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/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ring_nf | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [pow_two, ←mul_assoc, mul_assoc c, mul_inv_cancel norm_n_sub_m_pos.ne.symm, mul_assoc (R * _), mul_inv_cancel norm_n_sub_m_pos.ne.symm] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | ring | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [sub_eq_add_neg (-((n - m) * R')), ←neg_add, abs_neg, ←two_mul, abs_mul, abs_mul, mul_comm |(n : ℝ) - m|, mul_assoc] | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | gcongr | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply le_abs_self | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply ConditionallyCompleteLattice.le_biSup | x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆ z ∈ Me... | case hfs
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . convert bddAbove_localOscillation (Metric.ball x R) (θ n) (θ m)
apply norm_integer_linear_eq.symm | case hfs
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
... | case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
... | Please generate a tactic in lean4 to solve the state.
STATE:
case hfs
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillatio... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | . use y
simp
rw [abs_of_nonneg]
exact hR'.2
exact hR'.1 | case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | convert bddAbove_localOscillation (Metric.ball x R) (θ n) (θ m) | case hfs
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
... | case h.e'_3.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hfs
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillatio... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | apply norm_integer_linear_eq.symm | case h.e'_3.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscil... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | use y | case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
... | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | simp | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | rw [abs_of_nonneg] | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | exact hR'.2 | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | localOscillation_of_integer_linear | [124, 1] | [218, 22] | exact hR'.1 | case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_eq :
localOscillation (Metric.ball x R) (θ n) (θ m) =
⨆... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x R : ℝ
R_nonneg : 0 ≤ R
n m : ℤ
n_ne_m : n ≠ m
norm_n_sub_m_pos : 0 < |↑n - ↑m|
norm_integer_linear_eq :
∀ {n m : ℤ} {z : ℝ × ℝ},
‖(θ n) z.1 - (θ m) z.1 - (θ n) z.2 + (θ m) z.2‖ = ‖(↑n - ↑m) * (z.1 - x) - (↑n - ↑m) * (z.2 - x)‖
localOscillation_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | rw [iSup] | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ ⨆ i ∈ ∅, f i = sSup ∅ | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ sSup (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = sSup ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ ⨆ i ∈ ∅, f i = sSup ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | convert csSup_singleton _ | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ sSup (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = sSup ∅ | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ sSup (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = sSup ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | have : ∀ i : ι, IsEmpty (i ∈ (∅ : Set ι)) := by
intro i
simp | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅} | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | have : (fun (i : ι) ↦ ⨆ (_ : i ∈ (∅ : Set ι)), f i) = fun i ↦ sSup ∅ := by
ext i
rw [iSup]
congr
rw [Set.range_eq_empty_iff]
simp | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅} | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this✝ : ∀ (i : ι), IsEmpty (i ∈ ∅)
this : (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | rw [this] | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this✝ : ∀ (i : ι), IsEmpty (i ∈ ∅)
this : (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅} | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this✝ : ∀ (i : ι), IsEmpty (i ∈ ∅)
this : (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
⊢ (Set.range fun i => sSup ∅) = {sSup ∅} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this✝ : ∀ (i : ι), IsEmpty (i ∈ ∅)
this : (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
⊢ (Set.range fun i => ⨆ (_ : i ∈ ∅), f i) = {sSup ∅}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | apply Set.range_const | case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this✝ : ∀ (i : ι), IsEmpty (i ∈ ∅)
this : (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
⊢ (Set.range fun i => sSup ∅) = {sSup ∅} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_3
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this✝ : ∀ (i : ι), IsEmpty (i ∈ ∅)
this : (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
⊢ (Set.range fun i => sSup ∅) = {sSup ∅}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | intro i | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ ∀ (i : ι), IsEmpty (i ∈ ∅) | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
i : ι
⊢ IsEmpty (i ∈ ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
⊢ ∀ (i : ι), IsEmpty (i ∈ ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | simp | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
i : ι
⊢ IsEmpty (i ∈ ∅) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
i : ι
⊢ IsEmpty (i ∈ ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | ext i | α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
⊢ (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅ | case h
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ ⨆ (_ : i ∈ ∅), f i = sSup ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
⊢ (fun i => ⨆ (_ : i ∈ ∅), f i) = fun i => sSup ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | rw [iSup] | case h
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ ⨆ (_ : i ∈ ∅), f i = sSup ∅ | case h
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ sSup (Set.range fun x => f i) = sSup ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ ⨆ (_ : i ∈ ∅), f i = sSup ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | congr | case h
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ sSup (Set.range fun x => f i) = sSup ∅ | case h.e_a
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ (Set.range fun x => f i) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ sSup (Set.range fun x => f i) = sSup ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | rw [Set.range_eq_empty_iff] | case h.e_a
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ (Set.range fun x => f i) = ∅ | case h.e_a
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ IsEmpty (i ∈ ∅) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_a
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ (Set.range fun x => f i) = ∅
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | bciSup_of_emptyset | [222, 1] | [237, 24] | simp | case h.e_a
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ IsEmpty (i ∈ ∅) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_a
α : Type
inst✝¹ : ConditionallyCompleteLattice α
ι : Type
inst✝ : Nonempty ι
f : ι → α
this : ∀ (i : ι), IsEmpty (i ∈ ∅)
i : ι
⊢ IsEmpty (i ∈ ∅)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | intro f x | ⊢ T' ≤ CarlesonOperator' K {x | ∃ n, θ n = x} | f : ℝ → ℂ
x : ℝ
⊢ T' f x ≤ CarlesonOperator' K {x | ∃ n, θ n = x} f x | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ T' ≤ CarlesonOperator' K {x | ∃ n, θ n = x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | rw [CarlesonOperator', CarlesonOperatorReal'] | f : ℝ → ℂ
x : ℝ
⊢ T' f x ≤ CarlesonOperator' K {x | ∃ n, θ n = x} f x | f : ℝ → ℂ
x : ℝ
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
x : ℝ
⊢ T' f x ≤ CarlesonOperator' K {x | ∃ n, θ n = x} f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply iSup_le | f : ℝ → ℂ
x : ℝ
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo... | case h
f : ℝ → ℂ
x : ℝ
⊢ ∀ (i : ℤ),
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑i * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
x : ℝ
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | intro n | case h
f : ℝ → ℂ
x : ℝ
⊢ ∀ (i : ℤ),
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑i * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y... | case h
f : ℝ → ℂ
x : ℝ
n : ℤ
⊢ ⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
x : ℝ
⊢ ∀ (i : ℤ),
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑i * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply iSup_le | case h
f : ℝ → ℂ
x : ℝ
n : ℤ
⊢ ⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R... | case h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
⊢ ∀ (i : ℝ),
⨆ (_ : 0 < i),
⨆ (_ : i < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo i 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
x : ℝ
n : ℤ
⊢ ⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | intro r | case h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
⊢ ∀ (i : ℝ),
⨆ (_ : 0 < i),
⨆ (_ : i < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo i 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist... | case h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
⊢ ⨆ (_ : 0 < r), ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
⊢ ∀ (i : ℝ),
⨆ (_ : 0 < i),
⨆ (_ : i < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo i 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 <... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply iSup_le | case h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
⊢ ⨆ (_ : 0 < r), ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x ... | case h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
⊢ 0 < r →
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
⊢ ⨆ (_ : 0 < r), ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | intro rpos | case h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
⊢ 0 < r →
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ ... | case h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
⊢ ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
⊢ 0 < r →
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply iSup_le | case h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
⊢ ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x ... | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
⊢ r < 1 →
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
⊢ ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | intro rle1 | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
⊢ r < 1 →
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ ... | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
⊢ r < 1 →
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply le_iSup₂_of_le (θ n) (by simp) | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x ... | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ Q ∈ {x | ∃ n, θ n = x},
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply le_iSup₂_of_le r 1 | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo R₁ R₂}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ (_ : 0 < r), ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ R₁,
⨆ R₂,
⨆ (_ : 0 < R₁),
⨆ (_ : R₁ < R₂), ↑‖∫ (y : ℝ) in {y | dist x y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply le_iSup₂_of_le rpos rle1 | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ (_ : 0 < r), ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
⨆ (_ : 0 < r), ⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | apply le_of_eq | case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | case h.h.h.h.a
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ =
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ ≤
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | rw [integer_linear, ContinuousMap.coe_mk] | case h.h.h.h.a
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ =
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊ | case h.h.h.h.a
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ =
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (↑n * ↑y)).exp‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h.a
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ =
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (θ n) y).exp‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | congr | case h.h.h.h.a
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ =
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (↑n * ↑y)).exp‖₊ | case h.h.h.h.a.e_a.e_a.e_f
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun y => f y * K x y * (Complex.I * ↑n * ↑y).exp) = fun y => K x y * f y * (Complex.I * (↑n * ↑y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h.a
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * (Complex.I * ↑n * ↑y).exp‖₊ =
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, K x y * f y * (Complex.I * (↑n * ↑y)).exp‖₊
TACTIC... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | ext y | case h.h.h.h.a.e_a.e_a.e_f
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun y => f y * K x y * (Complex.I * ↑n * ↑y).exp) = fun y => K x y * f y * (Complex.I * (↑n * ↑y)).exp | case h.h.h.h.a.e_a.e_a.e_f.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ f y * K x y * (Complex.I * ↑n * ↑y).exp = K x y * f y * (Complex.I * (↑n * ↑y)).exp | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h.a.e_a.e_a.e_f
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun y => f y * K x y * (Complex.I * ↑n * ↑y).exp) = fun y => K x y * f y * (Complex.I * (↑n * ↑y)).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | ring_nf | case h.h.h.h.a.e_a.e_a.e_f.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ f y * K x y * (Complex.I * ↑n * ↑y).exp = K x y * f y * (Complex.I * (↑n * ↑y)).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.h.a.e_a.e_a.e_f.h
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ f y * K x y * (Complex.I * ↑n * ↑y).exp = K x y * f y * (Complex.I * (↑n * ↑y)).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | CarlesonOperatorReal'_le_CarlesonOperator' | [481, 1] | [499, 10] | simp | f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ θ n ∈ {x | ∃ n, θ n = x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
x : ℝ
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ θ n ∈ {x | ∃ n, θ n = x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | rcarleson' | [502, 1] | [516, 69] | calc ∫⁻ x in G, T' f x
_ ≤ ∫⁻ x in G, CarlesonOperator' K Θ f x := by
apply MeasureTheory.lintegral_mono
apply CarlesonOperatorReal'_le_CarlesonOperator'
_ ≤ ENNReal.ofReal (C1_2 4 2) * (MeasureTheory.volume G) ^ (2 : ℝ)⁻¹ * (MeasureTheory.volume F) ^ (2 : ℝ)⁻¹ := by
convert theorem1_2C' K (by simp) h1 ... | F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ ∫⁻ (x : ℝ) in G, T' f x ≤ ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume G ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ ∫⁻ (x : ℝ) in G, T' f x ≤ ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume G ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | rcarleson' | [502, 1] | [516, 69] | apply MeasureTheory.lintegral_mono | F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ ∫⁻ (x : ℝ) in G, T' f x ≤ ∫⁻ (x : ℝ) in G, CarlesonOperator' K {x | ∃ n, θ n = x} f x | case hfg
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ (fun a => T' f a) ≤ fun a => CarlesonOperator' K {x | ∃ n, θ n = x} f a | Please generate a tactic in lean4 to solve the state.
STATE:
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ ∫⁻ (x : ℝ) in G, T' f x ≤ ∫⁻ (x : ℝ) in G, CarlesonOperator' K {x | ∃ n, θ n = x} f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | rcarleson' | [502, 1] | [516, 69] | apply CarlesonOperatorReal'_le_CarlesonOperator' | case hfg
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ (fun a => T' f a) ≤ fun a => CarlesonOperator' K {x | ∃ n, θ n = x} f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hfg
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ (fun a => T' f a) ≤ fun a => CarlesonOperator' K {x | ∃ n, θ n = x} f a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | rcarleson' | [502, 1] | [516, 69] | convert theorem1_2C' K (by simp) h1 h2 hF hG h3 f hf <;> sorry | F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ ∫⁻ (x : ℝ) in G, CarlesonOperator' K {x | ∃ n, θ n = x} f x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume G ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ ∫⁻ (x : ℝ) in G, CarlesonOperator' K {x | ∃ n, θ n = x} f x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume G ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Carleson_on_the_real_line.lean | rcarleson' | [502, 1] | [516, 69] | simp | F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ 4 ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
F G : Set ℝ
hF : MeasurableSet F
hG : MeasurableSet G
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ F.indicator 1 x
⊢ 4 ≤ 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Proposition3.lean | prop2_3 | [35, 1] | [53, 8] | sorry | X : Type u_1
A : ℝ
inst✝⁶ : MetricSpace X
inst✝⁵ : IsSpaceOfHomogeneousType X A
inst✝⁴ : Inhabited X
τ q D κ ε δ C₀ C t : ℝ
Θ : Set C(X, ℂ)
inst✝³ : IsCompatible Θ
inst✝² : IsCancellative τ Θ
inst✝¹ : TileStructure Θ D κ C₀
F G : Set X
σ σ' : X → ℤ
Q' : X → C(X, ℂ)
K : X → X → ℂ
inst✝ : IsCZKernel τ K
ψ : ℝ → ℝ
n : ℕ
�... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
inst✝⁶ : MetricSpace X
inst✝⁵ : IsSpaceOfHomogeneousType X A
inst✝⁴ : Inhabited X
τ q D κ ε δ C₀ C t : ℝ
Θ : Set C(X, ℂ)
inst✝³ : IsCompatible Θ
inst✝² : IsCancellative τ Θ
inst✝¹ : TileStructure Θ D κ C₀
F G : Set X
σ σ' : X → ℤ
Q' : X → C... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Defs.lean | Memℒp_T | [249, 1] | [250, 11] | sorry | X : Type u_1
A : ℝ
inst✝³ : PseudoMetricSpace X
inst✝² : IsSpaceOfHomogeneousType X A
D κ C : ℝ
inst✝¹ : Inhabited X
Θ : Set C(X, ℂ)
inst✝ : TileStructure Θ D κ C
K : X → X → ℂ
Q' : X → C(X, ℂ)
σ σ' : X → ℤ
ψ : ℝ → ℝ
p : 𝔓 X
F : Set X
f : X → ℂ
q : ℝ≥0∞
hf : Memℒp f q volume
⊢ Memℒp (T K Q' σ σ' ψ p F f) q volume | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
A : ℝ
inst✝³ : PseudoMetricSpace X
inst✝² : IsSpaceOfHomogeneousType X A
D κ C : ℝ
inst✝¹ : Inhabited X
Θ : Set C(X, ℂ)
inst✝ : TileStructure Θ D κ C
K : X → X → ℂ
Q' : X → C(X, ℂ)
σ σ' : X → ℤ
ψ : ℝ → ℝ
p : 𝔓 X
F : Set X
f : X → ℂ
q : ℝ≥0∞
hf :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | Finset.sum_measure_singleton | [30, 1] | [37, 9] | change ∑ x in s, μ (id ⁻¹' {x}) = _ | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x ∈ s, μ {x} = μ ↑s | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x ∈ s, μ (id ⁻¹' {x}) = μ ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x ∈ s, μ {x} = μ ↑s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | Finset.sum_measure_singleton | [30, 1] | [37, 9] | rw [sum_measure_preimage_singleton] | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x ∈ s, μ (id ⁻¹' {x}) = μ ↑s | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ μ (id ⁻¹' ↑s) = μ ↑s
case hf
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∀ y ∈ s, MeasurableSet (id ⁻¹' {y}) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x ∈ s, μ (id ⁻¹' {x}) = μ ↑s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | Finset.sum_measure_singleton | [30, 1] | [37, 9] | simp | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ μ (id ⁻¹' ↑s) = μ ↑s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ μ (id ⁻¹' ↑s) = μ ↑s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | Finset.sum_measure_singleton | [30, 1] | [37, 9] | simp | case hf
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∀ y ∈ s, MeasurableSet (id ⁻¹' {y}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∀ y ∈ s, MeasurableSet (id ⁻¹' {y})
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | Finset.sum_toReal_measure_singleton | [39, 1] | [44, 7] | rw [← ENNReal.toReal_sum (fun _ _ ↦ measure_ne_top _ _)] | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ ∑ x ∈ s, (μ {x}).toReal = (μ ↑s).toReal | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ (∑ a ∈ s, μ {a}).toReal = (μ ↑s).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ ∑ x ∈ s, (μ {x}).toReal = (μ ↑s).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | Finset.sum_toReal_measure_singleton | [39, 1] | [44, 7] | simp | ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ (∑ a ∈ s, μ {a}).toReal = (μ ↑s).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
s : Finset S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ (∑ a ∈ s, μ {a}).toReal = (μ ↑s).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | sum_measure_singleton | [48, 1] | [51, 7] | simp | ι : Type u_1
S : Type u_2
inst✝¹ : Fintype S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x : S, μ {x} = μ univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
inst✝¹ : Fintype S
x✝ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ : Measure S
⊢ ∑ x : S, μ {x} = μ univ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | sum_toReal_measure_singleton | [55, 1] | [58, 7] | simp | ι : Type u_1
S : Type u_2
inst✝² : Fintype S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ ∑ x : S, (μ {x}).toReal = (μ univ).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
S : Type u_2
inst✝² : Fintype S
x✝ : MeasurableSpace S
inst✝¹ : MeasurableSingletonClass S
μ : Measure S
inst✝ : IsFiniteMeasure μ
⊢ ∑ x : S, (μ {x}).toReal = (μ univ).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measure_diff_eq_top | [80, 1] | [84, 52] | contrapose! hs | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : μ s = ⊤
ht : μ t ≠ ⊤
⊢ μ (s \ t) = ⊤ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ s ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : μ s = ⊤
ht : μ t ≠ ⊤
⊢ μ (s \ t) = ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measure_diff_eq_top | [80, 1] | [84, 52] | apply ((measure_mono (subset_diff_union s t)).trans_lt _).ne | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ s ≠ ⊤ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ (s \ t ∪ t) < ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ s ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measure_diff_eq_top | [80, 1] | [84, 52] | apply (measure_union_le _ _).trans_lt | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ (s \ t ∪ t) < ⊤ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ (s \ t) + μ t < ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ (s \ t ∪ t) < ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measure_diff_eq_top | [80, 1] | [84, 52] | exact ENNReal.add_lt_top.2 ⟨hs.lt_top, ht.lt_top⟩ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ (s \ t) + μ t < ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : μ t ≠ ⊤
hs : μ (s \ t) ≠ ⊤
⊢ μ (s \ t) + μ t < ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_eq_zero_iff | [91, 1] | [94, 11] | rw [Measure.real, ENNReal.toReal_eq_zero_iff] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real s = 0 ↔ μ s = 0 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ s = 0 ∨ μ s = ⊤ ↔ μ s = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real s = 0 ↔ μ s = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_eq_zero_iff | [91, 1] | [94, 11] | simp [h] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ s = 0 ∨ μ s = ⊤ ↔ μ s = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ s = 0 ∨ μ s = ⊤ ↔ μ s = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_zero | [96, 9] | [97, 25] | simp [measureReal_def] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
⊢ Measure.real 0 s = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
⊢ Measure.real 0 s = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_empty | [101, 9] | [102, 25] | simp [Measure.real] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
⊢ μ.real ∅ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
⊢ μ.real ∅ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.IsProbabilityMeasure.measureReal_univ | [104, 9] | [106, 22] | simp [Measure.real] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsProbabilityMeasure μ
⊢ μ.real univ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsProbabilityMeasure μ
⊢ μ.real univ = 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_univ_pos | [108, 1] | [112, 13] | rw [measureReal_def] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ 0 < μ.real univ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ 0 < (μ univ).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ 0 < μ.real univ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_univ_pos | [108, 1] | [112, 13] | apply ENNReal.toReal_pos | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ 0 < (μ univ).toReal | case ha₀
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ μ univ ≠ 0
case ha_top
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ 0 < (μ univ).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_univ_pos | [108, 1] | [112, 13] | exact NeZero.ne (μ Set.univ) | case ha₀
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ μ univ ≠ 0
case ha_top
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α... | case ha_top
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ μ univ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case ha₀
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ μ univ ≠ 0
case ha_top
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_univ_pos | [108, 1] | [112, 13] | finiteness | case ha_top
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ μ univ ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha_top
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
⊢ μ univ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_smul_apply | [120, 9] | [122, 6] | rw [measureReal_def, smul_apply, smul_eq_mul, ENNReal.toReal_mul] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
c : ℝ≥0∞
⊢ (c • μ).real s = c.toReal • μ.real s | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
c : ℝ≥0∞
⊢ c.toReal * (μ s).toReal = c.toReal • μ.real s | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
c : ℝ≥0∞
⊢ (c • μ).real s = c.toReal • μ.real s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_smul_apply | [120, 9] | [122, 6] | rfl | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
c : ℝ≥0∞
⊢ c.toReal * (μ s).toReal = c.toReal • μ.real s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
c : ℝ≥0∞
⊢ c.toReal * (μ s).toReal = c.toReal • μ.real s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.map_measureReal_apply | [124, 1] | [127, 6] | rw [measureReal_def, map_apply hf hs] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (map f μ).real s = μ.real (f ⁻¹' s) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (μ (f ⁻¹' s)).toReal = μ.real (f ⁻¹' s) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (map f μ).real s = μ.real (f ⁻¹' s)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.map_measureReal_apply | [124, 1] | [127, 6] | rfl | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (μ (f ⁻¹' s)).toReal = μ.real (f ⁻¹' s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : α → β
hf : Measurable f
s : Set β
hs : MeasurableSet s
⊢ (μ (f ⁻¹' s)).toReal = μ.real (f ⁻¹' s)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_mono_null | [133, 1] | [136, 46] | rw [measureReal_eq_zero_iff h'₂] at h₂ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₁ ⊆ s₂
h₂ : μ.real s₂ = 0
h'₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ = 0 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₁ ⊆ s₂
h₂ : μ s₂ = 0
h'₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₁ ⊆ s₂
h₂ : μ.real s₂ = 0
h'₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_mono_null | [133, 1] | [136, 46] | simp [Measure.real, measure_mono_null h h₂] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₁ ⊆ s₂
h₂ : μ s₂ = 0
h'₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₁ ⊆ s₂
h₂ : μ s₂ = 0
h'₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_le_measureReal_union_left | [138, 1] | [142, 83] | rcases eq_top_or_lt_top (μ s) with hs|hs | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real s ≤ μ.real (s ∪ t) | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
hs : μ s = ⊤
⊢ μ.real s ≤ μ.real (s ∪ t)
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real s ≤ μ.real (s ∪ t)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_le_measureReal_union_left | [138, 1] | [142, 83] | simp [Measure.real, hs] | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
hs : μ s = ⊤
⊢ μ.real s ≤ μ.real (s ∪ t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
hs : μ s = ⊤
⊢ μ.real s ≤ μ.real (s ∪ t)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_le_measureReal_union_left | [138, 1] | [142, 83] | exact measureReal_mono subset_union_left (measure_union_lt_top hs h.lt_top).ne | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
hs : μ s < ⊤
⊢ μ.real s ≤ μ.real (s ∪ t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ t ≠ ⊤) _auto✝
hs : μ s < ⊤
⊢ μ.real s ≤ μ.real (s ∪ t)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_le_measureReal_union_right | [144, 1] | [147, 48] | rw [union_comm] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real t ≤ μ.real (s ∪ t) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real t ≤ μ.real (t ∪ s) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real t ≤ μ.real (s ∪ t)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_le_measureReal_union_right | [144, 1] | [147, 48] | exact measureReal_le_measureReal_union_left h | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real t ≤ μ.real (t ∪ s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real t ≤ μ.real (t ∪ s)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | rcases eq_top_or_lt_top (μ (s₁ ∪ s₂)) with h|h | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | simp only [Measure.real, h, ENNReal.top_toReal] | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ⊤
⊢ 0 ≤ (μ s₁).toReal + (μ s₂).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ⊤
⊢ 0 ≤ (μ s₁).toReal + (μ s₂).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ⊤
⊢ 0 ≤ (μ s₁).toReal + (μ s₂).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | have A : μ s₁ ≠ ∞ := measure_ne_top_of_subset subset_union_left h.ne | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | have B : μ s₂ ≠ ∞ := measure_ne_top_of_subset subset_union_right h.ne | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | simp only [Measure.real, ← ENNReal.toReal_add A B] | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂ | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ (μ (s₁ ∪ s₂)).toReal ≤ (μ s₁ + μ s₂).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | exact ENNReal.toReal_mono (by simp [A, B]) (measure_union_le _ _) | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ (μ (s₁ ∪ s₂)).toReal ≤ (μ s₁ + μ s₂).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ (μ (s₁ ∪ s₂)).toReal ≤ (μ s₁ + μ s₂).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_le | [149, 1] | [156, 70] | simp [A, B] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ μ s₁ + μ s₂ ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) < ⊤
A : μ s₁ ≠ ⊤
B : μ s₂ ≠ ⊤
⊢ μ s₁ + μ s₂ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_biUnion_finset_le | [158, 1] | [165, 55] | induction' s using Finset.induction_on with x s hx IH | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
s : Finset β
f : β → Set α
⊢ μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p) | case empty
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
f : β → Set α
⊢ μ.real (⋃ b ∈ ∅, f b) ≤ ∑ p ∈ ∅, μ.real (f p)
case insert
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t :... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
s : Finset β
f : β → Set α
⊢ μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_biUnion_finset_le | [158, 1] | [165, 55] | simp | case empty
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
f : β → Set α
⊢ μ.real (⋃ b ∈ ∅, f b) ≤ ∑ p ∈ ∅, μ.real (f p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
f : β → Set α
⊢ μ.real (⋃ b ∈ ∅, f b) ≤ ∑ p ∈ ∅, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_biUnion_finset_le | [158, 1] | [165, 55] | simp only [hx, Finset.mem_insert, iUnion_iUnion_eq_or_left, not_false_eq_true,
Finset.sum_insert] | case insert
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (⋃ b ∈ insert x s, f b) ≤ ∑ p ∈ insert x s, μ.real (f p) | case insert
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (f x ∪ ⋃ x ∈ s, f x) ≤ μ.real (f x) + ∑ p ∈ s, μ.real (f p) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (⋃ b ∈ insert x s, f b... |
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