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/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | calc (2 / Real.pi) * η
_ ≤ 0 := mul_nonpos_of_nonneg_of_nonpos (div_nonneg zero_le_two Real.pi_pos.le) ηpos
_ ≤ ‖1 - Complex.exp (Complex.I * x)‖ := by apply norm_nonneg | case pos
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply norm_nonneg | η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 0 ≤ ‖1 - (Complex.I * ↑x).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : η ≤ 0
⊢ 0 ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | convert (@this (-x) _ (by simpa) (by linarith)) using 1 | case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case h.e'_4
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(-x)).exp‖
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . rw [Complex.norm_eq_abs, ←Complex.abs_conj, map_sub, map_one, Complex.ofReal_neg, mul_neg, Complex.norm_eq_abs,
←Complex.exp_conj, map_mul, Complex.conj_I, neg_mul, Complex.conj_ofReal] | case h.e'_4
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(-x)).exp‖
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)| | case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)| | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(-x)).exp‖
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . rwa [abs_neg] | case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simpa | η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ |(-x)| ≤ 2 * Real.pi - η | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ |(-x)| ≤ 2 * Real.pi - η
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | linarith | η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 0 ≤ -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ 0 ≤ -x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [Complex.norm_eq_abs, ←Complex.abs_conj, map_sub, map_one, Complex.ofReal_neg, mul_neg, Complex.norm_eq_abs,
←Complex.exp_conj, map_mul, Complex.conj_I, neg_mul, Complex.conj_ofReal] | case h.e'_4
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(-x)).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(-x)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rwa [abs_neg] | case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
η x : ℝ
le_abs_x : η ≤ |x|
abs_x_le : |x| ≤ 2 * Real.pi - η
ηpos : 0 < η
this : ∀ {x : ℝ}, η ≤ |x| → |x| ≤ 2 * Real.pi - η → 0 ≤ x → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_nonneg : ¬0 ≤ x
⊢ η ≤ |(-x)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | convert (@this (2 * Real.pi - x) _ _ _ _) using 1 | case inr
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖
case inr.convert_1
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ η ≤ 2 * Real.pi - x
case inr.convert_2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ 2 * Real.pi - η
case inr.convert_3
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 0 ≤ 2 * Real.pi - x
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . rw [Complex.norm_eq_abs, ←Complex.abs_conj]
simp
rw [←Complex.exp_conj]
simp
rw [mul_sub, Complex.conj_ofReal, Complex.exp_sub, mul_comm Complex.I (2 * Real.pi), Complex.exp_two_pi_mul_I, ←inv_eq_one_div, ←Complex.exp_neg] | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖
case inr.convert_1
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ η ≤ 2 * Real.pi - x
case inr.convert_2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ 2 * Real.pi - η
case inr.convert_3
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 0 ≤ 2 * Real.pi - x
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi | case inr.convert_1
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ η ≤ 2 * Real.pi - x
case inr.convert_2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ 2 * Real.pi - η
case inr.convert_3
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 0 ≤ 2 * Real.pi - x
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖
case inr.convert_1
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ η ≤ 2 * Real.pi - x
case inr.convert_2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ 2 * Real.pi - η
case inr.convert_3
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 0 ≤ 2 * Real.pi - x
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | all_goals linarith | case inr.convert_1
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ η ≤ 2 * Real.pi - x
case inr.convert_2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ 2 * Real.pi - η
case inr.convert_3
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 0 ≤ 2 * Real.pi - x
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.convert_1
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ η ≤ 2 * Real.pi - x
case inr.convert_2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ 2 * Real.pi - η
case inr.convert_3
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 0 ≤ 2 * Real.pi - x
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [Complex.norm_eq_abs, ←Complex.abs_conj] | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖ | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs ((starRingEnd ℂ) (1 - (Complex.I * ↑x).exp)) = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ ‖1 - (Complex.I * ↑x).exp‖ = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs ((starRingEnd ℂ) (1 - (Complex.I * ↑x).exp)) = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖ | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - (starRingEnd ℂ) (Complex.I * ↑x).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs ((starRingEnd ℂ) (1 - (Complex.I * ↑x).exp)) = ‖1 - (Complex.I * ↑(2 * Real.pi - x)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [←Complex.exp_conj] | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - (starRingEnd ℂ) (Complex.I * ↑x).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp) | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - ((starRingEnd ℂ) (Complex.I * ↑x)).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - (starRingEnd ℂ) (Complex.I * ↑x).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - ((starRingEnd ℂ) (Complex.I * ↑x)).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp) | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - (-(Complex.I * (starRingEnd ℂ) ↑x)).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - ((starRingEnd ℂ) (Complex.I * ↑x)).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [mul_sub, Complex.conj_ofReal, Complex.exp_sub, mul_comm Complex.I (2 * Real.pi), Complex.exp_two_pi_mul_I, ←inv_eq_one_div, ←Complex.exp_neg] | case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - (-(Complex.I * (starRingEnd ℂ) ↑x)).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ Complex.abs (1 - (-(Complex.I * (starRingEnd ℂ) ↑x)).exp) = Complex.abs (1 - (Complex.I * (2 * ↑Real.pi - ↑x)).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | linarith | case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.convert_4
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
this : ∀ {x : ℝ}, η ≤ x → x ≤ 2 * Real.pi - η → 0 ≤ x → x ≤ Real.pi → 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
x_le_pi : ¬x ≤ Real.pi
⊢ 2 * Real.pi - x ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | calc (2 / Real.pi) * η
_ ≤ (2 / Real.pi) * x := by gcongr
_ = (1 - (2 / Real.pi) * x) * Real.sin 0 + ((2 / Real.pi) * x) * Real.sin (Real.pi / 2) := by simp
_ ≤ Real.sin ((1 - (2 / Real.pi) * x) * 0 + ((2 / Real.pi) * x) * (Real.pi / 2)) := by
apply (strictConcaveOn_sin_Icc.concaveOn).2 (by simp [Real.pi_nonneg])
. simp
constructor <;> linarith [Real.pi_nonneg]
. rw [sub_nonneg, mul_comm]
apply mul_le_of_nonneg_of_le_div (by norm_num) (div_nonneg (by norm_num) Real.pi_nonneg) (by simpa)
. exact mul_nonneg (div_nonneg (by norm_num) Real.pi_nonneg) x_nonneg
. simp
_ = Real.sin x := by
congr
field_simp
_ ≤ Real.sqrt ((Real.sin x) ^ 2) := by
rw [Real.sqrt_sq_eq_abs]
apply le_abs_self
_ ≤ ‖1 - Complex.exp (Complex.I * ↑x)‖ := by
rw [mul_comm, Complex.exp_mul_I, Complex.norm_eq_abs, Complex.abs_eq_sqrt_sq_add_sq]
simp
rw [Complex.cos_ofReal_re, Complex.sin_ofReal_re]
apply (Real.sqrt_le_sqrt_iff _).mpr
. simp [pow_two_nonneg]
. linarith [pow_two_nonneg (1 - Real.cos x), pow_two_nonneg (Real.sin x)] | case pos
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | gcongr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ 2 / Real.pi * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ 2 / Real.pi * x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * x = (1 - 2 / Real.pi * x) * Real.sin 0 + 2 / Real.pi * x * (Real.pi / 2).sin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 2 / Real.pi * x = (1 - 2 / Real.pi * x) * Real.sin 0 + 2 / Real.pi * x * (Real.pi / 2).sin
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply (strictConcaveOn_sin_Icc.concaveOn).2 (by simp [Real.pi_nonneg]) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ (1 - 2 / Real.pi * x) * Real.sin 0 + 2 / Real.pi * x * (Real.pi / 2).sin ≤
((1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2)).sin | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ Real.pi / 2 ∈ Set.Icc 0 Real.pi
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ (1 - 2 / Real.pi * x) * Real.sin 0 + 2 / Real.pi * x * (Real.pi / 2).sin ≤
((1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2)).sin
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . simp
constructor <;> linarith [Real.pi_nonneg] | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ Real.pi / 2 ∈ Set.Icc 0 Real.pi
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ Real.pi / 2 ∈ Set.Icc 0 Real.pi
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . rw [sub_nonneg, mul_comm]
apply mul_le_of_nonneg_of_le_div (by norm_num) (div_nonneg (by norm_num) Real.pi_nonneg) (by simpa) | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . exact mul_nonneg (div_nonneg (by norm_num) Real.pi_nonneg) x_nonneg | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . simp | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp [Real.pi_nonneg] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ∈ Set.Icc 0 Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ∈ Set.Icc 0 Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ Real.pi / 2 ∈ Set.Icc 0 Real.pi | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ Real.pi / 2 ∧ 0 ≤ Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ Real.pi / 2 ∈ Set.Icc 0 Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | constructor <;> linarith [Real.pi_nonneg] | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ Real.pi / 2 ∧ 0 ≤ Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ Real.pi / 2 ∧ 0 ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [sub_nonneg, mul_comm] | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x * (2 / Real.pi) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 - 2 / Real.pi * x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply mul_le_of_nonneg_of_le_div (by norm_num) (div_nonneg (by norm_num) Real.pi_nonneg) (by simpa) | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x * (2 / Real.pi) ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x * (2 / Real.pi) ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | norm_num | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | norm_num | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simpa | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x ≤ 1 / (2 / Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x ≤ 1 / (2 / Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | exact mul_nonneg (div_nonneg (by norm_num) Real.pi_nonneg) x_nonneg | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ 2 / Real.pi * x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 1 - 2 / Real.pi * x + 2 / Real.pi * x = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | congr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ ((1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2)).sin = x.sin | case e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ (1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2) = x | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ ((1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2)).sin = x.sin
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | field_simp | case e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ (1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ (1 - 2 / Real.pi * x) * 0 + 2 / Real.pi * x * (Real.pi / 2) = x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [Real.sqrt_sq_eq_abs] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ≤ √(x.sin ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ≤ |x.sin| | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ≤ √(x.sin ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply le_abs_self | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ≤ |x.sin| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ≤ |x.sin|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [mul_comm, Complex.exp_mul_I, Complex.norm_eq_abs, Complex.abs_eq_sqrt_sq_add_sq] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ ‖1 - (Complex.I * ↑x).exp‖ | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - ((↑x).cos + (↑x).sin * Complex.I)).re ^ 2 + (1 - ((↑x).cos + (↑x).sin * Complex.I)).im ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - ((↑x).cos + (↑x).sin * Complex.I)).re ^ 2 + (1 - ((↑x).cos + (↑x).sin * Complex.I)).im ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - (↑x).cos.re) ^ 2 + (↑x).sin.re ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - ((↑x).cos + (↑x).sin * Complex.I)).re ^ 2 + (1 - ((↑x).cos + (↑x).sin * Complex.I)).im ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [Complex.cos_ofReal_re, Complex.sin_ofReal_re] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - (↑x).cos.re) ^ 2 + (↑x).sin.re ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - x.cos) ^ 2 + x.sin ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - (↑x).cos.re) ^ 2 + (↑x).sin.re ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply (Real.sqrt_le_sqrt_iff _).mpr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - x.cos) ^ 2 + x.sin ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ √(x.sin ^ 2) ≤ √((1 - x.cos) ^ 2 + x.sin ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . simp [pow_two_nonneg] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . linarith [pow_two_nonneg (1 - Real.cos x), pow_two_nonneg (Real.sin x)] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp [pow_two_nonneg] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ x.sin ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | linarith [pow_two_nonneg (1 - Real.cos x), pow_two_nonneg (Real.sin x)] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : x ≤ Real.pi / 2
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | push_neg at h | case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : ¬x ≤ Real.pi / 2
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | calc (2 / Real.pi) * η
_ ≤ (2 / Real.pi) * x := by gcongr
_ = 1 - ((1 - (2 / Real.pi) * (x - Real.pi / 2)) * Real.cos (Real.pi / 2) + ((2 / Real.pi) * (x - Real.pi / 2)) * Real.cos (Real.pi)) := by
field_simp
ring
_ ≤ 1 - (Real.cos ((1 - (2 / Real.pi) * (x - Real.pi / 2)) * (Real.pi / 2) + (((2 / Real.pi) * (x - Real.pi / 2)) * (Real.pi)))) := by
gcongr
apply (strictConvexOn_cos_Icc.convexOn).2 (by simp [Real.pi_nonneg])
. simp
constructor <;> linarith [Real.pi_nonneg]
. rw [sub_nonneg, mul_comm]
apply mul_le_of_nonneg_of_le_div (by norm_num) (div_nonneg (by norm_num) Real.pi_nonneg) (by simpa)
. exact mul_nonneg (div_nonneg (by norm_num) Real.pi_nonneg) (by linarith [h])
. simp
_ = 1 - Real.cos x := by
congr
field_simp
ring
_ ≤ Real.sqrt ((1 - Real.cos x) ^ 2) := by
rw [Real.sqrt_sq_eq_abs]
apply le_abs_self
_ ≤ ‖1 - Complex.exp (Complex.I * ↑x)‖ := by
rw [mul_comm, Complex.exp_mul_I, Complex.norm_eq_abs, Complex.abs_eq_sqrt_sq_add_sq]
simp
rw [Complex.cos_ofReal_re, Complex.sin_ofReal_re]
apply (Real.sqrt_le_sqrt_iff _).mpr
. simp [pow_two_nonneg]
. linarith [pow_two_nonneg (1 - Real.cos x), pow_two_nonneg (Real.sin x)] | case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | gcongr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * η ≤ 2 / Real.pi * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * η ≤ 2 / Real.pi * x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | field_simp | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * x =
1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 * x * (Real.pi * 2) = (Real.pi * 2 + 2 * (x * 2 - Real.pi)) * Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 / Real.pi * x =
1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | ring | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 * x * (Real.pi * 2) = (Real.pi * 2 + 2 * (x * 2 - Real.pi)) * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 2 * x * (Real.pi * 2) = (Real.pi * 2 + 2 * (x * 2 - Real.pi)) * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | gcongr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos) ≤
1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos | case h
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos ≤
(1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos) ≤
1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply (strictConvexOn_cos_Icc.convexOn).2 (by simp [Real.pi_nonneg]) | case h
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos ≤
(1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos ≤
(1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2).cos + 2 / Real.pi * (x - Real.pi / 2) * Real.pi.cos
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . simp
constructor <;> linarith [Real.pi_nonneg] | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . rw [sub_nonneg, mul_comm]
apply mul_le_of_nonneg_of_le_div (by norm_num) (div_nonneg (by norm_num) Real.pi_nonneg) (by simpa) | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . exact mul_nonneg (div_nonneg (by norm_num) Real.pi_nonneg) (by linarith [h]) | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . simp | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp [Real.pi_nonneg] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi / 2 ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi / 2 ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2) | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ Real.pi ∧ 0 ≤ Real.pi / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ Real.pi ∈ Set.Icc (Real.pi / 2) (Real.pi + Real.pi / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | constructor <;> linarith [Real.pi_nonneg] | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ Real.pi ∧ 0 ≤ Real.pi / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ Real.pi ∧ 0 ≤ Real.pi / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [sub_nonneg, mul_comm] | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2) | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (x - Real.pi / 2) * (2 / Real.pi) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 - 2 / Real.pi * (x - Real.pi / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply mul_le_of_nonneg_of_le_div (by norm_num) (div_nonneg (by norm_num) Real.pi_nonneg) (by simpa) | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (x - Real.pi / 2) * (2 / Real.pi) ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (x - Real.pi / 2) * (2 / Real.pi) ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | norm_num | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | norm_num | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simpa | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ x - Real.pi / 2 ≤ 1 / (2 / Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ x - Real.pi / 2 ≤ 1 / (2 / Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | exact mul_nonneg (div_nonneg (by norm_num) Real.pi_nonneg) (by linarith [h]) | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ 2 / Real.pi * (x - Real.pi / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | linarith [h] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ x - Real.pi / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ x - Real.pi / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - 2 / Real.pi * (x - Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | congr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos =
1 - x.cos | case e_a.e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi = x | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - ((1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi).cos =
1 - x.cos
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | field_simp | case e_a.e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi = x | case e_a.e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (Real.pi * 2 - 2 * (x * 2 - Real.pi)) * Real.pi * (Real.pi * 2) +
2 * (x * 2 - Real.pi) * Real.pi * (Real.pi * 2 * 2) =
x * (Real.pi * 2 * 2 * (Real.pi * 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - 2 / Real.pi * (x - Real.pi / 2)) * (Real.pi / 2) + 2 / Real.pi * (x - Real.pi / 2) * Real.pi = x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | ring | case e_a.e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (Real.pi * 2 - 2 * (x * 2 - Real.pi)) * Real.pi * (Real.pi * 2) +
2 * (x * 2 - Real.pi) * Real.pi * (Real.pi * 2 * 2) =
x * (Real.pi * 2 * 2 * (Real.pi * 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_x
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (Real.pi * 2 - 2 * (x * 2 - Real.pi)) * Real.pi * (Real.pi * 2) +
2 * (x * 2 - Real.pi) * Real.pi * (Real.pi * 2 * 2) =
x * (Real.pi * 2 * 2 * (Real.pi * 2))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [Real.sqrt_sq_eq_abs] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - x.cos ≤ √((1 - x.cos) ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - x.cos ≤ |1 - x.cos| | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - x.cos ≤ √((1 - x.cos) ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply le_abs_self | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - x.cos ≤ |1 - x.cos| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 1 - x.cos ≤ |1 - x.cos|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [mul_comm, Complex.exp_mul_I, Complex.norm_eq_abs, Complex.abs_eq_sqrt_sq_add_sq] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ ‖1 - (Complex.I * ↑x).exp‖ | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤
√((1 - ((↑x).cos + (↑x).sin * Complex.I)).re ^ 2 + (1 - ((↑x).cos + (↑x).sin * Complex.I)).im ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤
√((1 - ((↑x).cos + (↑x).sin * Complex.I)).re ^ 2 + (1 - ((↑x).cos + (↑x).sin * Complex.I)).im ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ √((1 - (↑x).cos.re) ^ 2 + (↑x).sin.re ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤
√((1 - ((↑x).cos + (↑x).sin * Complex.I)).re ^ 2 + (1 - ((↑x).cos + (↑x).sin * Complex.I)).im ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | rw [Complex.cos_ofReal_re, Complex.sin_ofReal_re] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ √((1 - (↑x).cos.re) ^ 2 + (↑x).sin.re ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ √((1 - x.cos) ^ 2 + x.sin ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ √((1 - (↑x).cos.re) ^ 2 + (↑x).sin.re ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | apply (Real.sqrt_le_sqrt_iff _).mpr | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ √((1 - x.cos) ^ 2 + x.sin ^ 2) | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - x.cos) ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ √((1 - x.cos) ^ 2) ≤ √((1 - x.cos) ^ 2 + x.sin ^ 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . simp [pow_two_nonneg] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - x.cos) ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - x.cos) ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | . linarith [pow_two_nonneg (1 - Real.cos x), pow_two_nonneg (Real.sin x)] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | simp [pow_two_nonneg] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - x.cos) ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ (1 - x.cos) ^ 2 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound' | [111, 1] | [185, 82] | linarith [pow_two_nonneg (1 - Real.cos x), pow_two_nonneg (Real.sin x)] | η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η : ℝ
ηpos : 0 < η
x : ℝ
le_abs_x : η ≤ x
abs_x_le : x ≤ 2 * Real.pi - η
x_nonneg : 0 ≤ x
x_le_pi : x ≤ Real.pi
h : Real.pi / 2 < x
⊢ 0 ≤ (1 - x.cos) ^ 2 + x.sin ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | by_cases ηpos : η < 0 | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | case pos
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : ¬η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | . calc η / 2
_ ≤ 0 := by linarith
_ ≤ ‖1 - Complex.exp (Complex.I * x)‖ := by apply norm_nonneg | case pos
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : ¬η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : ¬η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : ¬η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | push_neg at ηpos | case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : ¬η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : ¬η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | calc η / 2
_ ≤ (2 / Real.pi) * η := by
ring_nf
rw [mul_assoc]
gcongr
field_simp
rw [div_le_div_iff (by norm_num) Real.pi_pos]
linarith [Real.pi_le_four]
_ ≤ ‖1 - Complex.exp (Complex.I * x)‖ := by
apply lower_secant_bound' xAbs
rw [abs_le, neg_sub', sub_neg_eq_add, neg_mul_eq_neg_mul]
exact xIcc | case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | calc η / 2
_ ≤ 0 := by linarith
_ ≤ ‖1 - Complex.exp (Complex.I * x)‖ := by apply norm_nonneg | case pos
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ Complex.abs (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | linarith | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ η / 2 ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | apply norm_nonneg | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ 0 ≤ ‖1 - (Complex.I * ↑x).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : η < 0
⊢ 0 ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | ring_nf | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η / 2 ≤ 2 / Real.pi * η | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η * (1 / 2) ≤ η * Real.pi⁻¹ * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η / 2 ≤ 2 / Real.pi * η
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | rw [mul_assoc] | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η * (1 / 2) ≤ η * Real.pi⁻¹ * 2 | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η * (1 / 2) ≤ η * (Real.pi⁻¹ * 2) | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η * (1 / 2) ≤ η * Real.pi⁻¹ * 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | gcongr | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η * (1 / 2) ≤ η * (Real.pi⁻¹ * 2) | case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 / 2 ≤ Real.pi⁻¹ * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ η * (1 / 2) ≤ η * (Real.pi⁻¹ * 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | field_simp | case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 / 2 ≤ Real.pi⁻¹ * 2 | case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 / 2 ≤ 2 / Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 / 2 ≤ Real.pi⁻¹ * 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | rw [div_le_div_iff (by norm_num) Real.pi_pos] | case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 / 2 ≤ 2 / Real.pi | case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 * Real.pi ≤ 2 * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 / 2 ≤ 2 / Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | linarith [Real.pi_le_four] | case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 * Real.pi ≤ 2 * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 1 * Real.pi ≤ 2 * 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | norm_num | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 0 < 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | apply lower_secant_bound' xAbs | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖ | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ |x| ≤ 2 * Real.pi - η | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ 2 / Real.pi * η ≤ ‖1 - (Complex.I * ↑x).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | rw [abs_le, neg_sub', sub_neg_eq_add, neg_mul_eq_neg_mul] | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ |x| ≤ 2 * Real.pi - η | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ -2 * Real.pi + η ≤ x ∧ x ≤ 2 * Real.pi - η | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ |x| ≤ 2 * Real.pi - η
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Basic.lean | lower_secant_bound | [189, 1] | [207, 15] | exact xIcc | η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ -2 * Real.pi + η ≤ x ∧ x ≤ 2 * Real.pi - η | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
η x : ℝ
xIcc : x ∈ Set.Icc (-2 * Real.pi + η) (2 * Real.pi - η)
xAbs : η ≤ |x|
ηpos : 0 ≤ η
⊢ -2 * Real.pi + η ≤ x ∧ x ≤ 2 * Real.pi - η
TACTIC:
|
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