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/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply MeasureTheory.setIntegral_mono_on | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . apply f'_cont.norm.integrableOn_uIcc.mono_set
apply Set.Ioc_subset_Icc_self | case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . apply MeasureTheory.integrableOn_const.mpr
right
rw [Real.volume_uIoc]
apply ENNReal.ofReal_lt_top | case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case hs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . apply measurableSet_uIoc | case hs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . intro t ht
rw [Set.mem_uIoc] at ht
have ht' : 0 < t ∧ t ≤ 1 := by
rcases ht with ht | ht <;> (constructor <;> linarith)
rw [f'def]
simp only [norm_div, Complex.norm_eq_abs, norm_pow]
gcongr
. calc Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t)) + Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t)... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply f'_cont.norm.integrableOn_uIcc.mono_set | case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply Set.Ioc_subset_Icc_self | case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply MeasureTheory.integrableOn_const.mpr | case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | right | case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [Real.volume_uIoc] | case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Comple... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply ENNReal.ofReal_lt_top | case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Comple... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply measurableSet_uIoc | case hs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | intro t ht | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [Set.mem_uIoc] at ht | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | have ht' : 0 < t ∧ t ≤ 1 := by
rcases ht with ht | ht <;> (constructor <;> linarith) | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [f'def] | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | simp only [norm_div, Complex.norm_eq_abs, norm_pow] | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | gcongr | case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f'... | case h.hac
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . calc Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t)) + Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t)))
_ ≤ Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by
apply Complex.abs.isAbsoluteValue.abv_add
_ ≤ Complex.abs (... | case h.hac
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hac
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . rw [mul_comm, ←neg_mul, mul_comm]
norm_cast
apply lower_secant_bound
. simp only [neg_mul, Set.mem_Icc, neg_add_le_iff_le_add, le_add_neg_iff_add_le,
neg_le_sub_iff_le_add]
constructor <;> linarith [Real.two_le_pi, Real.two_pi_pos]
. rw [abs_neg, le_abs]
left
rcases ht with ht | ht <;> linarit... | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(C... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rcases ht with ht | ht <;> (constructor <;> linarith) | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | constructor <;> linarith | case inr
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | calc Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t)) + Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t)))
_ ≤ Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by
apply Complex.abs.isAbsoluteValue.abv_add
_ ≤ Complex.abs (-1) + Co... | case h.hac
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hac
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply Complex.abs.isAbsoluteValue.abv_add | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | gcongr | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | case bc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply Complex.abs.isAbsoluteValue.abv_add | case bc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | gcongr | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | case h₁.h₁
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . simp | case h₁.h₁
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₁
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . rw [mul_comm, ←neg_mul]
norm_cast
apply le_of_eq
apply Complex.abs_exp_ofReal_mul_I | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . simp
apply mul_le_one
norm_cast
rw [abs_of_nonpos] <;> linarith
simp
rw [mul_comm, ←neg_mul]
norm_cast
apply le_of_eq
apply Complex.abs_exp_ofReal_mul_I | case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | simp | case h₁.h₁
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₁
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [mul_comm, ←neg_mul] | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | norm_cast | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply le_of_eq | case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₁.h₂.a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply Complex.abs_exp_ofReal_mul_I | case h₁.h₂.a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂.a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Com... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | simp | case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply mul_le_one | case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f... | case h₂.ha
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | norm_cast | case h₂.ha
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₂.ha
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.ha
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [abs_of_nonpos] <;> linarith | case h₂.ha
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₂.hb'
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.ha
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | simp | case h₂.hb'
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :... | case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hb'
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Comp... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [mul_comm, ←neg_mul] | case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | norm_cast | case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply le_of_eq | case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | case h₂.hb.a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hb
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Compl... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply Complex.abs_exp_ofReal_mul_I | case h₂.hb.a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hb.a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Com... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | norm_num | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [mul_comm, ←neg_mul, mul_comm] | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(C... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | norm_cast | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(C... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply lower_secant_bound | case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'de... | case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(C... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . simp only [neg_mul, Set.mem_Icc, neg_add_le_iff_le_add, le_add_neg_iff_add_le,
neg_le_sub_iff_le_add]
constructor <;> linarith [Real.two_le_pi, Real.two_pi_pos] | case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . rw [abs_neg, le_abs]
left
rcases ht with ht | ht <;> linarith [ht.1] | case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | simp only [neg_mul, Set.mem_Icc, neg_add_le_iff_le_add, le_add_neg_iff_add_le,
neg_le_sub_iff_le_add] | case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | constructor <;> linarith [Real.two_le_pi, Real.two_pi_pos] | case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xIcc
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [abs_neg, le_abs] | case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | left | case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2... | case h.hbd.hab.xAbs.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xAbs
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rcases ht with ht | ht <;> linarith [ht.1] | case h.hbd.hab.xAbs.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hbd.hab.xAbs.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | apply MeasureTheory.setIntegral_const | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | congr | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | case e_a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [Real.volume_uIoc, ENNReal.toReal_ofReal (abs_nonneg (y - y'))] | case e_a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | ring | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | gcongr | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
f' = fun ... | case h.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t))... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | norm_num | case h.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2
f'def :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
f' : ℝ → ℂ :=
fun t =>
(-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [abs_neg, abs_of_nonneg yy'nonneg.2] | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1 | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y' ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | assumption | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y' ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y' ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [abs_neg, abs_of_nonneg yy'nonneg.1] | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | assumption | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ y ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [K, K] | x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖K x y - K x y'‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖K x y - K x y'‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | wlog x_eq_zero : x = 0 generalizing x y y' | x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |... | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . intro h
set x_ := (0 : ℝ) with x_def
set y_ := y - x with y_def
set y'_ := y' - x with y'_def
have h_ : 2 * |y_ - y'_| ≤ |x_ - y_| := by
rw [x_def, y_def, y'_def]
simpa
have := this x_def h_
rw [x_def, y_def, y'_def] at this
simpa | case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |... | x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [x_eq_zero] | x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |0 - y| → ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | intro h | x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |0 - y| → ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|) | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |0 - y|
⊢ ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
⊢ 2 * |y - y'| ≤ |0 - y| → ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp at h | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |0 - y|
⊢ ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|) | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
⊢ ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |0 - y|
⊢ ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp only [zero_sub, abs_neg] | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
⊢ ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|) | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
⊢ ‖k (0 - y) - k (0 - y')‖ ≤ 2 ^ 8 * (1 / |0 - y|) * (|y - y'| / |0 - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | wlog yy'nonneg : 0 ≤ y ∧ 0 ≤ y' generalizing y y' | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case inr
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . by_cases yge0 : 0 ≤ y
. push_neg at yy'nonneg
exfalso
rw [abs_of_nonneg yge0, abs_of_nonneg] at h <;> linarith [yy'nonneg yge0]
. push_neg at yge0
by_cases y'ge0 : 0 ≤ y'
. exfalso
rw [abs_of_neg yge0, abs_of_neg] at h <;> linarith
. push_neg at y'ge0
set! y_ := -y with y_def
... | case inr
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤... | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | by_cases ypos : y ≤ 0 | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ¬y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y -... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . have y_eq_zero : y = 0 := by
exact le_antisymm ypos yy'nonneg.1
have y'_eq_zero : y' = 0 := by
rw [y_eq_zero] at h
simp at h
rw [abs_of_nonneg yy'nonneg.2] at h
linarith
rw [y_eq_zero, y'_eq_zero]
simp | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ¬y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y -... | case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ¬y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | push_neg at ypos | case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ¬y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ¬y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have y2ley' : y / 2 ≤ y' := by
rw [div_le_iff two_pos]
calc y
_ = 2 * (y - y') - y + 2 * y' := by ring
_ ≤ 2 * |y - y'| - y + 2 * y' := by
gcongr
apply le_abs_self
_ ≤ y - y + 2 * y' := by
gcongr
rw [abs_eq_self.mpr yy'nonneg.1] at h
exact h
_ = y' * 2 := by ring | case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rcases le_or_gt y 1, le_or_gt y' 1 with ⟨hy | hy, hy' | hy'⟩ | case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . apply le_trans (Hilbert_kernel_regularity_main_part yy'nonneg ypos y2ley' hy hy')
gcongr <;> norm_num | case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [@k_of_one_le_abs (-y')]
. calc ‖k (-y) - 0‖
_ = ‖k (-y) - k (-1)‖ := by
congr
apply (k_of_one_le_abs _).symm
simp
_ ≤ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|) := by
apply Hilbert_kernel_regularity_main_part
constructor
all_goals linarith
_ ≤ 2 ^ 6 * (... | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [@k_of_one_le_abs (-y)]
. calc ‖0 - k (-y')‖
_ = ‖k (-1) - k (-y')‖ := by
congr
apply (k_of_one_le_abs _).symm
simp
_ = ‖k (-y') - k (-1)‖ := by
rw [norm_sub_rev]
_ ≤ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|) := by
apply Hilbert_kernel_regularity_main_part
... | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... | case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . calc ‖k (-y) - k (-y')‖
_ = 0 := by
simp
rw [k_of_one_le_abs, k_of_one_le_abs] <;> (rw [abs_neg, abs_of_nonneg] <;> linarith)
_ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) := by
apply mul_nonneg
apply mul_nonneg
. norm_num
. simp
. apply div_nonneg <;> simp | case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | intro h | case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
h : 2 * |y - y'| ≤ |x - y|
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
⊢ 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | set x_ := (0 : ℝ) with x_def | case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
h : 2 * |y - y'| ≤ |x - y|
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
this :
∀ {x y y' : ℝ},
x = 0 → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = 0
h : 2 * |y - y'| ≤ |x - y|
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | set y_ := y - x with y_def | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|) | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | set y'_ := y' - x with y'_def | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y... | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
⊢ ‖k (x - y) - k... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have h_ : 2 * |y_ - y'_| ≤ |x_ - y_| := by
rw [x_def, y_def, y'_def]
simpa | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
⊢ ‖k (x - y) - k... | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
h_ : 2 * |y_ - y... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have := this x_def h_ | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
h_ : 2 * |y_ - y... | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this✝ :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
h_ : 2 * |y_ - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [x_def, y_def, y'_def] at this | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this✝ :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
h_ : 2 * |y_ - ... | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this✝ :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
h_ : 2 * |y_ - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this✝ :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simpa | case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this✝ :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
h_ : 2 * |y_ - ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this✝ :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [x_def, y_def, y'_def] | x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
⊢ 2 * |y_ - y'_| ≤ |x_ - ... | x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
⊢ 2 * |y - x - (y' - x)| ... | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simpa | x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ : ℝ := y' - x
y'_def : y'_ = y' - x
⊢ 2 * |y - x - (y' - x)| ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
h : 2 * |y - y'| ≤ |x - y|
x_ : ℝ := 0
this :
∀ {x y y' : ℝ},
x = x_ → 2 * |y - y'| ≤ |x - y| → ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / |x - y|) * (|y - y'| / |x - y|)
x_eq_zero : ¬x = x_
x_def : x_ = 0
y_ : ℝ := y - x
y_def : y_ = y - x
y'_ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | by_cases yge0 : 0 ≤ y | case inr
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : 0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x y y' : ℝ
x_eq_zero : x = 0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . push_neg at yy'nonneg
exfalso
rw [abs_of_nonneg yge0, abs_of_nonneg] at h <;> linarith [yy'nonneg yge0] | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : 0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x y y' : ℝ
x_eq_zero : x = 0
... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : ¬0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : 0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . push_neg at yge0
by_cases y'ge0 : 0 ≤ y'
. exfalso
rw [abs_of_neg yge0, abs_of_neg] at h <;> linarith
. push_neg at y'ge0
set! y_ := -y with y_def
set! y'_ := -y' with y'_def
have h_ : 2 * |y_ - y'_| ≤ |y_| := by
rw [y_def, y'_def, ← abs_neg]
simpa [neg_add_eq_sub]
have y_y'_nonn... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : ¬0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : ¬0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | push_neg at yy'nonneg | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : 0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yge0 : 0 ≤ y
yy'nonneg : 0 ≤ y → y' < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : 0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | exfalso | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yge0 : 0 ≤ y
yy'nonneg : 0 ≤ y → y' < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yge0 : 0 ≤ y
yy'nonneg : 0 ≤ y → y' < 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yge0 : 0 ≤ y
yy'nonneg : 0 ≤ y → y' < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg yge0, abs_of_nonneg] at h <;> linarith [yy'nonneg yge0] | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yge0 : 0 ≤ y
yy'nonneg : 0 ≤ y → y' < 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yge0 : 0 ≤ y
yy'nonneg : 0 ≤ y → y' < 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | push_neg at yge0 | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : ¬0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : ¬0 ≤ y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |... |
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