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]	set f' : ℝ → ℂ := fun t ↦ (-1 + Complex.exp (-(Complex.I * t)) + Complex.I * (t - 1) * Complex.exp (-(Complex.I * t))) / (1 - Complex.exp (-(Complex.I * t))) ^ 2 with f'def	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	set c : ℝ → ℂ := fun t ↦ (1 - t) with cdef	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	set c' : ℝ → ℂ := fun t ↦ -1 with c'def	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	set d : ℝ → ℂ := fun t ↦ (1 - Complex.exp (-(Complex.I * t))) with ddef	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	set d' : ℝ → ℂ := fun t ↦ Complex.I * Complex.exp (-(Complex.I * t)) with d'def	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	have d_nonzero {t : ℝ} (ht : t ∈ Set.uIcc y' y) : d t ≠ 0 := by rw [Set.mem_uIcc] at ht have ht' : 0 < t ∧ t ≤ 1 := by rcases ht with ht \| ht <;> (constructor <;> linarith) rw [ddef] simp rw [←norm_eq_zero] apply ne_of_gt calc ‖1 - Complex.exp (-(Complex.I * ↑t))‖ _ ≥ \|(1 - Complex.exp (-(Complex.I * ↑t))).im\| := by simp only [Complex.norm_eq_abs, ge_iff_le] apply Complex.abs_im_le_abs _ = Real.sin t := by simp rw [Complex.exp_im] simp apply Real.sin_nonneg_of_nonneg_of_le_pi . linarith . linarith [Real.two_le_pi] _ > 0 := by apply Real.sin_pos_of_pos_of_lt_pi . linarith . linarith [Real.two_le_pi]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	have f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t := by intro t ht have : f = fun t ↦ c t / d t := by simp rw [this] have : f' = fun t ↦ ((c' t * d t - c t * d' t) / d t ^ 2) := by ext t rw [f'def, cdef, c'def, ddef, d'def] simp congr ring rw [this] apply HasDerivAt.div . rw [cdef, c'def] simp apply HasDerivAt.const_sub apply HasDerivAt.ofReal_comp apply hasDerivAt_id' . rw [ddef, d'def] simp rw [←neg_neg (Complex.I * Complex.exp (-(Complex.I * ↑t)))] apply HasDerivAt.const_sub rw [←neg_mul, mul_comm] apply HasDerivAt.cexp apply HasDerivAt.neg conv in fun (x : ℝ) ↦ Complex.I * (x : ℝ) => ext rw [mul_comm] set e : ℂ → ℂ := fun t ↦ t * Complex.I with edef have : (fun (t : ℝ) ↦ t * Complex.I) = fun (t : ℝ) ↦ e t := by rw [edef] rw [this] apply HasDerivAt.comp_ofReal rw [edef] apply hasDerivAt_mul_const . exact d_nonzero ht	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	have f'_cont : ContinuousOn (fun t ↦ f' t) (Set.uIcc y' y) := by apply ContinuousOn.div . apply Continuous.continuousOn continuity . apply Continuous.continuousOn apply Continuous.pow apply Continuous.sub . apply continuous_const . apply Continuous.cexp apply Continuous.neg apply Continuous.mul . apply continuous_const . apply Complex.continuous_ofReal . intro t ht simp apply d_nonzero ht	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (1 / y) * (\|y - 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 ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	calc ‖(1 - ↑y) / (1 - Complex.exp (-(Complex.I * ↑y))) - (1 - ↑y') / (1 - Complex.exp (-(Complex.I * ↑y')))‖ _ = ‖f y - f y'‖ := by simp _ = ‖∫ (t : ℝ) in y'..y, f' t‖ := by congr 1 rw [intervalIntegral.integral_eq_sub_of_hasDerivAt] . exact f_deriv . apply f'_cont.intervalIntegrable _ = ‖∫ (t : ℝ) in Ι y' y, f' t‖ := by apply intervalIntegral.norm_intervalIntegral_eq _ ≤ ∫ (t : ℝ) in Ι y' y, ‖f' t‖ := by apply MeasureTheory.norm_integral_le_integral_norm _ ≤ ∫ (t : ℝ) in Ι y' y, 3 / ((y / 2) / 2) ^ 2 := by apply MeasureTheory.setIntegral_mono_on . apply f'_cont.norm.integrableOn_uIcc.mono_set apply Set.Ioc_subset_Icc_self . apply MeasureTheory.integrableOn_const.mpr right rw [Real.volume_uIoc] apply ENNReal.ofReal_lt_top . apply measurableSet_uIoc . 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] _ = (MeasureTheory.volume (Ι y' y)).toReal * (3 / ((y / 2) / 2) ^ 2) := by apply MeasureTheory.setIntegral_const _ = \|y - y'\| * (3 / ((y / 2) / 2) ^ 2) := by congr rw [Real.volume_uIoc, ENNReal.toReal_ofReal (abs_nonneg (y - y'))] _ = (3 * (2 * 2) ^ 2) * (1 / y) * (\|y - y'\| / y) := by ring _ ≤ 2 ^ 6 * (1 / y) * (\|y - y'\| / y) := by gcongr 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) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 2 ^ 6 * (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) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤ 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]	rw [Set.mem_uIcc] at ht	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ d t ≠ 0	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ⊢ d t ≠ 0	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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ d t ≠ 0 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)	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ⊢ d t ≠ 0	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ d t ≠ 0	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ⊢ d t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	rw [ddef]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ d t ≠ 0	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ (fun t => 1 - (-(Complex.I * ↑t)).exp) t ≠ 0	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ d t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	simp	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ (fun t => 1 - (-(Complex.I * ↑t)).exp) t ≠ 0	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ¬1 - (-(Complex.I * ↑t)).exp = 0	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ (fun t => 1 - (-(Complex.I * ↑t)).exp) t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	rw [←norm_eq_zero]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ¬1 - (-(Complex.I * ↑t)).exp = 0	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ¬‖1 - (-(Complex.I * ↑t)).exp‖ = 0	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ¬1 - (-(Complex.I * ↑t)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply ne_of_gt	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ¬‖1 - (-(Complex.I * ↑t)).exp‖ = 0	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < ‖1 - (-(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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ¬‖1 - (-(Complex.I * ↑t)).exp‖ = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	calc ‖1 - Complex.exp (-(Complex.I * ↑t))‖ _ ≥ \|(1 - Complex.exp (-(Complex.I * ↑t))).im\| := by simp only [Complex.norm_eq_abs, ge_iff_le] apply Complex.abs_im_le_abs _ = Real.sin t := by simp rw [Complex.exp_im] simp apply Real.sin_nonneg_of_nonneg_of_le_pi . linarith . linarith [Real.two_le_pi] _ > 0 := by apply Real.sin_pos_of_pos_of_lt_pi . linarith . linarith [Real.two_le_pi]	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < ‖1 - (-(Complex.I * ↑t)).exp‖	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < ‖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 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 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 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 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]	simp only [Complex.norm_eq_abs, ge_iff_le]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖1 - (-(Complex.I * ↑t)).exp‖ ≥ \|(1 - (-(Complex.I * ↑t)).exp).im\|	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(1 - (-(Complex.I * ↑t)).exp).im\| ≤ Complex.abs (1 - (-(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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ ‖1 - (-(Complex.I * ↑t)).exp‖ ≥ \|(1 - (-(Complex.I * ↑t)).exp).im\| 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_im_le_abs	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(1 - (-(Complex.I * ↑t)).exp).im\| ≤ Complex.abs (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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(1 - (-(Complex.I * ↑t)).exp).im\| ≤ 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	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(1 - (-(Complex.I * ↑t)).exp).im\| = t.sin	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(-(Complex.I * ↑t)).exp.im\| = t.sin	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(1 - (-(Complex.I * ↑t)).exp).im\| = t.sin TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	rw [Complex.exp_im]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(-(Complex.I * ↑t)).exp.im\| = t.sin	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(-(Complex.I * ↑t)).re.exp * (-(Complex.I * ↑t)).im.sin\| = t.sin	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(-(Complex.I * ↑t)).exp.im\| = t.sin TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	simp	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(-(Complex.I * ↑t)).re.exp * (-(Complex.I * ↑t)).im.sin\| = t.sin	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t.sin	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ \|(-(Complex.I * ↑t)).re.exp * (-(Complex.I * ↑t)).im.sin\| = t.sin TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply Real.sin_nonneg_of_nonneg_of_le_pi	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t.sin	case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t.sin TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. linarith	case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi	case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi	Please generate a tactic in lean4 to solve the state. STATE: case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. linarith [Real.two_le_pi]	case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	linarith	case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 ≤ t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	linarith [Real.two_le_pi]	case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t ≤ Real.pi TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply Real.sin_pos_of_pos_of_lt_pi	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t.sin > 0	case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < t case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi	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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t.sin > 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. linarith	case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < t case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi	case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi	Please generate a tactic in lean4 to solve the state. STATE: case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < t case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. linarith [Real.two_le_pi]	case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	linarith	case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h0x y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ 0 < t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	linarith [Real.two_le_pi]	case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hxp y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : y' ≤ t ∧ t ≤ y ∨ y ≤ t ∧ t ≤ y' ht' : 0 < t ∧ t ≤ 1 ⊢ t < Real.pi 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	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ HasDerivAt f (f' t) t	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 ⊢ ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	have : f = fun t ↦ c t / d t := by simp	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ HasDerivAt f (f' t) t	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ HasDerivAt f (f' t) t	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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ HasDerivAt f (f' t) 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 [this]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ HasDerivAt f (f' t) t	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ HasDerivAt (fun t => c t / d t) (f' t) t	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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ HasDerivAt f (f' t) t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	have : f' = fun t ↦ ((c' t * d t - c t * d' t) / d t ^ 2) := by ext t rw [f'def, cdef, c'def, ddef, d'def] simp congr 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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ HasDerivAt (fun t => c t / d t) (f' t) t	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun t => c t / d t) (f' t) t	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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ HasDerivAt (fun t => c t / d t) (f' t) 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 [this]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun t => c t / d t) (f' t) t	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun t => c t / d t) ((fun t => (c' t * d t - c t * d' t) / d t ^ 2) t) t	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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun t => c t / d t) (f' t) 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 HasDerivAt.div	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun t => c t / d t) ((fun t => (c' t * d t - c t * d' t) / d t ^ 2) t) t	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => c y) (c' t) t case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun t => c t / d t) ((fun t => (c' t * d t - c t * d' t) / d t ^ 2) t) 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 [cdef, c'def] simp apply HasDerivAt.const_sub apply HasDerivAt.ofReal_comp apply hasDerivAt_id'	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => c y) (c' t) t case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => c y) (c' t) t case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. rw [ddef, d'def] simp rw [←neg_neg (Complex.I * Complex.exp (-(Complex.I * ↑t)))] apply HasDerivAt.const_sub rw [←neg_mul, mul_comm] apply HasDerivAt.cexp apply HasDerivAt.neg conv in fun (x : ℝ) ↦ Complex.I * (x : ℝ) => ext rw [mul_comm] set e : ℂ → ℂ := fun t ↦ t * Complex.I with edef have : (fun (t : ℝ) ↦ t * Complex.I) = fun (t : ℝ) ↦ e t := by rw [edef] rw [this] apply HasDerivAt.comp_ofReal rw [edef] apply hasDerivAt_mul_const	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. exact d_nonzero ht	case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	simp	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ f = fun t => c t / d t	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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ f = fun t => c t / d t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	ext t	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ f' = fun t => (c' t * d t - c t * d' t) / d t ^ 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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ f' t = (c' t * d t - c t * d' t) / d t ^ 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 t : ℝ ht : t ∈ Set.uIcc y' y this : f = fun t => c t / d t ⊢ f' = fun t => (c' t * d t - c t * d' t) / d t ^ 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, cdef, c'def, ddef, d'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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ f' t = (c' t * d t - c t * d' t) / d t ^ 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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ (fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2) t = ((fun t => -1) t * (fun t => 1 - (-(Complex.I * ↑t)).exp) t - (fun t => 1 - ↑t) t * (fun t => Complex.I * (-(Complex.I * ↑t)).exp) t) / (fun t => 1 - (-(Complex.I * ↑t)).exp) t ^ 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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ f' t = (c' t * d t - c t * d' t) / d t ^ 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	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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ (fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2) t = ((fun t => -1) t * (fun t => 1 - (-(Complex.I * ↑t)).exp) t - (fun t => 1 - ↑t) t * (fun t => Complex.I * (-(Complex.I * ↑t)).exp) t) / (fun t => 1 - (-(Complex.I * ↑t)).exp) t ^ 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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 = ((-(Complex.I * ↑t)).exp - 1 - (1 - ↑t) * (Complex.I * (-(Complex.I * ↑t)).exp)) / (1 - (-(Complex.I * ↑t)).exp) ^ 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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ (fun t => (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2) t = ((fun t => -1) t * (fun t => 1 - (-(Complex.I * ↑t)).exp) t - (fun t => 1 - ↑t) t * (fun t => Complex.I * (-(Complex.I * ↑t)).exp) t) / (fun t => 1 - (-(Complex.I * ↑t)).exp) t ^ 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	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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 = ((-(Complex.I * ↑t)).exp - 1 - (1 - ↑t) * (Complex.I * (-(Complex.I * ↑t)).exp)) / (1 - (-(Complex.I * ↑t)).exp) ^ 2	case h.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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp = (-(Complex.I * ↑t)).exp - 1 - (1 - ↑t) * (Complex.I * (-(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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ (-1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 = ((-(Complex.I * ↑t)).exp - 1 - (1 - ↑t) * (Complex.I * (-(Complex.I * ↑t)).exp)) / (1 - (-(Complex.I * ↑t)).exp) ^ 2 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	ring	case h.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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp = (-(Complex.I * ↑t)).exp - 1 - (1 - ↑t) * (Complex.I * (-(Complex.I * ↑t)).exp)	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h.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 t✝ : ℝ ht : t✝ ∈ Set.uIcc y' y this : f = fun t => c t / d t t : ℝ ⊢ -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp = (-(Complex.I * ↑t)).exp - 1 - (1 - ↑t) * (Complex.I * (-(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 [cdef, c'def]	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => c y) (c' t) t	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => (fun t => 1 - ↑t) y) ((fun t => -1) t) t	Please generate a tactic in lean4 to solve the state. STATE: case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => c y) (c' t) 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	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => (fun t => 1 - ↑t) y) ((fun t => -1) t) t	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - ↑y) (-1) t	Please generate a tactic in lean4 to solve the state. STATE: case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => (fun t => 1 - ↑t) y) ((fun t => -1) t) 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 HasDerivAt.const_sub	case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - ↑y) (-1) t	case hc.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => ↑x) 1 t	Please generate a tactic in lean4 to solve the state. STATE: case hc y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - ↑y) (-1) 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 HasDerivAt.ofReal_comp	case hc.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => ↑x) 1 t	case hc.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => y) 1 t	Please generate a tactic in lean4 to solve the state. STATE: case hc.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => ↑x) 1 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 hasDerivAt_id'	case hc.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => y) 1 t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hc.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => y) 1 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 [ddef, d'def]	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) t	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => (fun t => 1 - (-(Complex.I * ↑t)).exp) y) ((fun t => Complex.I * (-(Complex.I * ↑t)).exp) t) t	Please generate a tactic in lean4 to solve the state. STATE: case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => d y) (d' t) 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	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => (fun t => 1 - (-(Complex.I * ↑t)).exp) y) ((fun t => Complex.I * (-(Complex.I * ↑t)).exp) t) t	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - (-(Complex.I * ↑y)).exp) (Complex.I * (-(Complex.I * ↑t)).exp) t	Please generate a tactic in lean4 to solve the state. STATE: case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => (fun t => 1 - (-(Complex.I * ↑t)).exp) y) ((fun t => Complex.I * (-(Complex.I * ↑t)).exp) t) 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 [←neg_neg (Complex.I * Complex.exp (-(Complex.I * ↑t)))]	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - (-(Complex.I * ↑y)).exp) (Complex.I * (-(Complex.I * ↑t)).exp) t	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - (-(Complex.I * ↑y)).exp) (- -(Complex.I * (-(Complex.I * ↑t)).exp)) t	Please generate a tactic in lean4 to solve the state. STATE: case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - (-(Complex.I * ↑y)).exp) (Complex.I * (-(Complex.I * ↑t)).exp) 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 HasDerivAt.const_sub	case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - (-(Complex.I * ↑y)).exp) (- -(Complex.I * (-(Complex.I * ↑t)).exp)) t	case hd.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => (-(Complex.I * ↑x)).exp) (-(Complex.I * (-(Complex.I * ↑t)).exp)) t	Please generate a tactic in lean4 to solve the state. STATE: case hd y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun y => 1 - (-(Complex.I * ↑y)).exp) (- -(Complex.I * (-(Complex.I * ↑t)).exp)) 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 [←neg_mul, mul_comm]	case hd.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => (-(Complex.I * ↑x)).exp) (-(Complex.I * (-(Complex.I * ↑t)).exp)) t	case hd.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => (-(Complex.I * ↑x)).exp) ((-(Complex.I * ↑t)).exp * -Complex.I) t	Please generate a tactic in lean4 to solve the state. STATE: case hd.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => (-(Complex.I * ↑x)).exp) (-(Complex.I * (-(Complex.I * ↑t)).exp)) 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 HasDerivAt.cexp	case hd.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => (-(Complex.I * ↑x)).exp) ((-(Complex.I * ↑t)).exp * -Complex.I) t	case hd.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => -(Complex.I * ↑x)) (-Complex.I) t	Please generate a tactic in lean4 to solve the state. STATE: case hd.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => (-(Complex.I * ↑x)).exp) ((-(Complex.I * ↑t)).exp * -Complex.I) 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 HasDerivAt.neg	case hd.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => -(Complex.I * ↑x)) (-Complex.I) t	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => Complex.I * ↑x) Complex.I t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => -(Complex.I * ↑x)) (-Complex.I) t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	conv in fun (x : ℝ) ↦ Complex.I * (x : ℝ) => ext rw [mul_comm]	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => Complex.I * ↑x) Complex.I t	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => Complex.I * ↑x) Complex.I t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	set e : ℂ → ℂ := fun t ↦ t * Complex.I with edef	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	have : (fun (t : ℝ) ↦ t * Complex.I) = fun (t : ℝ) ↦ e t := by rw [edef]	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I 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 [this]	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I t	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun t => e ↑t) Complex.I t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun x => ↑x * Complex.I) Complex.I 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 HasDerivAt.comp_ofReal	case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun t => e ↑t) Complex.I t	case hd.hf.hf.h.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt e Complex.I ↑t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun t => e ↑t) Complex.I 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 [edef]	case hd.hf.hf.h.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt e Complex.I ↑t	case hd.hf.hf.h.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun t => t * Complex.I) Complex.I ↑t	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.h.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt e Complex.I ↑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 hasDerivAt_mul_const	case hd.hf.hf.h.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun t => t * Complex.I) Complex.I ↑t	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hd.hf.hf.h.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 t : ℝ ht : t ∈ Set.uIcc y' y this✝¹ : f = fun t => c t / d t this✝ : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I this : (fun t => ↑t * Complex.I) = fun t => e ↑t ⊢ HasDerivAt (fun t => t * Complex.I) Complex.I ↑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 [edef]	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I ⊢ (fun t => ↑t * Complex.I) = fun t => e ↑t	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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 e : ℂ → ℂ := fun t => t * Complex.I edef : e = fun t => t * Complex.I ⊢ (fun t => ↑t * Complex.I) = fun t => e ↑t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	exact d_nonzero ht	case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hx y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y this✝ : f = fun t => c t / d t this : f' = fun t => (c' t * d t - c t * d' t) / d t ^ 2 ⊢ d t ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply ContinuousOn.div	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ContinuousOn (fun t => f' t) (Set.uIcc y' y)	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 ⊢ ContinuousOn (fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) (Set.uIcc y' y) 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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc y' y) case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	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 ⊢ ContinuousOn (fun t => f' t) (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 Continuous.continuousOn continuity	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 ⊢ ContinuousOn (fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) (Set.uIcc y' y) 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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc y' y) case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc y' y) case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	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 ⊢ ContinuousOn (fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) (Set.uIcc y' y) 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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc y' y) case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. apply Continuous.continuousOn apply Continuous.pow apply Continuous.sub . apply continuous_const . apply Continuous.cexp apply Continuous.neg apply Continuous.mul . apply continuous_const . apply Complex.continuous_ofReal	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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc y' y) case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc y' y) case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0 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 simp apply d_nonzero ht	case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply Continuous.continuousOn	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 ⊢ ContinuousOn (fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) (Set.uIcc y' y)	case hf.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 ⊢ Continuous fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp	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 ⊢ ContinuousOn (fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp) (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]	continuity	case hf.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 ⊢ Continuous fun t => -1 + (-(Complex.I * ↑t)).exp + Complex.I * (↑t - 1) * (-(Complex.I * ↑t)).exp	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hf.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 ⊢ Continuous fun t => -1 + (-(Complex.I * ↑t)).exp + 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 Continuous.continuousOn	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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (Set.uIcc 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 ⊢ Continuous fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 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 ⊢ ContinuousOn (fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2) (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 Continuous.pow	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 ⊢ Continuous fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 2	case hg.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 ⊢ Continuous fun b => 1 - (-(Complex.I * ↑b)).exp	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 ⊢ Continuous fun t => (1 - (-(Complex.I * ↑t)).exp) ^ 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 Continuous.sub	case hg.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 ⊢ Continuous fun b => 1 - (-(Complex.I * ↑b)).exp	case hg.h.h.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 ⊢ Continuous fun x => 1 case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).exp	Please generate a tactic in lean4 to solve the state. STATE: case hg.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 ⊢ Continuous fun b => 1 - (-(Complex.I * ↑b)).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 continuous_const	case hg.h.h.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 ⊢ Continuous fun x => 1 case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).exp	case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).exp	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.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 ⊢ Continuous fun x => 1 case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).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 Continuous.cexp apply Continuous.neg apply Continuous.mul . apply continuous_const . apply Complex.continuous_ofReal	case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).exp	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).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 continuous_const	case hg.h.h.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 ⊢ Continuous fun x => 1	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.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 ⊢ Continuous fun x => 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 Continuous.cexp	case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).exp	case hg.h.h.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 ⊢ Continuous fun y => -(Complex.I * ↑y)	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.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 ⊢ Continuous fun x => (-(Complex.I * ↑x)).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 Continuous.neg	case hg.h.h.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 ⊢ Continuous fun y => -(Complex.I * ↑y)	case hg.h.h.hg.h.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 ⊢ Continuous fun x => Complex.I * ↑x	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.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 ⊢ Continuous fun y => -(Complex.I * ↑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 Continuous.mul	case hg.h.h.hg.h.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 ⊢ Continuous fun x => Complex.I * ↑x	case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => Complex.I case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.hg.h.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 ⊢ Continuous fun x => Complex.I * ↑x TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. apply continuous_const	case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => Complex.I case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x	case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => Complex.I case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x 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.continuous_ofReal	case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply continuous_const	case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => Complex.I	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => Complex.I 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.continuous_ofReal	case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case hg.h.h.hg.h.hf.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 ⊢ Continuous fun x => ↑x 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 h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0	case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≠ 0	Please generate a tactic in lean4 to solve the state. STATE: case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ⊢ ∀ x ∈ Set.uIcc y' y, (1 - (-(Complex.I * ↑x)).exp) ^ 2 ≠ 0 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 h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≠ 0	case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ ¬1 - (-(Complex.I * ↑t)).exp = 0	Please generate a tactic in lean4 to solve the state. STATE: case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ (1 - (-(Complex.I * ↑t)).exp) ^ 2 ≠ 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	apply d_nonzero ht	case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ ¬1 - (-(Complex.I * ↑t)).exp = 0	no goals	Please generate a tactic in lean4 to solve the state. STATE: case h₀ y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 t : ℝ ht : t ∈ Set.uIcc y' y ⊢ ¬1 - (-(Complex.I * ↑t)).exp = 0 TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	simp	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ = ‖f y - f 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) ⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ = ‖f y - f y'‖ TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	congr 1	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ ‖f y - f y'‖ = ‖∫ (t : ℝ) in y'..y, f' t‖	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) ⊢ f y - f y' = ∫ (t : ℝ) in y'..y, f' t	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) ⊢ ‖f y - f y'‖ = ‖∫ (t : ℝ) in y'..y, f' 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 [intervalIntegral.integral_eq_sub_of_hasDerivAt]	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) ⊢ f y - f y' = ∫ (t : ℝ) in y'..y, f' t	case e_a.hderiv y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ∈ Set.uIcc y' y, HasDerivAt f (f' x) x case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume y' y	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) ⊢ f y - f y' = ∫ (t : ℝ) in y'..y, f' t TACTIC:
https://github.com/fpvandoorn/carleson.git	6d448ddfa1ff78506367ab09a8caac5351011ead	Carleson/Theorem1_1/Hilbert_kernel.lean	Hilbert_kernel_regularity_main_part	[100, 1]	[259, 15]	. exact f_deriv	case e_a.hderiv y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ∈ Set.uIcc y' y, HasDerivAt f (f' x) x case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume y' y	case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume y' y	Please generate a tactic in lean4 to solve the state. STATE: case e_a.hderiv y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ∈ Set.uIcc y' y, HasDerivAt f (f' x) x case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume 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 f'_cont.intervalIntegrable	case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume y' y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume 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]	exact f_deriv	case e_a.hderiv y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ∈ Set.uIcc y' y, HasDerivAt f (f' x) x	no goals	Please generate a tactic in lean4 to solve the state. STATE: case e_a.hderiv y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 ∈ Set.uIcc y' y, HasDerivAt f (f' x) x 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.intervalIntegrable	case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume y' y	no goals	Please generate a tactic in lean4 to solve the state. STATE: case e_a.hint y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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) ⊢ IntervalIntegrable (fun t => f' t) MeasureTheory.volume 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 intervalIntegral.norm_intervalIntegral_eq	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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 y'..y, f' t‖ = ‖∫ (t : ℝ) in Ι y' y, f' t‖	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 y'..y, f' t‖ = ‖∫ (t : ℝ) in Ι y' y, f' 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.norm_integral_le_integral_norm	y y' : ℝ yy'nonneg : 0 ≤ y ∧ 0 ≤ y' ypos : 0 < y y2ley' : y / 2 ≤ y' hy : y ≤ 1 hy' : y' ≤ 1 f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp) f' : ℝ → 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, ‖f' t‖	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, f' t‖ ≤ ∫ (t : ℝ) in Ι y' y, ‖f' t‖ TACTIC:

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