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/ToMathlib/MeasureReal.lean | MeasureTheory.ext_iff_measureReal_singleton | [467, 1] | [476, 16] | rw [measureReal_def, measureReal_def, ENNReal.toReal_eq_toReal_iff] | case a.h.a
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
h2 : μ2 {x} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1.real {x} = μ2.real {x} | case a.h.a
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
h2 : μ2 {x} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1 {x} = μ2 {x} ∨ μ1 {x} = 0 ∧ μ2 {x} = ⊤ ∨ μ1 {x} = ⊤ ∧ μ2 {x} = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
h2 : μ2 {x} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1.real {x} = μ2.real {x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.ext_iff_measureReal_singleton | [467, 1] | [476, 16] | simp [h1, h2] | case a.h.a
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
h2 : μ2 {x} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1 {x} = μ2 {x} ∨ μ1 {x} = 0 ∧ μ2 {x} = ⊤ ∨ μ1 {x} = ⊤ ∧ μ2 {x} = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
h2 : μ2 {x} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1 {x} = μ2 {x} ∨ μ1 {x} = 0 ∧ μ2 {x} = ⊤ ∨ μ1 {x} = ⊤ ∧ μ2 {x} = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.ext_iff_measureReal_singleton | [467, 1] | [476, 16] | finiteness | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
⊢ μ1 {x} ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
⊢ μ1 {x} ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.ext_iff_measureReal_singleton | [467, 1] | [476, 16] | finiteness | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
⊢ μ2 {x} ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝⁴ : Fintype S
inst✝³ : MeasurableSpace S
inst✝² : MeasurableSingletonClass S
μ1 μ2 : Measure S
inst✝¹ : IsFiniteMeasure μ1
inst✝ : IsFiniteMeasure μ2
x : S
h1 : μ1 {x} ≠ ⊤
⊢ μ2 {x} ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | set Ef := E ∩ f⁻¹' (Set.Ici (a / 2)) with Ef_def | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | set Eg := E ∩ g⁻¹' (Set.Ici (a / 2)) with Eg_def | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | have : E ⊆ Ef ∪ Eg := by
intro x hx
rw [Ef_def, Eg_def]
simp
by_contra hx'
push_neg at hx'
absurd le_refl a
push_neg
calc a
_ ≤ f x + g x := h x hx
_ < a / 2 + a / 2 := by
gcongr
. exact hx'.1 hx
. exact hx'.2 hx
_ = a := by
ring_nf
apply ENNReal.div_mul_cancel <;> norm_num | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | have : μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg := by
by_contra hEfg
push_neg at hEfg
absurd le_refl (2 * μ E)
push_neg
calc 2 * μ E
_ ≤ 2 * μ (Ef ∪ Eg) := by
gcongr
_ ≤ 2 * (μ Ef + μ Eg) := by
gcongr
apply MeasureTheory.measure_union_le
_ = 2 * μ Ef + 2 * μ Eg := by ring
_ < μ E + μ E := by
gcongr
. exact hEfg.1
. exact hEfg.2
_ = 2 * μ E := by ring | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this✝ : E ⊆ Ef ∪ Eg
this : μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | rcases this with hEf | hEg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this✝ : E ⊆ Ef ∪ Eg
this : μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | case inl
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this✝ : E ⊆ Ef ∪ Eg
this : μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . use Ef
constructor
. apply Set.inter_subset_left
constructor
. apply MeasurableSet.inter hE
apply hf measurableSet_Ici
use hEf
left
rw [Ef_def]
simp | case inl
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . use Eg
constructor
. apply Set.inter_subset_left
constructor
. apply MeasurableSet.inter hE
apply hg measurableSet_Ici
use hEg
right
rw [Eg_def]
simp | case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | intro x hx | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
⊢ E ⊆ Ef ∪ Eg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ Ef ∪ Eg | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
⊢ E ⊆ Ef ∪ Eg
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | rw [Ef_def, Eg_def] | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ Ef ∪ Eg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ E ∩ f ⁻¹' Set.Ici (a / 2) ∪ E ∩ g ⁻¹' Set.Ici (a / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ Ef ∪ Eg
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | simp | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ E ∩ f ⁻¹' Set.Ici (a / 2) ∪ E ∩ g ⁻¹' Set.Ici (a / 2) | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ E ∧ a / 2 ≤ f x ∨ x ∈ E ∧ a / 2 ≤ g x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ E ∩ f ⁻¹' Set.Ici (a / 2) ∪ E ∩ g ⁻¹' Set.Ici (a / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | by_contra hx' | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ E ∧ a / 2 ≤ f x ∨ x ∈ E ∧ a / 2 ≤ g x | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : ¬(x ∈ E ∧ a / 2 ≤ f x ∨ x ∈ E ∧ a / 2 ≤ g x)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
⊢ x ∈ E ∧ a / 2 ≤ f x ∨ x ∈ E ∧ a / 2 ≤ g x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | push_neg at hx' | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : ¬(x ∈ E ∧ a / 2 ≤ f x ∨ x ∈ E ∧ a / 2 ≤ g x)
⊢ False | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : ¬(x ∈ E ∧ a / 2 ≤ f x ∨ x ∈ E ∧ a / 2 ≤ g x)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | absurd le_refl a | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ False | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ ¬a ≤ a | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | push_neg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ ¬a ≤ a | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a < a | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ ¬a ≤ a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | calc a
_ ≤ f x + g x := h x hx
_ < a / 2 + a / 2 := by
gcongr
. exact hx'.1 hx
. exact hx'.2 hx
_ = a := by
ring_nf
apply ENNReal.div_mul_cancel <;> norm_num | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a < a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a < a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | gcongr | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x + g x < a / 2 + a / 2 | case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x < a / 2
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x + g x < a / 2 + a / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . exact hx'.1 hx | case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x < a / 2
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2 | case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x < a / 2
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . exact hx'.2 hx | case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | exact hx'.1 hx | case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x < a / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ f x < a / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | exact hx'.2 hx | case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ g x < a / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | ring_nf | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a / 2 + a / 2 = a | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a / 2 * 2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a / 2 + a / 2 = a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply ENNReal.div_mul_cancel <;> norm_num | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a / 2 * 2 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
x : X
hx : x ∈ E
hx' : (x ∈ E → f x < a / 2) ∧ (x ∈ E → g x < a / 2)
⊢ a / 2 * 2 = a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | by_contra hEfg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
⊢ μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : ¬(μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
⊢ μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | push_neg at hEfg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : ¬(μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg)
⊢ False | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : ¬(μ E ≤ 2 * μ Ef ∨ μ E ≤ 2 * μ Eg)
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | absurd le_refl (2 * μ E) | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ False | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ ¬2 * μ E ≤ 2 * μ E | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | push_neg | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ ¬2 * μ E ≤ 2 * μ E | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ E < 2 * μ E | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ ¬2 * μ E ≤ 2 * μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | calc 2 * μ E
_ ≤ 2 * μ (Ef ∪ Eg) := by
gcongr
_ ≤ 2 * (μ Ef + μ Eg) := by
gcongr
apply MeasureTheory.measure_union_le
_ = 2 * μ Ef + 2 * μ Eg := by ring
_ < μ E + μ E := by
gcongr
. exact hEfg.1
. exact hEfg.2
_ = 2 * μ E := by ring | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ E < 2 * μ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ E < 2 * μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | gcongr | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ E ≤ 2 * μ (Ef ∪ Eg) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ E ≤ 2 * μ (Ef ∪ Eg)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | gcongr | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ (Ef ∪ Eg) ≤ 2 * (μ Ef + μ Eg) | case bc
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ μ (Ef ∪ Eg) ≤ μ Ef + μ Eg | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ (Ef ∪ Eg) ≤ 2 * (μ Ef + μ Eg)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply MeasureTheory.measure_union_le | case bc
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ μ (Ef ∪ Eg) ≤ μ Ef + μ Eg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ μ (Ef ∪ Eg) ≤ μ Ef + μ Eg
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | ring | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * (μ Ef + μ Eg) = 2 * μ Ef + 2 * μ Eg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * (μ Ef + μ Eg) = 2 * μ Ef + 2 * μ Eg
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | gcongr | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef + 2 * μ Eg < μ E + μ E | case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef < μ E
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef + 2 * μ Eg < μ E + μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . exact hEfg.1 | case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef < μ E
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E | case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E | Please generate a tactic in lean4 to solve the state.
STATE:
case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef < μ E
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . exact hEfg.2 | case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | exact hEfg.1 | case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef < μ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ac
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Ef < μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | exact hEfg.2 | case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bd
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ 2 * μ Eg < μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | ring | X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ μ E + μ E = 2 * μ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEfg : 2 * μ Ef < μ E ∧ 2 * μ Eg < μ E
⊢ μ E + μ E = 2 * μ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | use Ef | case inl
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | case h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E ∧ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | constructor | case h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E ∧ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E ∧ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . apply Set.inter_subset_left | case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | constructor | case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef ∧ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . apply MeasurableSet.inter hE
apply hf measurableSet_Ici | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | use hEf | case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x) | case right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ (∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ μ E ≤ 2 * μ Ef ∧ ((∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | left | case right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ (∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∀ x ∈ Ef, a / 2 ≤ f x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ (∀ x ∈ Ef, a / 2 ≤ f x) ∨ ∀ x ∈ Ef, a / 2 ≤ g x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | rw [Ef_def] | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∀ x ∈ Ef, a / 2 ≤ f x | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∀ x ∈ E ∩ f ⁻¹' Set.Ici (a / 2), a / 2 ≤ f x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∀ x ∈ Ef, a / 2 ≤ f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | simp | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∀ x ∈ E ∩ f ⁻¹' Set.Ici (a / 2), a / 2 ≤ f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ ∀ x ∈ E ∩ f ⁻¹' Set.Ici (a / 2), a / 2 ≤ f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply Set.inter_subset_left | case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ Ef ⊆ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply MeasurableSet.inter hE | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet (f ⁻¹' Set.Ici (a / 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet Ef
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply hf measurableSet_Ici | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet (f ⁻¹' Set.Ici (a / 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEf : μ E ≤ 2 * μ Ef
⊢ MeasurableSet (f ⁻¹' Set.Ici (a / 2))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | use Eg | case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x) | case h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E ∧ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∃ E' ⊆ E, MeasurableSet E' ∧ μ E ≤ 2 * μ E' ∧ ((∀ x ∈ E', a / 2 ≤ f x) ∨ ∀ x ∈ E', a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | constructor | case h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E ∧ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E ∧ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . apply Set.inter_subset_left | case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | constructor | case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg ∧ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | . apply MeasurableSet.inter hE
apply hg measurableSet_Ici | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | use hEg | case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x) | case right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ (∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ μ E ≤ 2 * μ Eg ∧ ((∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | right | case right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ (∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∀ x ∈ Eg, a / 2 ≤ g x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ (∀ x ∈ Eg, a / 2 ≤ f x) ∨ ∀ x ∈ Eg, a / 2 ≤ g x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | rw [Eg_def] | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∀ x ∈ Eg, a / 2 ≤ g x | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∀ x ∈ E ∩ g ⁻¹' Set.Ici (a / 2), a / 2 ≤ g x | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∀ x ∈ Eg, a / 2 ≤ g x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | simp | case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∀ x ∈ E ∩ g ⁻¹' Set.Ici (a / 2), a / 2 ≤ g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ ∀ x ∈ E ∩ g ⁻¹' Set.Ici (a / 2), a / 2 ≤ g x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply Set.inter_subset_left | case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ Eg ⊆ E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply MeasurableSet.inter hE | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet (g ⁻¹' Set.Ici (a / 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet Eg
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | ENNReal.le_on_subset | [19, 1] | [78, 9] | apply hg measurableSet_Ici | case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet (g ⁻¹' Set.Ici (a / 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.left
X : Type
inst✝ : MeasurableSpace X
μ : MeasureTheory.Measure X
f g : X → ℝ≥0∞
E : Set X
hE : MeasurableSet E
hf : Measurable f
hg : Measurable g
a : ℝ≥0∞
h : ∀ x ∈ E, a ≤ f x + g x
Ef : Set X := E ∩ f ⁻¹' Set.Ici (a / 2)
Ef_def : Ef = E ∩ f ⁻¹' Set.Ici (a / 2)
Eg : Set X := E ∩ g ⁻¹' Set.Ici (a / 2)
Eg_def : Eg = E ∩ g ⁻¹' Set.Ici (a / 2)
this : E ⊆ Ef ∪ Eg
hEg : μ E ≤ 2 * μ Eg
⊢ MeasurableSet (g ⁻¹' Set.Ici (a / 2))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | intervalIntegral.integral_conj' | [84, 1] | [89, 7] | rw [intervalIntegral_eq_integral_uIoc, integral_conj, intervalIntegral_eq_integral_uIoc,
RCLike.real_smul_eq_coe_mul, RCLike.real_smul_eq_coe_mul, map_mul] | μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ∫ (x : ℝ) in a..b, (starRingEnd 𝕜) (f x) ∂μ = (starRingEnd 𝕜) (∫ (x : ℝ) in a..b, f x ∂μ) | μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ↑(if a ≤ b then 1 else -1) * (starRingEnd 𝕜) (∫ (x : ℝ) in Ι a b, f x ∂μ) =
(starRingEnd 𝕜) ↑(if a ≤ b then 1 else -1) * (starRingEnd 𝕜) (∫ (x : ℝ) in Ι a b, f x ∂μ) | Please generate a tactic in lean4 to solve the state.
STATE:
μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ∫ (x : ℝ) in a..b, (starRingEnd 𝕜) (f x) ∂μ = (starRingEnd 𝕜) (∫ (x : ℝ) in a..b, f x ∂μ)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | intervalIntegral.integral_conj' | [84, 1] | [89, 7] | congr | μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ↑(if a ≤ b then 1 else -1) * (starRingEnd 𝕜) (∫ (x : ℝ) in Ι a b, f x ∂μ) =
(starRingEnd 𝕜) ↑(if a ≤ b then 1 else -1) * (starRingEnd 𝕜) (∫ (x : ℝ) in Ι a b, f x ∂μ) | case e_a
μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ↑(if a ≤ b then 1 else -1) = (starRingEnd 𝕜) ↑(if a ≤ b then 1 else -1) | Please generate a tactic in lean4 to solve the state.
STATE:
μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ↑(if a ≤ b then 1 else -1) * (starRingEnd 𝕜) (∫ (x : ℝ) in Ι a b, f x ∂μ) =
(starRingEnd 𝕜) ↑(if a ≤ b then 1 else -1) * (starRingEnd 𝕜) (∫ (x : ℝ) in Ι a b, f x ∂μ)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | intervalIntegral.integral_conj' | [84, 1] | [89, 7] | simp | case e_a
μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ↑(if a ≤ b then 1 else -1) = (starRingEnd 𝕜) ↑(if a ≤ b then 1 else -1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
μ : MeasureTheory.Measure ℝ
𝕜 : Type
inst✝ : RCLike 𝕜
f : ℝ → 𝕜
a b : ℝ
⊢ ↑(if a ≤ b then 1 else -1) = (starRingEnd 𝕜) ↑(if a ≤ b then 1 else -1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | rw [dirichletKernel', K, k] | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) =
cexp (I * (-↑N * ↑x)) * K x y * cexp (I * ↑N * ↑y) +
(starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * K x y * cexp (I * ↑N * ↑y)) | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * ↑(x - y)) / (1 - cexp (-I * ↑(x - y))) + cexp (-I * ↑N * ↑(x - y)) / (1 - cexp (I * ↑(x - y)))) =
cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y) +
(starRingEnd ℂ)
(cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) =
cexp (I * (-↑N * ↑x)) * K x y * cexp (I * ↑N * ↑y) +
(starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * K x y * cexp (I * ↑N * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | rw [map_mul, map_mul, map_div₀, conj_ofReal, map_sub, ←exp_conj, ←exp_conj, ←exp_conj,
map_mul, map_mul, map_mul, map_mul, map_mul, conj_ofReal, conj_ofReal, conj_ofReal] | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * ↑(x - y)) / (1 - cexp (-I * ↑(x - y))) + cexp (-I * ↑N * ↑(x - y)) / (1 - cexp (I * ↑(x - y)))) =
cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y) +
(starRingEnd ℂ)
(cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y)) | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * ↑(x - y)) / (1 - cexp (-I * ↑(x - y))) + cexp (-I * ↑N * ↑(x - y)) / (1 - cexp (I * ↑(x - y)))) =
cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y) +
cexp ((starRingEnd ℂ) I * ((starRingEnd ℂ) (-↑N) * ↑x)) *
(↑(max (1 - |x - y|) 0) / ((starRingEnd ℂ) 1 - cexp ((starRingEnd ℂ) I * ↑(x - y)))) *
cexp ((starRingEnd ℂ) I * (starRingEnd ℂ) ↑N * ↑y) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * ↑(x - y)) / (1 - cexp (-I * ↑(x - y))) + cexp (-I * ↑N * ↑(x - y)) / (1 - cexp (I * ↑(x - y)))) =
cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y) +
(starRingEnd ℂ)
(cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | simp | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * ↑(x - y)) / (1 - cexp (-I * ↑(x - y))) + cexp (-I * ↑N * ↑(x - y)) / (1 - cexp (I * ↑(x - y)))) =
cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y) +
cexp ((starRingEnd ℂ) I * ((starRingEnd ℂ) (-↑N) * ↑x)) *
(↑(max (1 - |x - y|) 0) / ((starRingEnd ℂ) 1 - cexp ((starRingEnd ℂ) I * ↑(x - y)))) *
cexp ((starRingEnd ℂ) I * (starRingEnd ℂ) ↑N * ↑y) | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) =
cexp (-(I * (↑N * ↑x))) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * (↑x - ↑y)))) * cexp (I * ↑N * ↑y) +
cexp (I * (↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (-(I * (↑x - ↑y))))) * cexp (-(I * ↑N * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * ↑(x - y)) / (1 - cexp (-I * ↑(x - y))) + cexp (-I * ↑N * ↑(x - y)) / (1 - cexp (I * ↑(x - y)))) =
cexp (I * (-↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * ↑(x - y)))) * cexp (I * ↑N * ↑y) +
cexp ((starRingEnd ℂ) I * ((starRingEnd ℂ) (-↑N) * ↑x)) *
(↑(max (1 - |x - y|) 0) / ((starRingEnd ℂ) 1 - cexp ((starRingEnd ℂ) I * ↑(x - y)))) *
cexp ((starRingEnd ℂ) I * (starRingEnd ℂ) ↑N * ↑y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | field_simp | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) =
cexp (-(I * (↑N * ↑x))) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * (↑x - ↑y)))) * cexp (I * ↑N * ↑y) +
cexp (I * (↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (-(I * (↑x - ↑y))))) * cexp (-(I * ↑N * ↑y)) | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) =
cexp (-(I * (↑N * ↑x))) * ↑(max (1 - |x - y|) 0) * cexp (I * ↑N * ↑y) / (1 - cexp (I * (↑x - ↑y))) +
cexp (I * (↑N * ↑x)) * ↑(max (1 - |x - y|) 0) * cexp (-(I * ↑N * ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) =
cexp (-(I * (↑N * ↑x))) * (↑(max (1 - |x - y|) 0) / (1 - cexp (I * (↑x - ↑y)))) * cexp (I * ↑N * ↑y) +
cexp (I * (↑N * ↑x)) * (↑(max (1 - |x - y|) 0) / (1 - cexp (-(I * (↑x - ↑y))))) * cexp (-(I * ↑N * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | symm | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) =
cexp (-(I * (↑N * ↑x))) * ↑(max (1 - |x - y|) 0) * cexp (I * ↑N * ↑y) / (1 - cexp (I * (↑x - ↑y))) +
cexp (I * (↑N * ↑x)) * ↑(max (1 - |x - y|) 0) * cexp (-(I * ↑N * ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) | N : ℕ
x y : ℝ
⊢ cexp (-(I * (↑N * ↑x))) * ↑(max (1 - |x - y|) 0) * cexp (I * ↑N * ↑y) / (1 - cexp (I * (↑x - ↑y))) +
cexp (I * (↑N * ↑x)) * ↑(max (1 - |x - y|) 0) * cexp (-(I * ↑N * ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) =
↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) =
cexp (-(I * (↑N * ↑x))) * ↑(max (1 - |x - y|) 0) * cexp (I * ↑N * ↑y) / (1 - cexp (I * (↑x - ↑y))) +
cexp (I * (↑N * ↑x)) * ↑(max (1 - |x - y|) 0) * cexp (-(I * ↑N * ↑y)) / (1 - cexp (-(I * (↑x - ↑y))))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | rw [mul_comm, ←mul_assoc, ←exp_add, mul_comm, add_comm, mul_comm, ←mul_assoc, ←exp_add, mul_comm, mul_add, mul_div_assoc, mul_div_assoc] | N : ℕ
x y : ℝ
⊢ cexp (-(I * (↑N * ↑x))) * ↑(max (1 - |x - y|) 0) * cexp (I * ↑N * ↑y) / (1 - cexp (I * (↑x - ↑y))) +
cexp (I * (↑N * ↑x)) * ↑(max (1 - |x - y|) 0) * cexp (-(I * ↑N * ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) =
↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) * (cexp (-(I * ↑N * ↑y) + I * (↑N * ↑x)) / (1 - cexp (-(I * (↑x - ↑y))))) +
↑(max (1 - |x - y|) 0) * (cexp (I * ↑N * ↑y + -(I * (↑N * ↑x))) / (1 - cexp (I * (↑x - ↑y)))) =
↑(max (1 - |x - y|) 0) * (cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y))))) +
↑(max (1 - |x - y|) 0) * (cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ cexp (-(I * (↑N * ↑x))) * ↑(max (1 - |x - y|) 0) * cexp (I * ↑N * ↑y) / (1 - cexp (I * (↑x - ↑y))) +
cexp (I * (↑N * ↑x)) * ↑(max (1 - |x - y|) 0) * cexp (-(I * ↑N * ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) =
↑(max (1 - |x - y|) 0) *
(cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y)))) +
cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y))))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | dirichlet_Hilbert_eq | [92, 1] | [101, 17] | congr <;> ring | N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) * (cexp (-(I * ↑N * ↑y) + I * (↑N * ↑x)) / (1 - cexp (-(I * (↑x - ↑y))))) +
↑(max (1 - |x - y|) 0) * (cexp (I * ↑N * ↑y + -(I * (↑N * ↑x))) / (1 - cexp (I * (↑x - ↑y)))) =
↑(max (1 - |x - y|) 0) * (cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y))))) +
↑(max (1 - |x - y|) 0) * (cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y)))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
N : ℕ
x y : ℝ
⊢ ↑(max (1 - |x - y|) 0) * (cexp (-(I * ↑N * ↑y) + I * (↑N * ↑x)) / (1 - cexp (-(I * (↑x - ↑y))))) +
↑(max (1 - |x - y|) 0) * (cexp (I * ↑N * ↑y + -(I * (↑N * ↑x))) / (1 - cexp (I * (↑x - ↑y)))) =
↑(max (1 - |x - y|) 0) * (cexp (I * ↑N * (↑x - ↑y)) / (1 - cexp (-(I * (↑x - ↑y))))) +
↑(max (1 - |x - y|) 0) * (cexp (-(I * ↑N * (↑x - ↑y))) / (1 - cexp (I * (↑x - ↑y))))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | apply le_of_forall_lt | α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ a ≤ iSup f | case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ ∀ c < a, c < iSup f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ a ≤ iSup f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | intro c hc | case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ ∀ c < a, c < iSup f | case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
⊢ c < iSup f | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
⊢ ∀ c < a, c < iSup f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | have : ∀ᶠ (x : β) in atTop, c < f x := by
apply eventually_gt_of_tendsto_gt hc ha | case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
⊢ c < iSup f | case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
⊢ c < iSup f | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
⊢ c < iSup f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | rcases this.exists with ⟨x, hx⟩ | case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
⊢ c < iSup f | case H.intro
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
x : β
hx : c < f x
⊢ c < iSup f | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
⊢ c < iSup f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | apply lt_of_lt_of_le hx | case H.intro
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
x : β
hx : c < f x
⊢ c < iSup f | case H.intro
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
x : β
hx : c < f x
⊢ f x ≤ iSup f | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
x : β
hx : c < f x
⊢ c < iSup f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | apply le_iSup | case H.intro
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
x : β
hx : c < f x
⊢ f x ≤ iSup f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H.intro
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
this : ∀ᶠ (x : β) in atTop, c < f x
x : β
hx : c < f x
⊢ f x ≤ iSup f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_iSup_of_tendsto | [108, 1] | [116, 16] | apply eventually_gt_of_tendsto_gt hc ha | α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
⊢ ∀ᶠ (x : β) in atTop, c < f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : CompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → α
a : α
ha : Tendsto f atTop (𝓝 a)
c : α
hc : c < a
⊢ ∀ᶠ (x : β) in atTop, c < f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | rw [← MeasureTheory.IntegrableOn, annulus_real_eq r_nonneg] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1}) | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.IntegrableOn (fun x => f x) (Set.Ioo (x - 1) (x - r) ∪ Set.Ioo (x + r) (x + 1)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | apply MeasureTheory.IntegrableOn.union <;>
. rw [← intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith)]
apply hf.mono_set
rw [Set.uIcc_of_le (by linarith), Set.uIcc_of_le (by linarith [Real.pi_pos])]
intro y hy
constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.IntegrableOn (fun x => f x) (Set.Ioo (x - 1) (x - r) ∪ Set.Ioo (x + r) (x + 1)) MeasureTheory.volume | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.IntegrableOn (fun x => f x) (Set.Ioo (x - 1) (x - r) ∪ Set.Ioo (x + r) (x + 1)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | rw [← intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith)] | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.IntegrableOn (fun x => f x) (Set.Ioo (x + r) (x + 1)) MeasureTheory.volume | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ IntervalIntegrable (fun x => f x) MeasureTheory.volume (x + r) (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ MeasureTheory.IntegrableOn (fun x => f x) (Set.Ioo (x + r) (x + 1)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | apply hf.mono_set | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ IntervalIntegrable (fun x => f x) MeasureTheory.volume (x + r) (x + 1) | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ Set.uIcc (x + r) (x + 1) ⊆ Set.uIcc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ IntervalIntegrable (fun x => f x) MeasureTheory.volume (x + r) (x + 1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | rw [Set.uIcc_of_le (by linarith), Set.uIcc_of_le (by linarith [Real.pi_pos])] | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ Set.uIcc (x + r) (x + 1) ⊆ Set.uIcc (-Real.pi) (3 * Real.pi) | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ Set.Icc (x + r) (x + 1) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ Set.uIcc (x + r) (x + 1) ⊆ Set.uIcc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | intro y hy | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ Set.Icc (x + r) (x + 1) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
y : ℝ
hy : y ∈ Set.Icc (x + r) (x + 1)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ Set.Icc (x + r) (x + 1) ⊆ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
y : ℝ
hy : y ∈ Set.Icc (x + r) (x + 1)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
y : ℝ
hy : y ∈ Set.Icc (x + r) (x + 1)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | linarith | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ x + r ≤ x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ x + r ≤ x + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrable_annulus | [118, 1] | [126, 70] | linarith [Real.pi_pos] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ -Real.pi ≤ 3 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
r : ℝ
r_nonneg : 0 ≤ r
rle1 : r < 1
⊢ -Real.pi ≤ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | conv => pattern ((f _) * _ * _); rw [mul_assoc, mul_comm] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith [Real.pi_pos])] at hf | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply MeasureTheory.Integrable.bdd_mul' | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . apply hf.mono_set
intro y hy
constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ | case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . apply Measurable.aestronglyMeasurable
apply Measurable.mul
. apply Measurable.comp Complex.measurable_ofReal
apply Measurable.max
apply Measurable.const_sub
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const
. apply Measurable.comp dirichletKernel'_measurable
apply Measurable.const_sub measurable_id | case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc]
apply eventually_of_forall
intro y _
calc ‖(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖
_ = ‖max (1 - |x - y|) 0‖ * ‖dirichletKernel' N (x - y)‖ := by
rw [norm_mul, Complex.norm_real]
_ ≤ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_nonneg]
apply max_le
linarith [abs_nonneg (x - y)]
norm_num
apply le_max_right
apply norm_dirichletKernel'_le | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | linarith [Real.pi_pos] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ -Real.pi ≤ 3 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ -Real.pi ≤ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply hf.mono_set | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi))) | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Set.Icc (x - Real.pi) (x + Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | intro y hy | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Set.Icc (x - Real.pi) (x + Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
hy : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Set.Icc (x - Real.pi) (x + Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
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