url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | by_cases y'ge0 : 0 ≤ y' | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x y y' : ℝ
x_e... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . exfalso
rw [abs_of_neg yge0, abs_of_neg] at h <;> linarith | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg
x y y' : ℝ
x_e... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : ¬0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . push_neg at y'ge0
set! y_ := -y with y_def
set! y'_ := -y' with y'_def
have h_ : 2 * |y_ - y'_| ≤ |y_| := by
rw [y_def, y'_def, ← abs_neg]
simpa [neg_add_eq_sub]
have y_y'_nonneg : 0 ≤ y_ ∧ 0 ≤ y'_ := by constructor <;> linarith
have := this h_ y_y'_nonneg
rw [y_def, y'_def] at this
simp only [n... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : ¬0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : ¬0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | exfalso | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_neg yge0, abs_of_neg] at h <;> linarith | case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : 0 ≤ y'
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | push_neg at y'ge0 | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : ¬0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : ¬0 ≤ y'
⊢ ‖k (-y) - k (-y')‖ ≤... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | set! y_ := -y with y_def | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
⊢ ‖k (-y) - k (-y')‖ ≤ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | set! y'_ := -y' with y'_def | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / ... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
⊢ ‖k (-y) - k (-y')‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have h_ : 2 * |y_ - y'_| ≤ |y_| := by
rw [y_def, y'_def, ← abs_neg]
simpa [neg_add_eq_sub] | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
⊢ ‖k (-y) - k (-y')‖ ... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have y_y'_nonneg : 0 ≤ y_ ∧ 0 ≤ y'_ := by constructor <;> linarith | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ≤... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have := this h_ y_y'_nonneg | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ≤... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [y_def, y'_def] at this | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp only [neg_neg, abs_neg, sub_neg_eq_add, neg_add_eq_sub] at this | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [← RCLike.norm_conj, map_sub, ← k_of_neg_eq_conj_k, ← k_of_neg_eq_conj_k, ←abs_neg (y' - y)] at this | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simpa | case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this✝ : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [y_def, y'_def, ← abs_neg] | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
⊢ 2 * |y_ - y'_| ≤ |y_| | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
⊢ 2 * |(-(-y - -y'))| ≤ |(-y)| | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simpa [neg_add_eq_sub] | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
⊢ 2 * |(-(-y - -y'))| ≤ |(-y)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | constructor <;> linarith | x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_ : ℝ := -y'
y'_def : y'_ = -y'
h_ : 2 * |y_ - y'_| ≤ |y_|
⊢ 0... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y y' : ℝ
x_eq_zero : x = 0
h : 2 * |y - y'| ≤ |y|
this : ∀ {y y' : ℝ}, 2 * |y - y'| ≤ |y| → 0 ≤ y ∧ 0 ≤ y' → ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
yy'nonneg : ¬(0 ≤ y ∧ 0 ≤ y')
yge0 : y < 0
y'ge0 : y' < 0
y_ : ℝ := -y
y_def : y_ = -y
y'_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have y_eq_zero : y = 0 := by
exact le_antisymm ypos yy'nonneg.1 | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | have y'_eq_zero : y' = 0 := by
rw [y_eq_zero] at h
simp at h
rw [abs_of_nonneg yy'nonneg.2] at h
linarith | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
y'_eq_zero : y' = 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [y_eq_zero, y'_eq_zero] | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
y'_eq_zero : y' = 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
y'_eq_zero : y' = 0
⊢ ‖k (-0) - k (-0)‖ ≤ 2 ^ 8 * (1 / |0|) * (|0 - 0| / |0|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
y'_eq_zero : y' = 0
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp | case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
y'_eq_zero : y' = 0
⊢ ‖k (-0) - k (-0)‖ ≤ 2 ^ 8 * (1 / |0|) * (|0 - 0| / |0|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
y'_eq_zero : y' = 0
⊢ ‖k (-0) - k (-0)‖ ≤ 2 ^ 8 * (1 / |0|) * (|0 - 0| / |0|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | exact le_antisymm ypos yy'nonneg.1 | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ y = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
⊢ y = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [y_eq_zero] at h | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ y' = 0 | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |0 - y'| ≤ |0|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ y' = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ y' = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp at h | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |0 - y'| ≤ |0|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ y' = 0 | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
h : 2 * |y'| ≤ 0
⊢ y' = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |0 - y'| ≤ |0|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
⊢ y' = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg yy'nonneg.2] at h | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
h : 2 * |y'| ≤ 0
⊢ y' = 0 | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
h : 2 * y' ≤ 0
⊢ y' = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
h : 2 * |y'| ≤ 0
⊢ y' = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | linarith | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
h : 2 * y' ≤ 0
⊢ y' = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : y ≤ 0
y_eq_zero : y = 0
h : 2 * y' ≤ 0
⊢ y' = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [div_le_iff two_pos] | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y / 2 ≤ y' | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y ≤ y' * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y / 2 ≤ y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | calc y
_ = 2 * (y - y') - y + 2 * y' := by ring
_ ≤ 2 * |y - y'| - y + 2 * y' := by
gcongr
apply le_abs_self
_ ≤ y - y + 2 * y' := by
gcongr
rw [abs_eq_self.mpr yy'nonneg.1] at h
exact h
_ = y' * 2 := by ring | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y ≤ y' * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y ≤ y' * 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | ring | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y = 2 * (y - y') - y + 2 * y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y = 2 * (y - y') - y + 2 * y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * (y - y') - y + 2 * y' ≤ 2 * |y - y'| - y + 2 * y' | case bc.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y - y' ≤ |y - y'| | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * (y - y') - y + 2 * y' ≤ 2 * |y - y'| - y + 2 * y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply le_abs_self | case bc.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y - y' ≤ |y - y'| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y - y' ≤ |y - y'|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| - y + 2 * y' ≤ y - y + 2 * y' | case bc.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| - y + 2 * y' ≤ y - y + 2 * y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_eq_self.mpr yy'nonneg.1] at h | case bc.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| ≤ y | case bc.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ y
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| ≤ y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | exact h | case bc.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ y
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ y
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ 2 * |y - y'| ≤ y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | ring | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y - y + 2 * y' = y' * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
⊢ y - y + 2 * y' = y' * 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply le_trans (Hilbert_kernel_regularity_main_part yy'nonneg ypos y2ley' hy hy') | case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr <;> norm_num | case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [@k_of_one_le_abs (-y')] | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . calc ‖k (-y) - 0‖
_ = ‖k (-y) - k (-1)‖ := by
congr
apply (k_of_one_le_abs _).symm
simp
_ ≤ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|) := by
apply Hilbert_kernel_regularity_main_part
constructor
all_goals linarith
_ ≤ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) := by
gcongr 2 ^... | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0... | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-y')| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [abs_neg, abs_of_nonneg] <;> linarith | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-y')| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-y')|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | calc ‖k (-y) - 0‖
_ = ‖k (-y) - k (-1)‖ := by
congr
apply (k_of_one_le_abs _).symm
simp
_ ≤ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|) := by
apply Hilbert_kernel_regularity_main_part
constructor
all_goals linarith
_ ≤ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) := by
gcongr 2 ^ 6 * (1 / |y|) * (?_ /... | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | congr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ = ‖k (-y) - k (-1)‖ | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 = k (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - 0‖ = ‖k (-y) - k (-1)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply (k_of_one_le_abs _).symm | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 = k (-1) | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-1)| | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 = k (-1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-1)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-1)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply Hilbert_kernel_regularity_main_part | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-1)‖ ≤ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|) | case yy'nonneg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 ≤ y ∧ 0 ≤ 1
case ypos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > ... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ ‖k (-y) - k (-1)‖ ≤ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | constructor | case yy'nonneg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 ≤ y ∧ 0 ≤ 1
case ypos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > ... | case yy'nonneg.left
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 ≤ y
case yy'nonneg.right
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy'... | Please generate a tactic in lean4 to solve the state.
STATE:
case yy'nonneg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 ≤ y ∧ 0 ≤ 1
case ypos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | all_goals linarith | case yy'nonneg.left
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 ≤ y
case yy'nonneg.right
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy'... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case yy'nonneg.left
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 0 ≤ y
case yy'nonneg.right
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | linarith | case hy'
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy'
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr 2 ^ 6 * (1 / |y|) * (?_ / |y|) | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|) ≤ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) | case h.hab
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ |y - 1| ≤ |y - y'| | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - 1| / |y|) ≤ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_sub_comm, abs_of_nonneg, abs_sub_comm, abs_of_nonneg] <;> linarith | case h.hab
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ |y - 1| ≤ |y - y'| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hab
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ |y - 1| ≤ |y - y'|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr <;> norm_num | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_neg, abs_of_nonneg] <;> linarith | case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-y')| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' > 1
⊢ 1 ≤ |(-y')|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [@k_of_one_le_abs (-y)] | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . calc ‖0 - k (-y')‖
_ = ‖k (-1) - k (-y')‖ := by
congr
apply (k_of_one_le_abs _).symm
simp
_ = ‖k (-y') - k (-1)‖ := by
rw [norm_sub_rev]
_ ≤ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|) := by
apply Hilbert_kernel_regularity_main_part
constructor
all_goals linarith
_... | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : ... | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-y)| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [abs_neg, abs_of_nonneg] <;> linarith | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-y)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-y)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | calc ‖0 - k (-y')‖
_ = ‖k (-1) - k (-y')‖ := by
congr
apply (k_of_one_le_abs _).symm
simp
_ = ‖k (-y') - k (-1)‖ := by
rw [norm_sub_rev]
_ ≤ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|) := by
apply Hilbert_kernel_regularity_main_part
constructor
all_goals linarith
_ = 2 ^ 6 * (1 / y') * ((... | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | congr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ = ‖k (-1) - k (-y')‖ | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 = k (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖0 - k (-y')‖ = ‖k (-1) - k (-y')‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply (k_of_one_le_abs _).symm | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 = k (-1) | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-1)| | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 = k (-1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-1)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-1)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [norm_sub_rev] | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-1) - k (-y')‖ = ‖k (-y') - k (-1)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-1) - k (-y')‖ = ‖k (-y') - k (-1)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply Hilbert_kernel_regularity_main_part | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y') - k (-1)‖ ≤ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|) | case yy'nonneg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' ∧ 0 ≤ 1
case ypos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ ‖k (-y') - k (-1)‖ ≤ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | constructor | case yy'nonneg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' ∧ 0 ≤ 1
case ypos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤... | case yy'nonneg.left
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y'
case yy'nonneg.right
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy... | Please generate a tactic in lean4 to solve the state.
STATE:
case yy'nonneg
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' ∧ 0 ≤ 1
case ypos
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | all_goals linarith | case yy'nonneg.left
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y'
case yy'nonneg.right
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case yy'nonneg.left
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y'
case yy'nonneg.right
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | linarith | case hy'
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy'
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | congr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|) = 2 ^ 6 * (1 / y') * ((1 - y') / y') | case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / |y'|) * (|y' - 1| / |y'|) = 2 ^ 6 * (1 / y') * ((1 - y') / y')
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [abs_of_nonneg]
exact yy'nonneg.2 | case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤... | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y' - 1| = 1 - y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [abs_of_nonpos]
simp
linarith | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y' - 1| = 1 - y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : ... | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y' | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y' - 1| = 1 - y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . rw [abs_of_nonneg]
exact yy'nonneg.2 | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg] | case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y' | case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | exact yy'nonneg.2 | case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonpos] | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y' - 1| = 1 - y' | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ -(y' - 1) = 1 - y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' :... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y' - 1| = 1 - y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ -(y' - 1) = 1 - y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' :... | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ y' - 1 ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ -(y' - 1) = 1 - y'
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | linarith | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ y' - 1 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ y' - 1 ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg] | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y' | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y'| = y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | exact yy'nonneg.2 | case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / y') * ((1 - y') / y') ≤ 2 ^ 6 * (1 / (y / 2)) * ((1 - y') / (y / 2)) | case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y'
case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / y') * ((1 - y') / y') ≤ 2 ^ 6 * (1 / (y / 2)) * ((1 - y') / (y / 2))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . apply div_nonneg <;> linarith | case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y'
case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
... | case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ 1 - y' | Please generate a tactic in lean4 to solve the state.
STATE:
case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y'
case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . linarith | case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ 1 - y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ 1 - y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply div_nonneg <;> linarith | case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | linarith | case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ 1 - y' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ 1 - y'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | ring | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / (y / 2)) * ((1 - y') / (y / 2)) = 2 ^ 6 * 2 * 2 * (1 / y) * ((1 - y') / y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * (1 / (y / 2)) * ((1 - y') / (y / 2)) = 2 ^ 6 * 2 * 2 * (1 / y) * ((1 - y') / y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 * (1 / y) * ((1 - y') / y) ≤ 2 ^ 6 * 2 * 2 * (1 / |y|) * (|y - y'| / |y|) | case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y
case h₁.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 * (1 / y) * ((1 - y') / y) ≤ 2 ^ 6 * 2 * 2 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply div_nonneg <;> linarith | case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y
case h₁.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
... | case h₁.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y| ≤ y
case h₂.hac
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 -... | Please generate a tactic in lean4 to solve the state.
STATE:
case c0
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 0 ≤ (1 - y') / y
case h₁.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg yy'nonneg.1] | case h₁.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y| ≤ y
case h₂.hac
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 -... | case h₂.hac
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 - y' ≤ |y - y'|
case h₂.hbd
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y'... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y| ≤ y
case h₂.hac
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg] <;> linarith | case h₂.hac
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 - y' ≤ |y - y'|
case h₂.hbd
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y'... | case h₂.hbd
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y| ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hac
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 - y' ≤ |y - y'|
case h₂.hbd
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_of_nonneg yy'nonneg.1] | case h₂.hbd
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y| ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.hbd
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ |y| ≤ y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | gcongr | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 ≤ 2 ^ 8 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 * (1 / |y|) * (|y - y'| / |y|) ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | norm_num | case h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 ≤ 2 ^ 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 2 ^ 6 * 2 * 2 ≤ 2 ^ 8
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_neg, abs_of_nonneg] <;> linarith | case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-y)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' ≤ 1
⊢ 1 ≤ |(-y)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | calc ‖k (-y) - k (-y')‖
_ = 0 := by
simp
rw [k_of_one_le_abs, k_of_one_le_abs] <;> (rw [abs_neg, abs_of_nonneg] <;> linarith)
_ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) := by
apply mul_nonneg
apply mul_nonneg
. norm_num
. simp
. apply div_nonneg <;> simp | case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | simp | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ = 0 | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ k (-y) = k (-y') | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ ‖k (-y) - k (-y')‖ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [k_of_one_le_abs, k_of_one_le_abs] <;> (rw [abs_neg, abs_of_nonneg] <;> linarith) | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ k (-y) = k (-y') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ k (-y) = k (-y')
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | rw [abs_neg, abs_of_nonneg] <;> linarith | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 1 ≤ |(-y)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 1 ≤ |(-y)|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply mul_nonneg | x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|) | case ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8 * (1 / |y|)
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | apply mul_nonneg | case ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8 * (1 / |y|)
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1... | case ha.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8
case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8 * (1 / |y|)
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . norm_num | case ha.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8
case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤... | case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 1 / |y|
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8
case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . simp | case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 1 / |y|
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ ... | case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ |y - y'| / |y| | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 1 / |y|
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
yp... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | . apply div_nonneg <;> simp | case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ |y - y'| / |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ |y - y'| / |y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity | [262, 1] | [397, 36] | norm_num | case ha.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.ha
x : ℝ
x_eq_zero : x = 0
y y' : ℝ
h : 2 * |y - y'| ≤ |y|
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y > 1
hy' : y' > 1
⊢ 0 ≤ 2 ^ 8
TACTIC:
|
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