Split (1)

train · 219k rows

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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∫ (t : ℝ) in Ι y' y, ‖f' t‖ ≤ ∫ (t : ℝ) in Ι y' y, 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => ‖f' x‖) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∫ (t : ℝ) in Ι y' y, ‖f' t‖ ≤ ∫ (t : ℝ) in Ι y' y, 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => ‖f' x‖) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => ‖f' x‖) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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))) _ ≤ 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) + Complex.abs (Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by gcongr apply Complex.abs.isAbsoluteValue.abv_add _ ≤ 1 + 1 + 1 := by gcongr . simp . rw [mul_comm, ←neg_mul] norm_cast apply le_of_eq apply Complex.abs_exp_ofReal_mul_I . 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 _ = 3 := by norm_num . 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 <;> linarith [ht.1]	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => ‖f' x‖) (Ι y' y) MeasureTheory.volume	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ Ι y' y ⊆ Set.uIcc y' y	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => ‖f' x‖) (Ι y' y) MeasureTheory.volume TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ Ι y' y ⊆ Set.uIcc y' y	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ Ι y' y ⊆ Set.uIcc y' y TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 / (y / 2 / 2) ^ 2 = 0 ∨ MeasureTheory.volume (Ι y' y) < ⊤	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.IntegrableOn (fun x => 3 / (y / 2 / 2) ^ 2) (Ι y' y) MeasureTheory.volume TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 / (y / 2 / 2) ^ 2 = 0 ∨ MeasureTheory.volume (Ι y' y) < ⊤	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.volume (Ι y' y) < ⊤	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 / (y / 2 / 2) ^ 2 = 0 ∨ MeasureTheory.volume (Ι y' y) < ⊤ TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.volume (Ι y' y) < ⊤	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ENNReal.ofReal \|y - y'\| < ⊤	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasureTheory.volume (Ι y' y) < ⊤ TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ENNReal.ofReal \|y - y'\| < ⊤	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ENNReal.ofReal \|y - y'\| < ⊤ TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y)	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ MeasurableSet (Ι y' y) TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : t ∈ Ι y' y ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∀ x ∈ Ι y' y, ‖f' x‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : t ∈ Ι y' y ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : t ∈ Ι y' y ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖(fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2) t‖ ≤ 3 / (y / 2 / 2) ^ 2	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖f' t‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖(fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2) t‖ ≤ 3 / (y / 2 / 2) ^ 2	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / Complex.abs (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≤ 3 / (y / 2 / 2) ^ 2	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖(fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2) t‖ ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / Complex.abs (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≤ 3 / (y / 2 / 2) ^ 2	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 3 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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp)	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 * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / Complex.abs (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≤ 3 / (y / 2 / 2) ^ 2 TACTIC:
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) + Complex.abs (Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by gcongr apply Complex.abs.isAbsoluteValue.abv_add _ ≤ 1 + 1 + 1 := by gcongr . simp . rw [mul_comm, ←neg_mul] norm_cast apply le_of_eq apply Complex.abs_exp_ofReal_mul_I . 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 _ = 3 := by norm_num	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 3 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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp)	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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp)	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 3 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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp) TACTIC:
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 <;> linarith [ht.1]	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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp)	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp) TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ⊢ 0 < t ∧ t ≤ 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 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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ⊢ 0 < t ∧ t ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y < t ∧ t ≤ y' ⊢ 0 < t ∧ t ≤ 1	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y < t ∧ t ≤ y' ⊢ 0 < t ∧ t ≤ 1 TACTIC:
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) + Complex.abs (Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by gcongr apply Complex.abs.isAbsoluteValue.abv_add _ ≤ 1 + 1 + 1 := by gcongr . simp . rw [mul_comm, ←neg_mul] norm_cast apply le_of_eq apply Complex.abs_exp_ofReal_mul_I . 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 _ = 3 := by norm_num	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 3	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 3 TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp)	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp)	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1 + (-(Complex.I * ↑t)).exp) ≤ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1 + 1 + 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 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 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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) + Complex.abs (-(Complex.I * ↑t)).exp + Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1 + 1 + 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 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 [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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-1) ≤ 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 [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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-↑t * Complex.I).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-↑t * Complex.I).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-↑t * Complex.I).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp = 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp = 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp = 1 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1	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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑t - 1) * Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) ≤ 1 TACTIC:
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' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑t - 1) * Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑t - 1) ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑t - 1) * Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑t - 1) ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|t - 1\| ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑t - 1) ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 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_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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|t - 1\| ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|t - 1\| ≤ 1 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ Complex.abs (-(Complex.I * ↑t)).exp 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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 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 [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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-↑t * Complex.I).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-(Complex.I * ↑t)).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-↑t * Complex.I).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp ≤ 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (-↑t * Complex.I).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp ≤ 1	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp = 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp ≤ 1 TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp = 1	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ Complex.abs (↑(-t) * Complex.I).exp = 1 TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 1 + 1 + 1 = 3	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 1 + 1 + 1 = 3 TACTIC:
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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp)	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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (Complex.I * -↑t).exp)	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (-(Complex.I * ↑t)).exp) TACTIC:
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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (Complex.I * -↑t).exp)	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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (Complex.I * ↑(-t)).exp)	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (Complex.I * -↑t).exp) TACTIC:
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'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (Complex.I * ↑(-t)).exp)	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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ -t ∈ Set.Icc (-2 * Real.pi + y / 2) (2 * Real.pi - y / 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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\|	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 / 2 ≤ Complex.abs (1 - (Complex.I * ↑(-t)).exp) TACTIC:
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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ -t ∈ Set.Icc (-2 * Real.pi + y / 2) (2 * Real.pi - y / 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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\|	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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\|	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ -t ∈ Set.Icc (-2 * Real.pi + y / 2) (2 * Real.pi - y / 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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\| 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, 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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\|	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\| TACTIC:
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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ -t ∈ Set.Icc (-2 * Real.pi + y / 2) (2 * Real.pi - y / 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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 + t ≤ 2 * Real.pi ∧ y / 2 ≤ 2 * Real.pi + t	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ -t ∈ Set.Icc (-2 * Real.pi + y / 2) (2 * Real.pi - y / 2) TACTIC:
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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 + t ≤ 2 * Real.pi ∧ y / 2 ≤ 2 * Real.pi + t	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 + t ≤ 2 * Real.pi ∧ y / 2 ≤ 2 * Real.pi + t 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, 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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\|	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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ t ∨ y / 2 ≤ -t	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ \|(-t)\| TACTIC:
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 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ t ∨ y / 2 ≤ -t	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) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ t	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ t ∨ y / 2 ≤ -t TACTIC:
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) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ t	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) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) t : ℝ ht : y' < t ∧ t ≤ y ∨ y < t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ y / 2 ≤ t TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∫ (t : ℝ) in Ι y' y, 3 / (y / 2 / 2) ^ 2 = (MeasureTheory.volume (Ι y' y)).toReal * (3 / (y / 2 / 2) ^ 2)	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ ∫ (t : ℝ) in Ι y' y, 3 / (y / 2 / 2) ^ 2 = (MeasureTheory.volume (Ι y' y)).toReal * (3 / (y / 2 / 2) ^ 2) TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ (MeasureTheory.volume (Ι y' y)).toReal * (3 / (y / 2 / 2) ^ 2) = \|y - y'\| * (3 / (y / 2 / 2) ^ 2)	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ (MeasureTheory.volume (Ι y' y)).toReal = \|y - y'\|	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ (MeasureTheory.volume (Ι y' y)).toReal * (3 / (y / 2 / 2) ^ 2) = \|y - y'\| * (3 / (y / 2 / 2) ^ 2) TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ (MeasureTheory.volume (Ι y' y)).toReal = \|y - y'\|	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ (MeasureTheory.volume (Ι y' y)).toReal = \|y - y'\| TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ \|y - y'\| * (3 / (y / 2 / 2) ^ 2) = 3 * (2 * 2) ^ 2 * (1 / y) * (\|y - y'\| / y)	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ \|y - y'\| * (3 / (y / 2 / 2) ^ 2) = 3 * (2 * 2) ^ 2 * (1 / y) * (\|y - y'\| / y) TACTIC:
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 t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 * (2 * 2) ^ 2 * (1 / y) * (\|y - y'\| / y) ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 * (2 * 2) ^ 2 ≤ 2 ^ 6	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)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 * (2 * 2) ^ 2 * (1 / y) * (\|y - y'\| / y) ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y) TACTIC:
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 : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 * (2 * 2) ^ 2 ≤ 2 ^ 6	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.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 f'def : f' = fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 c : ℝ → ℂ := fun t => 1 - ↑t cdef : c = fun t => 1 - ↑t c' : ℝ → ℂ := fun t => -1 c'def : c' = fun t => -1 d : ℝ → ℂ := fun t => 1 - (-(Complex.I * ↑t)).exp ddef : d = fun t => 1 - (-(Complex.I * ↑t)).exp d' : ℝ → ℂ := fun t => Complex.I * (-(Complex.I * ↑t)).exp d'def : d' = fun t => Complex.I * (-(Complex.I * ↑t)).exp d_nonzero : ∀ {t : ℝ}, t ∈ Set.uIcc y' y → d t ≠ 0 f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t f'_cont : ContinuousOn (fun t => f' t) (Set.uIcc y' y) ⊢ 3 * (2 * 2) ^ 2 ≤ 2 ^ 6 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.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'\| ≤ \|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]	. 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\| → ‖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\| → ‖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'\| / \|x - 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\|) TACTIC:
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'\| ≤ \|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: 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 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'_nonneg : 0 ≤ y_ ∧ 0 ≤ y'_ := by constructor <;> linarith have := this h_ y_y'_nonneg rw [y_def, y'_def] at this simp only [neg_neg, abs_neg, sub_neg_eq_add, neg_add_eq_sub] at this rw [← RCLike.norm_conj, map_sub, ← k_of_neg_eq_conj_k, ← k_of_neg_eq_conj_k, ←abs_neg (y' - y)] at this simpa	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'\| ≤ \|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'\| ≤ \|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'\| / \|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\|) TACTIC:
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 - 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 - 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' 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]	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' 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' 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' 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 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' 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' 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' 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' 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' 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' 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.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' 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' 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' 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]	. 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 * (1 / \|y\|) * (\|y - y'\| / \|y\|) := by gcongr 2 ^ 6 * (1 / \|y\|) * (?_ / \|y\|) rw [abs_sub_comm, abs_of_nonneg, abs_sub_comm, abs_of_nonneg] <;> linarith _ ≤ 2 ^ 8 * (1 / \|y\|) * (\|y - y'\| / \|y\|) := by gcongr <;> norm_num . rw [abs_neg, abs_of_nonneg] <;> linarith	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' 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' 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' 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' 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.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' 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' 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]	. 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 constructor all_goals linarith _ = 2 ^ 6 * (1 / y') * ((1 - y') / y') := by congr . rw [abs_of_nonneg] exact yy'nonneg.2 . rw [abs_of_nonpos] simp linarith . rw [abs_of_nonneg] exact yy'nonneg.2 _ ≤ 2 ^ 6 * (1 / (y / 2)) * ((1 - y') / (y / 2)) := by gcongr . apply div_nonneg <;> linarith . linarith _ = (2 ^ 6 * 2 * 2) * (1 / y) * ((1 - y') / y) := by ring _ ≤ (2 ^ 6 * 2 * 2) * (1 / \|y\|) * (\|y - y'\| / \|y\|) := by gcongr apply div_nonneg <;> linarith rw [abs_of_nonneg yy'nonneg.1] rw [abs_of_nonneg] <;> linarith rw [abs_of_nonneg yy'nonneg.1] _ ≤ 2 ^ 8 * (1 / \|y\|) * (\|y - y'\| / \|y\|) := by gcongr norm_num . rw [abs_neg, abs_of_nonneg] <;> linarith	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' 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' 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 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]	. 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'\| / \|x - y\|) TACTIC:
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 - y'\| / \|x - y\|) TACTIC:
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 - y'\| / \|x - 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 ^ 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]	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 - 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 y'_ : ℝ := y' - x y'_def : y'_ = y' - x ⊢ ‖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 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 - y'\| / \|x - y\|) TACTIC:
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 (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 y'_ : ℝ := y' - x y'_def : y'_ = y' - x 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 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 (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]	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'_\| ≤ \|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 y_ : ℝ := y - x y_def : y_ = y - x y'_ : ℝ := y' - x y'_def : y'_ = y' - x h_ : 2 * \|y_ - y'_\| ≤ \|x_ - y_\| this : ‖k (x_ - y_) - k (x_ - y'_)‖ ≤ 2 ^ 8 * (1 / \|x_ - y_\|) * (\|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 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'_\| ≤ \|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]	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_ - y'_\| ≤ \|x_ - y_\| this : ‖k (x_ - y_) - k (x_ - y'_)‖ ≤ 2 ^ 8 * (1 / \|x_ - y_\|) * (\|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 y_ : ℝ := y - x y_def : y_ = y - x y'_ : ℝ := y' - x y'_def : y'_ = y' - x h_ : 2 * \|y_ - y'_\| ≤ \|x_ - y_\| this : ‖k (0 - (y - x)) - k (0 - (y' - x))‖ ≤ 2 ^ 8 * (1 / \|0 - (y - x)\|) * (\|y - x - (y' - x)\| / \|0 - (y - x)\|) ⊢ ‖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 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'_\| ≤ \|x_ - y_\| this : ‖k (x_ - y_) - k (x_ - y'_)‖ ≤ 2 ^ 8 * (1 / \|x_ - y_\|) * (\|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]	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_ - y'_\| ≤ \|x_ - y_\| this : ‖k (0 - (y - x)) - k (0 - (y' - x))‖ ≤ 2 ^ 8 * (1 / \|0 - (y - x)\|) * (\|y - x - (y' - x)\| / \|0 - (y - x)\|) ⊢ ‖k (x - y) - k (x - y')‖ ≤ 2 ^ 8 * (1 / \|x - y\|) * (\|y - y'\| / \|x - 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_ = y - x y'_ : ℝ := y' - x y'_def : y'_ = y' - x h_ : 2 * \|y_ - y'_\| ≤ \|x_ - y_\| this : ‖k (0 - (y - x)) - k (0 - (y' - x))‖ ≤ 2 ^ 8 * (1 / \|0 - (y - x)\|) * (\|y - x - (y' - x)\| / \|0 - (y - x)\|) ⊢ ‖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]	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_ - y_\|	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)\| ≤ \|0 - (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'_ : ℝ := y' - x y'_def : y'_ = y' - x ⊢ 2 * \|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]	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)\| ≤ \|0 - (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'_ : ℝ := y' - x y'_def : y'_ = y' - x ⊢ 2 * \|y - x - (y' - x)\| ≤ \|0 - (y - x)\| 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	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 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 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\|) TACTIC:
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 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 : ¬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\|) * (\|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 : ¬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]	. 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'_nonneg : 0 ≤ y_ ∧ 0 ≤ y'_ := by constructor <;> linarith have := this h_ y_y'_nonneg rw [y_def, y'_def] at this simp only [neg_neg, abs_neg, sub_neg_eq_add, neg_add_eq_sub] at this rw [← RCLike.norm_conj, map_sub, ← k_of_neg_eq_conj_k, ← k_of_neg_eq_conj_k, ←abs_neg (y' - y)] at this simpa	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 / \|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]	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\|) * (\|y - y'\| / \|y\|) TACTIC:
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\|) * (\|y - y'\| / \|y\|) TACTIC:
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 / \|y\|) * (\|y - y'\| / \|y\|) TACTIC:

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