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 | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply Measurable.comp continuous_star.measurable measurable₁ | case e_a.hg.hf.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet] | case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s... | case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply eventually_of_forall | case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s... | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro y hy | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n... | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [RCLike.norm_conj] | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n... | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | exact boundedness₁ hy.1.le | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply nnnorm_add_le | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr 1 | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . congr
ext y
ring | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . rw [←nnnorm_star, ←starRingEnd_apply, ←integral_conj]
congr
ext y
simp
ring | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
t... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ext y | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
t... | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ring | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [←nnnorm_star, ←starRingEnd_apply, ←integral_conj] | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
t... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ext y | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
t... | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n... | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ring | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [← NNReal.coe_inj] | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | push_cast | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | norm_cast | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr 1 <;>
. rw [MeasureTheory.integral_mul_left, norm_mul, norm_eq_abs, mul_comm I, abs_exp_ofReal_mul_I, one_mul] | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [MeasureTheory.integral_mul_left, norm_mul, norm_eq_abs, mul_comm I, abs_exp_ofReal_mul_I, one_mul] | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [CarlesonOperatorReal', CarlesonOperatorReal'] | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply iSup₂_le | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Te... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro n r | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Te... | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Te... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply iSup₂_le | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Te... | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro rpos rle1 | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | gcongr <;>
. apply le_iSup₂_of_le n r
apply le_iSup₂_of_le rpos rle1
trivial | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply le_iSup₂_of_le n r | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :... | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply le_iSup₂_of_le rpos rle1 | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :... | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | trivial | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | calc ε * MeasureTheory.volume E
_ = ∫⁻ _ in E, ε := by
symm
apply MeasureTheory.set_lintegral_const
_ ≤ ∫⁻ x in E, T' f x := by
apply MeasureTheory.set_lintegral_mono' measurableSetE hE
_ = ENNReal.ofReal δ * ∫⁻ x in E, T' (fun x ↦ (1 / δ) * f x) x := by
rw [← MeasureTheory.lintegral_const_mul']
... | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | symm | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E = ∫⁻ (x : ℝ) in E, ε | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε = ε * MeasureTheory.volume E | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E = ∫⁻ (x : ℝ) in E, ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | apply MeasureTheory.set_lintegral_const | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε = ε * MeasureTheory.volume E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε = ε * MeasureTheory.volume E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | apply MeasureTheory.set_lintegral_mono' measurableSetE hE | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε ≤ ∫⁻ (x : ℝ) in E, T' f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε ≤ ∫⁻ (x : ℝ) in E, T' f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [← MeasureTheory.lintegral_const_mul'] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ∫⁻ (a : ℝ) in E, ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f :... | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E,... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | congr | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ∫⁻ (a : ℝ) in E, ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f :... | case e_f
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ (fun x => T' f x) = fun a => ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ∫⁻ (a : ℝ) in E, ENNReal.ofReal δ *... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | ext x | case e_f
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ (fun x => T' f x) = fun a => ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ... | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ T' f x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ (fun x => T' f x) = fun a => ENNReal.ofReal δ * T' (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [CarlesonOperatorReal'_mul δpos] | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ T' f x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
... | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ↑δ.toNNReal * T' (fun x => 1 / ↑δ * f x) x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ T' f x = ENNReal.ofReal δ * T' (fun x => 1 /... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | congr | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ↑δ.toNNReal * T' (fun x => 1 / ↑δ * f x) x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ :... | case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ↑δ.toNNReal * T' (fun x => 1 / ↑δ * f x) x =... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | exact ENNReal.ofReal_ne_top | case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | gcongr | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * Mea... | case bc
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * Mea... | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | apply rcarleson' measurableSetF measurableSetE | case bc
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * Mea... | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∀ (x : ℝ), ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
E... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | intro x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∀ (x : ℝ), ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∀ (x : ℝ), ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x
TACTIC... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | simp | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ |δ|⁻¹ * Complex.abs (f x) ≤ F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [_root_.abs_of_nonneg δpos.le, inv_mul_le_iff δpos] | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ |δ|⁻¹ * Complex.abs (f x) ≤ F.indicator 1 x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ Complex.abs (f x) ≤ δ * F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ |δ|⁻¹ * Complex.abs (f x) ≤ F.indicator 1 x
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | exact hf x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ Complex.abs (f x) ≤ δ * F.indicator 1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ Complex.abs (f x) ≤ δ * F.indicator 1 x
TACT... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [ENNReal.ofReal_mul δpos.le] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal... | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal... | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | ring | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | rw [ENNReal.ofReal_mul (by apply mul_nonneg δpos.le; rw [C1_2]; norm_num), ← ENNReal.ofReal_rpow_of_pos (by linarith [Real.pi_pos])] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤ ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E ^ 2⁻¹ | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * ENNReal.ofReal (4 * Real.pi) ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤ ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.p... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | convert rcarleson_exceptional_set_estimate δpos measurableSet_Icc hf measurableSetE hE | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * ENNReal.ofReal (4 * Real.pi) ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = MeasureTheory.volume (Set.Icc (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * ENNRea... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | rw [Real.volume_Icc] | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = MeasureTheory.volume (Set.Icc (-Real.pi) (3 * Real.pi)) | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = ENNReal.ofReal (3 * Real.pi - -Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = Measure... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | ring_nf | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = ENNReal.ofReal (3 * Real.pi - -Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = ENNReal... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | apply mul_nonneg δpos.le | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ δ * C1_2 4 2 | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ C1_2 4 2 | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ δ * C1_2 4 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | rw [C1_2] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ C1_2 4 2 | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ C1_2 4 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | norm_num | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | linarith [Real.pi_pos] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 < 4 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 < 4 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | rw [C_control_approximation_effect] | ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C_control_approximation_effect ε | ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C_control_approximation_effect ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply lt_add_of_pos_of_le _ (by rfl) | ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi | ε : ℝ
εpos : 0 < ε
⊢ 0 < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply mul_pos (C1_2_pos (by norm_num)) | ε : ℝ
εpos : 0 < ε
⊢ 0 < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | ε : ℝ
εpos : 0 < ε
⊢ 0 < (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 0 < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply Real.rpow_pos_of_pos | ε : ℝ
εpos : 0 < ε
⊢ 0 < (8 / (Real.pi * ε)) ^ 2⁻¹ | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 / (Real.pi * ε) | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 0 < (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply div_pos (by norm_num) | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 / (Real.pi * ε) | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < Real.pi * ε | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 / (Real.pi * ε)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply mul_pos Real.pi_pos εpos | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < Real.pi * ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < Real.pi * ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | rfl | ε : ℝ
εpos : 0 < ε
⊢ Real.pi ≤ Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ Real.pi ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | norm_num | ε : ℝ
εpos : 0 < ε
⊢ 1 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 1 < 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | norm_num | ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | symm | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C_control_approximation_effect ε * δ = δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ = C_control_approximation_effect ε * δ | Please generate a tactic in lean4 to solve the state.
STATE:
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C_control_approximation_effect ε * δ = δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [C_control_approximation_effect, mul_comm, mul_div_right_comm, mul_comm δ, mul_assoc, mul_comm δ, ← mul_assoc, ← mul_assoc, ← add_mul] | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ = C_control_approximation_effect ε * δ | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ ((2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ + Real.pi) * δ =
(C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi) * δ | Please generate a tactic in lean4 to solve the state.
STATE:
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ = C_control_approximation_effect ε * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | congr 2 | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ ((2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ + Real.pi) * δ =
(C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi) * δ | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ ((2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ + Real.pi) * δ =
(C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi) * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [mul_comm _ (C1_2 4 2), mul_assoc] | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C1_2 4 2 * ((2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹) = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | congr | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C1_2 4 2 * ((2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹) = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹ = (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C1_2 4 2 * ((2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹) = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [Real.div_rpow, Real.div_rpow _ (mul_nonneg _ _), Real.mul_rpow, Real.mul_rpow] | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹ = (8 / (Real.pi * ε)) ^ 2⁻¹ | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ 2⁻¹ / ε ^ 2⁻¹ / Real.pi * (4 ^ 2⁻¹ * Real.pi ^ 2⁻¹) = 8 ^ 2⁻¹ / (Real.pi ^ 2⁻¹ * ε ^ 2⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹ = (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | ring_nf | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ 2⁻¹ / ε ^ 2⁻¹ / Real.pi * (4 ^ 2⁻¹ * Real.pi ^ 2⁻¹) = 8 ^ 2⁻¹ / (Real.pi ^ 2⁻¹ * ε ^ 2⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.... | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ (1 / 2) * (ε ^ (1 / 2))⁻¹ * Real.pi⁻¹ * 4 ^ (1 / 2) * Real.pi ^ (1 / 2) =
(ε ^ (1 / 2))⁻¹ * 8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ 2⁻¹ / ε ^ 2⁻¹ / Real.pi * (4 ^ 2⁻¹ * Real.pi ^ 2⁻¹) = 8 ^ 2⁻¹ / (Real.pi ^ 2⁻¹ * ε ^ 2⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [mul_assoc, mul_comm (2 ^ _), mul_assoc, mul_assoc, mul_assoc, mul_comm (4 ^ _), ← mul_assoc Real.pi⁻¹,
← Real.rpow_neg_one Real.pi, ← Real.rpow_add, mul_comm (Real.pi ^ _), ← mul_assoc (2 ^ _), ← Real.mul_rpow] | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ (1 / 2) * (ε ^ (1 / 2))⁻¹ * Real.pi⁻¹ * 4 ^ (1 / 2) * Real.pi ^ (1 / 2) =
(ε ^ (1 / 2))⁻¹ * 8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ :... | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (ε ^ (1 / 2))⁻¹ * ((2 * 4) ^ (1 / 2) * Real.pi ^ (-1 + 1 / 2)) =
(ε ^ (1 / 2))⁻¹ * (8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ (1 / 2) * (ε ^ (1 / 2))⁻¹ * Real.pi⁻¹ * 4 ^ (1 / 2) * Real.pi ^ (1 / 2) =
(ε ^ (1 / 2))⁻¹ * 8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.h... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | congr | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (ε ^ (1 / 2))⁻¹ * ((2 * 4) ^ (1 / 2) * Real.pi ^ (-1 + 1 / 2)) =
(ε ^ (1 / 2))⁻¹ * (8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤... | case e_a.e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 * 4 = 8
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (ε ^ (1 / 2))⁻¹ * ((2 * 4) ^ (1 / 2) * Real.pi ^ (-1 + 1 / 2)) =
(ε ^ (1 / 2))⁻¹ * (8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonne... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | norm_num | case e_a.e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 * 4 = 8
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
... | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ R... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 * 4 = 8
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | ring_nf | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ R... | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [neg_div, Real.rpow_neg] | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.... | case e_a.e_a.e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | all_goals linarith [Real.pi_pos] | case e_a.e_a.e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | linarith [Real.pi_pos] | case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set ε' := C_control_approximation_effect ε * δ with ε'def | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Se... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
⊢ ∃ E ⊆ Set.Ic... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set E := {x ∈ Set.Icc 0 (2 * Real.pi) | ∃ N, ε' < abs (partialFourierSum h N x)} with Edef | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
⊢ ∃ E ⊆ Set.Ic... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'd... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have E_eq: E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N : ℕ, {x | ε' < ‖partialFourierSum h N x‖} := by
rw [Edef]
ext x
simp | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'd... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have measurableSetE : MeasurableSet E := by
rw [E_eq]
apply MeasurableSet.inter
. apply measurableSet_Icc
apply MeasurableSet.iUnion
intro N
apply measurableSet_lt
. apply measurable_const
apply Measurable.norm
apply partialFourierSum_uniformContinuous.continuous.measurable | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'd... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | use E | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {... | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'd... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | constructor | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set... | case h.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . intro x hx
rw [Edef] at hx
simp at hx
exact hx.1 | case h.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E ... | case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effe... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | use measurableSetE | case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E... | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_eff... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [and_comm] | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E :... | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effec... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | constructor | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E :... | case right.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * ... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effec... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . rw [Edef]
simp
exact fun x x_nonneg x_le_two_pi h ↦ h x_nonneg x_le_two_pi | case right.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * ... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε *... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi) := by
apply @IntervalIntegrable.mono_fun' _ _ _ _ _ _ (fun _ ↦ δ)
apply intervalIntegrable_const
exact h_measurable.aestronglyMeasurable
rw [Filter.EventuallyLE, ae_restrict_iff_subtype]
apply Filter.eventually_of_f... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε *... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε *... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set F := Set.Icc (-Real.pi) (3 * Real.pi) with Fdef | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε *... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set f := fun x ↦ h x * F.indicator 1 x with fdef | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have f_measurable : Measurable f := by
apply Measurable.mul h_measurable
apply Measurable.indicator measurable_const measurableSet_Icc | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi) := by
rw [fdef, intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith [Real.pi_pos])]
conv => pattern (h _) * _; rw [mul_comm]
apply MeasureTheory.Integrable.bdd_mul'
rwa [← MeasureTheory.IntegrableOn, ← intervalIntegrab... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' ((starRingEnd ℂ) ∘ f) x := by
have h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi) := by
apply h_intervalIntegrable.mono_set
rw [Set.uIcc_of_le (by linarith [Real.pi_pos]), Set.u... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have Evolume : MeasureTheory.volume E < ⊤ := by
calc MeasureTheory.volume E
_ ≤ MeasureTheory.volume (Set.Icc 0 (2 * Real.pi)) := by
apply MeasureTheory.measure_mono
rw [E_eq]
apply Set.inter_subset_left
_ = ENNReal.ofReal (2 * Real.pi) := by
rw [Real.volume_Icc, sub_zero]
_ < ⊤ :=... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | obtain ⟨E', E'subset, measurableSetE', E'measure, h⟩ := ENNReal.le_on_subset MeasureTheory.volume measurableSetE (CarlesonOperatorReal'_measurable f_measurable) (CarlesonOperatorReal'_measurable (Measurable.comp continuous_star.measurable f_measurable)) le_operator_add | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs... | case right.right.intro.intro.intro.intro
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ... |
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