x : ℝ
h : ¬cexp (I * ↑x) ≠ 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 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 : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 1 | case neg
N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ 0 ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ ‖dirichletKernel' N x‖ ≤ 2 * ↑N + 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 : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ 0 ≤ 2 * ↑N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : ℕ
x : ℝ
h : cexp (I * ↑x) = 1
⊢ 0 ≤ 2 * ↑N + 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
_ = ∑ n in Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x := by
rw [partialFourierSum]
_ = ∑ n in Icc (-Int.ofNat N) ↑N, (1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in (0 : ℝ)..2 * Real.pi, (fourier (-n) ↑y • f y)) * (fourier n) ↑x) := by
congr 1
ext n
... | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum f N x = 1 / (2 * ↑Real.pi) * ∫ (y : ℝ) 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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum f N x = 1 / (2 * ↑Real.pi) * ∫ (y : ℝ) 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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum f N x = ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ partialFourierSum f N x = ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x =
∑ n ∈ Icc (-Int.ofNat N) ↑N,
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) | case e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x) = fun n =>
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x =
∑ n ∈ Icc (-Int.ofNat N) ↑N,
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Re... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x) = fun n =>
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) | case e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x =
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun n => fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x) = fun n =>
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x =
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ fourierCoeffOn Real.two_pi_pos f n * (fourier n) ↑x =
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [← smul_sum, real_smul, sub_zero] | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N,
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) =
1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (four... | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ↑(1 / (2 * Real.pi)) *
∑ x_1 ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-x_1)) ↑y • f y) * (fourier x_1) ↑x =
1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fou... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N,
(1 / (2 * Real.pi - 0)) • ((∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) =
1 / (2 * ↑Real.pi) *
∑ n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | norm_cast | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ↑(1 / (2 * Real.pi)) *
∑ x_1 ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-x_1)) ↑y • f y) * (fourier x_1) ↑x =
1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fou... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ↑(1 / (2 * Real.pi)) *
∑ x_1 ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-x_1)) ↑y • f y) * (fourier x_1) ↑x =
1 / (2 * ↑Real.pi) *
∑ n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x =
1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, ∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)... | case e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun n => (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) = fun n =>
∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x =
1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun n => (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) = fun n =>
∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x | case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x =
∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun n => (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x) = fun n =>
∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (four... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x =
∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x | case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ ∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
(∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ (∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x =
∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x
TA... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ ∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
(∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
⊢ ∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
(∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y) * (fourier n) ↑x
TA... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | rw [← intervalIntegral.integral_finset_sum] | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, ∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in 0 ..2 * Real.pi, ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ... | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∀ i ∈ Icc (-Int.ofNat N) ↑N,
IntervalIntegrable (fun y => (fourier (-i)) ↑y • f y * (fourier i) ↑x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∑ n ∈ Icc (-Int.ofNat N) ↑N, ∫ (y : ℝ) in 0 ..2 * Real.pi, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | intro n _ | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∀ i ∈ Icc (-Int.ofNat N) ↑N,
IntervalIntegrable (fun y => (fourier (-i)) ↑y • f y * (fourier i) ↑x) MeasureTheory.volume 0 (2 * Real.pi) | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ IntervalIntegrable (fun y => (fourier (-n)) ↑y • f y * (fourier n) ↑x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∀ i ∈ Icc (-Int.ofNat N) ↑N,
IntervalIntegrable (fun y => (fourier (-i)) ↑y • f y * (fourier i) ↑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] | apply IntervalIntegrable.mul_const | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ IntervalIntegrable (fun y => (fourier (-n)) ↑y • f y * (fourier n) ↑x) MeasureTheory.volume 0 (2 * Real.pi) | case hf
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ IntervalIntegrable (fun x => (fourier (-n)) ↑x • f x) MeasureTheory.volume 0 (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ IntervalIntegrable (fun y => (fourier (-n)) ↑y • f y * (fourier n) ↑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] | apply IntervalIntegrable.continuousOn_mul h fourier_uniformContinuous.continuous.continuousOn | case hf
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ IntervalIntegrable (fun x => (fourier (-n)) ↑x • f x) MeasureTheory.volume 0 (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
n : ℤ
a✝ : n ∈ Icc (-Int.ofNat N) ↑N
⊢ IntervalIntegrable (fun x => (fourier (-n)) ↑x • 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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in 0 ..2 * Real.pi, ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in 0 ..2 * Real.pi, f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier ... | case e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun y => ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x) = fun y =>
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in 0 ..2 * Real.pi, ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in ... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun y => ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x) = fun y =>
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x | case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun y => ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x) = fun y =>
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier ... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x | case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
∑ i ∈ Icc (-Int.ofNat N) ↑N, f y * ((fourier (-i)) ↑y * (fourier i) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.vol... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
∑ i ∈ Icc (-Int.ofNat N) ↑N, f y * ((fourier (-i)) ↑y * (fourier i) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.vol... | case e_a.e_f.h.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ (fun n => (fourier (-n)) ↑y • f y * (fourier n) ↑x) = fun i => f y * ((fourier (-i)) ↑y * (fourier i) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ℝ → ℤ → ℝ
f : ℝ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y • f y * (fourier n) ↑x =
∑ i ∈ Icc (-Int.ofNat N) ↑N, f y * ((fourier (-i)) ↑y * (fourier i) ↑x)
f :... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ (fun n => (fourier (-n)) ↑y • f y * (fourier n) ↑x) = fun i => f y * ((fourier (-i)) ↑y * (fourier i) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ℝ → ℤ → ℝ
f : ℝ... | case e_a.e_f.h.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
n : ℤ
⊢ (fourier (-n)) ↑y • f y * (fourier n) ↑x = f y * ((fourier (-n)) ↑y * (fourier n) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ℝ → ℤ → ℝ
f : ℝ → ℂ
N : ℕ
x... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ (fun n => (fourier (-n)) ↑y • f y * (fourier n) ↑x) = fun i => f y * ((fourier (-i)) ↑y * (fourier i) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegra... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
n : ℤ
⊢ (fourier (-n)) ↑y • f y * (fourier n) ↑x = f y * ((fourier (-n)) ↑y * (fourier n) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ℝ → ℤ → ℝ
f : ℝ → ℂ
N : ℕ
x... | case e_a.e_f.h.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
n : ℤ
⊢ (fourier (-n)) ↑y * f y * (fourier n) ↑x = f y * ((fourier (-n)) ↑y * (fourier n) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ℝ → ℤ → ℝ
f : ℝ → ℂ
N : ℕ
x... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
n : ℤ
⊢ (fourier (-n)) ↑y • f y * (fourier n) ↑x = f y * ((fourier (-n)) ↑y * (fourier n) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f Measur... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
n : ℤ
⊢ (fourier (-n)) ↑y * f y * (fourier n) ↑x = f y * ((fourier (-n)) ↑y * (fourier n) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ ℝ → ℤ → ℝ
f : ℝ → ℂ
N : ℕ
x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
n : ℤ
⊢ (fourier (-n)) ↑y * f y * (fourier n) ↑x = f y * ((fourier (-n)) ↑y * (fourier n) ↑x)
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f Measur... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Dirichlet_kernel.lean | partialFourierSum_eq_conv_dirichletKernel | [152, 1] | [191, 35] | congr | f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in 0 ..2 * Real.pi, f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x =
1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, f y * dirichletKernel N (x - y) | case e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun y => f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x) = fun y =>
f y * dirichletKernel N (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ 1 / (2 * ↑Real.pi) *
∫ (y : ℝ) in 0 ..2 * Real.pi, f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x =
1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 ... |
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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun y => f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x) = fun y =>
f y * dirichletKernel N (x - y) | case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x = f y * dirichletKernel N (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
⊢ (fun y => f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x) = 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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x = f y * dirichletKernel N (x - y) | case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x =
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑(x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x = 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 : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x =
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑(x - y) | case e_a.e_f.h.e_a.e_f
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ (fun n => (fourier (-n)) ↑y * (fourier n) ↑x) = fun n => (fourier n) ↑(x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
f : ℝ → ℂ
N : ℕ
x : ℝ
h : IntervalIntegrable f MeasureTheory.volume 0 (2 * Real.pi)
y : ℝ
⊢ f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier (-n)) ↑y * (fourier n) ↑x =
f y * ∑ n ∈ Icc (-Int.ofNat N) ↑N, (fourier n) ↑(x - y)
TACTIC:
|
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