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/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
hy : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ y β Set.Icc (-Real.pi) (3 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
hy : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ y β Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.aestronglyMeasurable | case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi))) | case hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => β(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.mul | case hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => β(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(max (1 - |x - a|) 0)
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => β(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . apply Measurable.comp Complex.measurable_ofReal
apply Measurable.max
apply Measurable.const_sub
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(max (1 - |x - a|) 0)
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(max (1 - |x - a|) 0)
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . apply Measurable.comp dirichletKernel'_measurable
apply Measurable.const_sub measurable_id | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.comp Complex.measurable_ofReal | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(max (1 - |x - a|) 0) | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => max (1 - |x - a|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(max (1 - |x - a|) 0)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.max | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => max (1 - |x - a|) 0 | case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 - |x - a|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => max (1 - |x - a|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.const_sub | case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 - |x - a|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | case hf.hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => |x - x_1|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 - |x - a|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.comp _root_.continuous_abs.measurable | case hf.hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => |x - x_1|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | case hf.hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => x - x_1
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => |x - x_1|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => x - x_1
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => x - x_1
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | exact measurable_const | case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.comp dirichletKernel'_measurable | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc] | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β),
x_1 β Set.Icc (x - Real.pi) (x + Real.pi) β ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply eventually_of_forall | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β),
x_1 β Set.Icc (x - Real.pi) (x + Real.pi) β ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β x_1 β Set.Icc (x - Real.pi) (x + Real.pi), ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β),
x_1 β Set.Icc (x - Real.pi) (x + Real.pi) β ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | intro y _ | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β x_1 β Set.Icc (x - Real.pi) (x + Real.pi), ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β x_1 β Set.Icc (x - Real.pi) (x + Real.pi), ββ(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | calc β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)β
_ = βmax (1 - |x - y|) 0β * βdirichletKernel' N (x - y)β := by
rw [norm_mul, Complex.norm_real]
_ β€ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_nonneg]
apply max_le
linarith [abs_nonneg (x - y)]
norm_num
apply le_max_right
apply norm_dirichletKernel'_le | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [norm_mul, Complex.norm_real] | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)β = βmax (1 - |x - y|) 0β * βdirichletKernel' N (x - y)β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)β = βmax (1 - |x - y|) 0β * βdirichletKernel' N (x - y)β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | gcongr | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β * βdirichletKernel' N (x - y)β β€ 1 * (2 * βN + 1) | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β * βdirichletKernel' N (x - y)β β€ 1 * (2 * βN + 1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . rw [Real.norm_of_nonneg]
apply max_le
linarith [abs_nonneg (x - y)]
norm_num
apply le_max_right | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply norm_dirichletKernel'_le | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [Real.norm_of_nonneg] | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β β€ 1 | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ max (1 - |x - y|) 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmax (1 - |x - y|) 0β β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply max_le | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ max (1 - |x - y|) 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 1 - |x - y| β€ 1
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ max (1 - |x - y|) 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | linarith [abs_nonneg (x - y)] | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 1 - |x - y| β€ 1
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 1 - |x - y| β€ 1
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | norm_num | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply le_max_right | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | conv => pattern ((f _) * _ * _); rw [mul_assoc, mul_comm] | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => f y * β(min |x - y| 1) * dirichletKernel' N (x - y))
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => β(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => f y * β(min |x - y| 1) * dirichletKernel' N (x - y))
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith [Real.pi_pos])] at hf | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => β(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.IntegrableOn (fun y => β(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => β(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply MeasureTheory.Integrable.bdd_mul' | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.IntegrableOn (fun y => β(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.IntegrableOn (fun y => β(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply hf.mono_set
intro y hy
constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply Measurable.aestronglyMeasurable
apply Measurable.mul
. apply Measurable.comp Complex.measurable_ofReal
apply Measurable.min
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const
. apply Measurable.comp dirichletKernel'_measurable
apply Measurable.const_sub measurable_id | case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc]
apply eventually_of_forall
intro y _
calc β(min |x - y| 1) * dirichletKernel' N (x - y)β
_ = βmin |x - y| 1β * βdirichletKernel' N (x - y)β := by
rw [norm_mul, Complex.norm_real]
_ β€ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_nonneg]
apply min_le_right
apply le_min
linarith [abs_nonneg (x - y)]
norm_num
apply norm_dirichletKernel'_le | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | linarith [Real.pi_pos] | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ -Real.pi β€ 3 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ -Real.pi β€ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply hf.mono_set | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi))) | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Set.Icc (x - Real.pi) (x + Real.pi) β Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | intro y hy | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Set.Icc (x - Real.pi) (x + Real.pi) β Set.Icc (-Real.pi) (3 * Real.pi) | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
hy : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ y β Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Set.Icc (x - Real.pi) (x + Real.pi) β Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
hy : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ y β Set.Icc (-Real.pi) (3 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
hy : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ y β Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.aestronglyMeasurable | case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi))) | case hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ MeasureTheory.AEStronglyMeasurable (fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.mul | case hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1) | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(min |x - a| 1)
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun x_1 => β(min |x - x_1| 1) * dirichletKernel' N (x - x_1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply Measurable.comp Complex.measurable_ofReal
apply Measurable.min
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(min |x - a| 1)
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(min |x - a| 1)
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply Measurable.comp dirichletKernel'_measurable
apply Measurable.const_sub measurable_id | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.comp Complex.measurable_ofReal | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(min |x - a| 1) | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => min |x - a| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => β(min |x - a| 1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.min | case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => min |x - a| 1 | case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => |x - a|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => min |x - a| 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.comp _root_.continuous_abs.measurable | case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => |x - a|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 | case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => |x - a|
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 | case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | exact measurable_const | case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.comp dirichletKernel'_measurable | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a) | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ Measurable fun a => x - a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc] | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β), x_1 β Set.Icc (x - Real.pi) (x + Real.pi) β ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β) βMeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply eventually_of_forall | case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β), x_1 β Set.Icc (x - Real.pi) (x + Real.pi) β ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β x_1 β Set.Icc (x - Real.pi) (x + Real.pi), ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ βα΅ (x_1 : β), x_1 β Set.Icc (x - Real.pi) (x + Real.pi) β ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | intro y _ | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β x_1 β Set.Icc (x - Real.pi) (x + Real.pi), ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(min |x - y| 1) * dirichletKernel' N (x - y)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β x_1 β Set.Icc (x - Real.pi) (x + Real.pi), ββ(min |x - x_1| 1) * dirichletKernel' N (x - x_1)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | calc β(min |x - y| 1) * dirichletKernel' N (x - y)β
_ = βmin |x - y| 1β * βdirichletKernel' N (x - y)β := by
rw [norm_mul, Complex.norm_real]
_ β€ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_nonneg]
apply min_le_right
apply le_min
linarith [abs_nonneg (x - y)]
norm_num
apply norm_dirichletKernel'_le | case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(min |x - y| 1) * dirichletKernel' N (x - y)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(min |x - y| 1) * dirichletKernel' N (x - y)β β€ ?c
case c
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
β’ β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [norm_mul, Complex.norm_real] | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(min |x - y| 1) * dirichletKernel' N (x - y)β = βmin |x - y| 1β * βdirichletKernel' N (x - y)β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ ββ(min |x - y| 1) * dirichletKernel' N (x - y)β = βmin |x - y| 1β * βdirichletKernel' N (x - y)β
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | gcongr | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β * βdirichletKernel' N (x - y)β β€ 1 * (2 * βN + 1) | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β * βdirichletKernel' N (x - y)β β€ 1 * (2 * βN + 1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . rw [Real.norm_of_nonneg]
apply min_le_right
apply le_min
linarith [abs_nonneg (x - y)]
norm_num | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply norm_dirichletKernel'_le | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βdirichletKernel' N (x - y)β β€ 2 * βN + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [Real.norm_of_nonneg] | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β β€ 1 | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ min |x - y| 1 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ min |x - y| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ βmin |x - y| 1β β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply min_le_right | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ min |x - y| 1 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ min |x - y| 1 | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ min |x - y| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ min |x - y| 1 β€ 1
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ min |x - y| 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply le_min | case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ min |x - y| 1 | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ |x - y|
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ min |x - y| 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | linarith [abs_nonneg (x - y)] | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ |x - y|
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1 | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ |x - y|
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | norm_num | case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.hβ
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : β
y : β
aβ : y β Set.Icc (x - Real.pi) (x + Real.pi)
β’ 0 β€ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | apply (integrableOn_mul_dirichletKernel'_max hx hf).mono_set | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
{y | dist x y β Set.Ioo 0 1} MeasureTheory.volume | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ {y | dist x y β Set.Ioo 0 1} β Set.Icc (x - Real.pi) (x + Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ MeasureTheory.IntegrableOn (fun y => f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
{y | dist x y β Set.Ioo 0 1} MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | intro y hy | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ {y | dist x y β Set.Ioo 0 1} β Set.Icc (x - Real.pi) (x + Real.pi) | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ y β Set.Icc (x - Real.pi) (x + Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ {y | dist x y β Set.Ioo 0 1} β Set.Icc (x - Real.pi) (x + Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | rw [annulus_real_eq (by rfl)] at hy | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ y β Set.Icc (x - Real.pi) (x + Real.pi) | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β Set.Ioo (x - 1) (x - 0) βͺ Set.Ioo (x + 0) (x + 1)
β’ y β Set.Icc (x - Real.pi) (x + Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ y β Set.Icc (x - Real.pi) (x + Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | rcases hy with h | h <;> constructor <;> linarith [h.1, h.2, hx.1, hx.2, Real.two_le_pi] | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β Set.Ioo (x - 1) (x - 0) βͺ Set.Ioo (x + 0) (x + 1)
β’ y β Set.Icc (x - Real.pi) (x + Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β Set.Ioo (x - 1) (x - 0) βͺ Set.Ioo (x + 0) (x + 1)
β’ y β Set.Icc (x - Real.pi) (x + Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | rfl | x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ 0 β€ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ 0 β€ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro x hx | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ β x β Set.Icc 0 (2 * Real.pi),
βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
β’ β x β Set.Icc 0 (2 * Real.pi),
βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | set s : β β Set β := fun n β¦ {y | dist x y β Set.Ioo (1 / (n + 2 : β)) 1} with sdef | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | have hs : {y | dist x y β Set.Ioo 0 1} = β n, s n := by
ext y
constructor
. intro hy
rw [Set.mem_setOf_eq, Set.mem_Ioo] at hy
obtain β¨n, hnβ© := exists_nat_gt (1 / dist x y)
simp
use n
rw [sdef]
simp
constructor
. rw [inv_lt, inv_eq_one_div]
apply lt_trans hn
linarith
linarith
exact hy.1
. exact hy.2
. intro hy
simp at hy
rcases hy with β¨n, hnβ©
rw [sdef] at hn
simp at hn
constructor
. apply lt_trans' hn.1
norm_num
linarith
. exact hn.2 | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
hs : {y | dist x y β Set.Ioo 0 1} = β n, s n
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | have : Tendsto (fun i => β« y in s i, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop (π (β« y in β n, s n, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y))) := by
apply MeasureTheory.tendsto_setIntegral_of_monotone
. intro n
exact annulus_measurableSet
. intro n m nlem
simp
intro y hy
rw [sdef]
rw [sdef] at hy
simp
simp at hy
constructor
. apply lt_of_le_of_lt _ hy.1
rw [inv_le_inv]
norm_cast
all_goals linarith
. exact hy.2
. rw [β hs]
exact integrableOn_mul_dirichletKernel'_specific hx hf | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
hs : {y | dist x y β Set.Ioo 0 1} = β n, s n
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
hs : {y | dist x y β Set.Ioo 0 1} = β n, s n
this :
Tendsto (fun i => β« (y : β) in s i, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(π (β« (y : β) in β n, s n, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
hs : {y | dist x y β Set.Ioo 0 1} = β n, s n
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | calc ENNReal.ofNNReal ββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ
_ = ββ« y in β n, s n, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ := by
congr
_ β€ β¨ (i : β), βββ« y in s i, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ := by
apply le_iSup_of_tendsto
rw [ENNReal.tendsto_coe]
apply Tendsto.nnnorm this
_ β€ β¨ (r : β) (_ : 0 < r) (_ : r < 1), βββ« y in {y | dist x y β Set.Ioo r 1}, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ := by
apply iSup_le
intro n
apply le_iSup_of_le (1 / (n + 2 : β))
apply le_iSupβ_of_le (by simp; linarith) (by rw [div_lt_iff] <;> linarith)
rfl
_ = β¨ (r : β) (_ : 0 < r) (_ : r < 1), βββ« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-(Int.ofNat N) * x)) * K x y * exp (I * N * y) + (starRingEnd β) (exp (I * (-(Int.ofNat N) * x)) * K x y * exp (I * (Int.ofNat N) * y)))ββ := by
apply iSup_congr
intro r
apply iSup_congr
intro _
apply iSup_congr
intro _
congr
ext y
rw [mul_assoc, dirichlet_Hilbert_eq]
norm_cast
_ β€ β¨ (n : β€) (r : β) (_ : 0 < r) (_ : r < 1), βββ« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-n * x)) * K x y * exp (I * n * y) + (starRingEnd β) (exp (I * (-n * x)) * K x y * exp (I * n * y)))ββ := by
let F : β€ β ENNReal := fun (n : β€) β¦ β¨ (r : β) (_ : 0 < r) (_ : r < 1), βββ« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-n * x)) * K x y * exp (I * n * y) + (starRingEnd β) (exp (I * (-n * x)) * K x y * exp (I * n * y)))ββ
apply le_iSup F ((Int.ofNat N))
_ β€ β¨ (n : β€) (r : β) (_ : 0 < r) (_ : r < 1), ( βββ« y in {y | dist x y β Set.Ioo r 1}, f y * K x y * exp (I * n * y)ββ
+ βββ« y in {y | dist x y β Set.Ioo r 1}, ((starRingEnd β) β f) y * K x y * exp (I * n * y)ββ) := by
apply iSupβ_mono
intro n r
apply iSupβ_mono
intro rpos rle1
norm_cast
push_cast
calc ββ« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-n * x)) * K x y * exp (I * n * y) + (starRingEnd β) (exp (I * (-n * x)) * K x y * exp (I * n * y)))ββ
_ = ββ« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-n * x)) * K x y * exp (I * n * y)) + f y * (starRingEnd β) (exp (I * (-n * x)) * K x y * exp (I * n * y))ββ := by
congr
ext y
rw [mul_add]
_ = β (β« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-n * x)) * K x y * exp (I * n * y)))
+ β« y in {y | dist x y β Set.Ioo r 1}, f y * (starRingEnd β) (exp (I * (-n * x)) * K x y * exp (I * n * y))ββ := by
congr
have measurableβ : Measurable fun x_1 β¦ (I * (-βn * βx)).exp * K x x_1 * (I * βn * βx_1).exp := by
apply Measurable.mul
apply Measurable.mul
measurability
apply Measurable.of_uncurry_left
exact Hilbert_kernel_measurable
measurability
have boundednessβ {y : β} (h : r β€ dist x y) : β(I * (-βn * βx)).exp * K x y * (I * βn * βy).expβ β€ (2 ^ (2 : β) / (2 * r)) := by
calc β(I * (-βn * βx)).exp * K x y * (I * βn * βy).expβ
_ = β(I * (-βn * βx)).expβ * βK x yβ * β(I * βn * βy).expβ := by
rw [norm_mul, norm_mul]
_ β€ 1 * (2 ^ (2 : β) / (2 * |x - y|)) * 1 := by
gcongr
. rw [norm_eq_abs, mul_comm]
norm_cast
rw [abs_exp_ofReal_mul_I]
. apply Hilbert_kernel_bound
. rw [norm_eq_abs, mul_assoc, mul_comm]
norm_cast
rw [abs_exp_ofReal_mul_I]
_ β€ (2 ^ (2 : β) / (2 * r)) := by
rw [one_mul, mul_one, β Real.dist_eq]
gcongr
have integrableβ := (integrable_annulus hx hf rpos.le rle1)
rw [MeasureTheory.integral_add]
. conv => pattern ((f _) * _); rw [mul_comm]
apply MeasureTheory.Integrable.bdd_mul' integrableβ measurableβ.aestronglyMeasurable
. rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet]
apply eventually_of_forall
intro y hy
exact boundednessβ hy.1.le
. conv => pattern ((f _) * _); rw [mul_comm]
apply MeasureTheory.Integrable.bdd_mul' integrableβ
. apply Measurable.aestronglyMeasurable
apply Measurable.comp continuous_star.measurable measurableβ
. rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet]
apply eventually_of_forall
intro y hy
rw [RCLike.norm_conj]
exact boundednessβ hy.1.le
_ β€ ββ« y in {y | dist x y β Set.Ioo r 1}, f y * (exp (I * (-n * x)) * K x y * exp (I * n * y))ββ
+ ββ« y in {y | dist x y β Set.Ioo r 1}, f y * (starRingEnd β) (exp (I * (-n * x)) * K x y * exp (I * n * y))ββ := by
apply nnnorm_add_le
_ = ββ« y in {y | dist x y β Set.Ioo r 1}, exp (I * (-n * x)) * (f y * K x y * exp (I * n * y))ββ
+ ββ« y in {y | dist x y β Set.Ioo r 1}, exp (I * (-n * x)) * (((starRingEnd β) β f) y * K x y * exp (I * n * y))ββ := by
congr 1
. congr
ext y
ring
. rw [βnnnorm_star, βstarRingEnd_apply, βintegral_conj]
congr
ext y
simp
ring
_ = ββ« y in {y | dist x y β Set.Ioo r 1}, f y * K x y * exp (I * n * y)ββ
+ ββ« y in {y | dist x y β Set.Ioo r 1}, ((starRingEnd β) β f) y * K x y * exp (I * n * y)ββ := by
rw [β NNReal.coe_inj]
push_cast
norm_cast
congr 1 <;>
. rw [MeasureTheory.integral_mul_left, norm_mul, norm_eq_abs, mul_comm I, abs_exp_ofReal_mul_I, one_mul]
_ β€ T' f x + T' ((starRingEnd β) β f) x := by
rw [CarlesonOperatorReal', CarlesonOperatorReal']
apply iSupβ_le
intro n r
apply iSupβ_le
intro rpos rle1
gcongr <;>
. apply le_iSupβ_of_le n r
apply le_iSupβ_of_le rpos rle1
trivial | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
hs : {y | dist x y β Set.Ioo 0 1} = β n, s n
this :
Tendsto (fun i => β« (y : β) in s i, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(π (β« (y : β) in β n, s n, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
hs : {y | dist x y β Set.Ioo 0 1} = β n, s n
this :
Tendsto (fun i => β« (y : β) in s i, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(π (β« (y : β) in β n, s n, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
β’ βββ« (y : β) in {y | dist x y β Set.Ioo 0 1}, f y * β(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)ββ β€
T' f x + T' (β(starRingEnd β) β f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ext y | f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
β’ {y | dist x y β Set.Ioo 0 1} = β n, s n | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
β’ {y | dist x y β Set.Ioo 0 1} = β n, s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | constructor | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . intro hy
rw [Set.mem_setOf_eq, Set.mem_Ioo] at hy
obtain β¨n, hnβ© := exists_nat_gt (1 / dist x y)
simp
use n
rw [sdef]
simp
constructor
. rw [inv_lt, inv_eq_one_div]
apply lt_trans hn
linarith
linarith
exact hy.1
. exact hy.2 | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1} | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . intro hy
simp at hy
rcases hy with β¨n, hnβ©
rw [sdef] at hn
simp at hn
constructor
. apply lt_trans' hn.1
norm_num
linarith
. exact hn.2 | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro hy | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ y β β n, s n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β {y | dist x y β Set.Ioo 0 1} β y β β n, s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [Set.mem_setOf_eq, Set.mem_Ioo] at hy | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ y β β n, s n | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
β’ y β β n, s n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : y β {y | dist x y β Set.Ioo 0 1}
β’ y β β n, s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | obtain β¨n, hnβ© := exists_nat_gt (1 / dist x y) | case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
β’ y β β n, s n | case h.mp.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β β n, s n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
β’ y β β n, s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp | case h.mp.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β β n, s n | case h.mp.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ β i, y β s i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β β n, s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | use n | case h.mp.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ β i, y β s i | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β s n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ β i, y β s i
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [sdef] | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β s n | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β (fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}) n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β s n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β (fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}) n | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ y β (fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}) n
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | constructor | case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1 | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y
case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . rw [inv_lt, inv_eq_one_div]
apply lt_trans hn
linarith
linarith
exact hy.1 | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y
case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1 | case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y
case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . exact hy.2 | case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [inv_lt, inv_eq_one_div] | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 1 / dist x y < βn + 2
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ (βn + 2)β»ΒΉ < dist x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply lt_trans hn | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 1 / dist x y < βn + 2
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ βn < βn + 2
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 1 / dist x y < βn + 2
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | linarith | case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ βn < βn + 2
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ βn < βn + 2
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | linarith | case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.ha
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < βn + 2
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | exact hy.1 | case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hb
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ 0 < dist x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | exact hy.2 | case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : 0 < dist x y β§ dist x y < 1
n : β
hn : 1 / dist x y < βn
β’ dist x y < 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro hy | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1} | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : y β β n, s n
β’ y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
β’ y β β n, s n β y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp at hy | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : y β β n, s n
β’ y β {y | dist x y β Set.Ioo 0 1} | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : β i, y β s i
β’ y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : y β β n, s n
β’ y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rcases hy with β¨n, hnβ© | case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : β i, y β s i
β’ y β {y | dist x y β Set.Ioo 0 1} | case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : y β s n
β’ y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
hy : β i, y β s i
β’ y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [sdef] at hn | case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : y β s n
β’ y β {y | dist x y β Set.Ioo 0 1} | case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : y β (fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}) n
β’ y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : y β s n
β’ y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp at hn | case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : y β (fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}) n
β’ y β {y | dist x y β Set.Ioo 0 1} | case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ y β {y | dist x y β Set.Ioo 0 1} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : y β (fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}) n
β’ y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | constructor | case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ y β {y | dist x y β Set.Ioo 0 1} | case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < dist x y
case h.mpr.intro.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ dist x y < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ y β {y | dist x y β Set.Ioo 0 1}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . apply lt_trans' hn.1
norm_num
linarith | case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < dist x y
case h.mpr.intro.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ dist x y < 1 | case h.mpr.intro.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ dist x y < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < dist x y
case h.mpr.intro.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ dist x y < 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . exact hn.2 | case h.mpr.intro.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ dist x y < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.right
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ dist x y < 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply lt_trans' hn.1 | case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < dist x y | case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < (βn + 2)β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < dist x y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | norm_num | case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < (βn + 2)β»ΒΉ | case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < βn + 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.left
f : β β β
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : β
x : β
hx : x β Set.Icc 0 (2 * Real.pi)
s : β β Set β := fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
sdef : s = fun n => {y | dist x y β Set.Ioo (1 / (βn + 2)) 1}
y : β
n : β
hn : (βn + 2)β»ΒΉ < dist x y β§ dist x y < 1
β’ 0 < (βn + 2)β»ΒΉ
TACTIC:
|
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