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.measureReal_biUnion_finset_le | [158, 1] | [165, 55] | exact (measureReal_union_le _ _).trans (by gcongr) | case insert
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (f x ∪ ⋃ x ∈ s, f x) ≤ μ.real (f x) + ∑ p ∈ s, μ.real (f p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (f x ∪ ⋃ x ∈ s, f x) ≤ μ.real (f x) + ∑ p ∈ s, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_biUnion_finset_le | [158, 1] | [165, 55] | gcongr | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (f x) + μ.real (⋃ x ∈ s, f x) ≤ μ.real (f x) + ∑ p ∈ s, μ.real (f p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
f : β → Set α
x : β
s : Finset β
hx : x ∉ s
IH : μ.real (⋃ b ∈ s, f b) ≤ ∑ p ∈ s, μ.real (f p)
⊢ μ.real (f x) + μ.real (⋃ x ∈ s, f x) ≤ μ.real (f x) + ∑ p ∈ s, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_iUnion_fintype_le | [167, 1] | [170, 7] | convert measureReal_biUnion_finset_le Finset.univ f | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
⊢ μ.real (⋃ b, f b) ≤ ∑ p : β, μ.real (f p) | case h.e'_3.h.e'_4.h.e'_3.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
x✝ : β
⊢ f x✝ = ⋃ (_ : x✝ ∈ Finset.univ), f x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
⊢ μ.real (⋃ b, f b) ≤ ∑ p : β, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_iUnion_fintype_le | [167, 1] | [170, 7] | simp | case h.e'_3.h.e'_4.h.e'_3.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
x✝ : β
⊢ f x✝ = ⋃ (_ : x✝ ∈ Finset.univ), f x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h.e'_4.h.e'_3.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝¹ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
x✝ : β
⊢ f x✝ = ⋃ (_ : x✝ ∈ Finset.univ), f x✝
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_iUnion_fintype | [172, 1] | [176, 6] | rw [measureReal_def, measure_iUnion hn h, tsum_fintype, ENNReal.toReal_sum (fun i _hi ↦ h' i)] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : autoParam (∀ (i : β), μ (f i) ≠ ⊤) _auto✝
⊢ μ.real (⋃ b, f b) = ∑ p : β, μ.real (f p) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : autoParam (∀ (i : β), μ (f i) ≠ ⊤) _auto✝
⊢ ∑ a : β, (μ (f a)).toReal = ∑ p : β, μ.real (f p) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : autoParam (∀ (i : β), μ (f i) ≠ ⊤) _auto✝
⊢ μ.real (⋃ b, f b) = ∑ p : β, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_iUnion_fintype | [172, 1] | [176, 6] | rfl | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : autoParam (∀ (i : β), μ (f i) ≠ ⊤) _auto✝
⊢ ∑ a : β, (μ (f a)).toReal = ∑ p : β, μ.real (f p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : Fintype β
f : β → Set α
hn : Pairwise (Disjoint on f)
h : ∀ (i : β), MeasurableSet (f i)
h' : autoParam (∀ (i : β), μ (f i) ≠ ⊤) _auto✝
⊢ ∑ a : β, (μ (f a)).toReal = ∑ p : β, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null | [178, 1] | [181, 62] | apply le_antisymm _ measureReal_nonneg | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real (s₁ ∪ s₂) = 0 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real (s₁ ∪ s₂) ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real (s₁ ∪ s₂) = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null | [178, 1] | [181, 62] | exact (measureReal_union_le s₁ s₂).trans (by simp [h₁, h₂]) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real (s₁ ∪ s₂) ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real (s₁ ∪ s₂) ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null | [178, 1] | [181, 62] | simp [h₁, h₂] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real s₁ + μ.real s₂ ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : μ.real s₁ = 0
h₂ : μ.real s₂ = 0
⊢ μ.real s₁ + μ.real s₂ ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null_iff | [184, 1] | [190, 58] | have : μ (s₁ ∪ s₂) ≠ ∞ := measure_union_ne_top h₁ h₂ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null_iff | [184, 1] | [190, 58] | refine ⟨fun h => ⟨?_, ?_⟩, fun h => measureReal_union_null h.1 h.2⟩ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0 | case refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
h : μ.real (s₁ ∪ s₂) = 0
⊢ μ.real s₁ = 0
case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
h : μ.real (s₁ ∪ s₂) = 0
⊢ μ.real s₂ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
⊢ μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null_iff | [184, 1] | [190, 58] | exact measureReal_mono_null subset_union_left h this | case refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
h : μ.real (s₁ ∪ s₂) = 0
⊢ μ.real s₁ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
h : μ.real (s₁ ∪ s₂) = 0
⊢ μ.real s₁ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_null_iff | [184, 1] | [190, 58] | exact measureReal_mono_null subset_union_right h this | case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
h : μ.real (s₁ ∪ s₂) = 0
⊢ μ.real s₂ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
h₂ : autoParam (μ s₂ ≠ ⊤) _auto✝
this : μ (s₁ ∪ s₂) ≠ ⊤
h : μ.real (s₁ ∪ s₂) = 0
⊢ μ.real s₂ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_congr | [193, 1] | [194, 39] | simp [Measure.real, measure_congr H] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
H : s =ᶠ[ae μ] t
⊢ μ.real s = μ.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
H : s =ᶠ[ae μ] t
⊢ μ.real s = μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff₀ | [196, 1] | [202, 49] | simp only [measureReal_def] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real (s ∩ t) + μ.real (s \ t) = μ.real s | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ (μ (s ∩ t)).toReal + (μ (s \ t)).toReal = (μ s).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real (s ∩ t) + μ.real (s \ t) = μ.real s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff₀ | [196, 1] | [202, 49] | rw [← ENNReal.toReal_add, measure_inter_add_diff₀ s ht] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ (μ (s ∩ t)).toReal + (μ (s \ t)).toReal = (μ s).toReal | case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s ∩ t) ≠ ⊤
case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ (μ (s ∩ t)).toReal + (μ (s \ t)).toReal = (μ s).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff₀ | [196, 1] | [202, 49] | exact measure_ne_top_of_subset inter_subset_left h | case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s ∩ t) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s ∩ t) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff₀ | [196, 1] | [202, 49] | exact measure_ne_top_of_subset diff_subset h | case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_add_inter₀ | [204, 1] | [211, 9] | have : μ (s ∪ t) ≠ ∞ :=
((measure_union_le _ _).trans_lt (ENNReal.add_lt_top.2 ⟨h₁.lt_top, h₂.lt_top⟩ )).ne | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
this : μ (s ∪ t) ≠ ⊤
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_add_inter₀ | [204, 1] | [211, 9] | rw [← measureReal_inter_add_diff₀ (s ∪ t) ht this, Set.union_inter_cancel_right, union_diff_right,
← measureReal_inter_add_diff₀ s ht h₁] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
this : μ (s ∪ t) ≠ ⊤
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
this : μ (s ∪ t) ≠ ⊤
⊢ μ.real t + μ.real (s \ t) + μ.real (s ∩ t) = μ.real (s ∩ t) + μ.real (s \ t) + μ.real t | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
this : μ (s ∪ t) ≠ ⊤
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_add_inter₀ | [204, 1] | [211, 9] | ac_rfl | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
this : μ (s ∪ t) ≠ ⊤
⊢ μ.real t + μ.real (s \ t) + μ.real (s ∩ t) = μ.real (s ∩ t) + μ.real (s \ t) + μ.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : NullMeasurableSet t μ
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
this : μ (s ∪ t) ≠ ⊤
⊢ μ.real t + μ.real (s \ t) + μ.real (s ∩ t) = μ.real (s ∩ t) + μ.real (s \ t) + μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_add_inter₀' | [213, 1] | [216, 81] | rw [union_comm, inter_comm, measureReal_union_add_inter₀ t hs h₂ h₁, add_comm] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t✝ : Set α
hs : NullMeasurableSet s μ
t : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t✝ : Set α
hs : NullMeasurableSet s μ
t : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union₀ | [218, 1] | [222, 54] | simp only [Measure.real] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) = μ.real s + μ.real t | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ (μ (s ∪ t)).toReal = (μ s).toReal + (μ t).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) = μ.real s + μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union₀ | [218, 1] | [222, 54] | rw [measure_union₀ ht hd, ENNReal.toReal_add h₁ h₂] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ (μ (s ∪ t)).toReal = (μ s).toReal + (μ t).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
ht : NullMeasurableSet t μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ (μ (s ∪ t)).toReal = (μ s).toReal + (μ t).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union₀' | [224, 1] | [227, 78] | rw [union_comm, measureReal_union₀ hs (AEDisjoint.symm hd) h₂ h₁, add_comm] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : NullMeasurableSet s μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) = μ.real s + μ.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : NullMeasurableSet s μ
hd : AEDisjoint μ s t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∪ t) = μ.real s + μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_add_measureReal_compl₀ | [229, 1] | [232, 73] | rw [← measureReal_union₀' hs aedisjoint_compl_right, union_compl_self] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
s : Set α
hs : NullMeasurableSet s μ
⊢ μ.real s + μ.real sᶜ = μ.real univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
s : Set α
hs : NullMeasurableSet s μ
⊢ μ.real s + μ.real sᶜ = μ.real univ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff | [244, 1] | [250, 49] | simp only [Measure.real] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real (s ∩ t) + μ.real (s \ t) = μ.real s | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ (μ (s ∩ t)).toReal + (μ (s \ t)).toReal = (μ s).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ.real (s ∩ t) + μ.real (s \ t) = μ.real s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff | [244, 1] | [250, 49] | rw [← ENNReal.toReal_add, measure_inter_add_diff _ ht] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ (μ (s ∩ t)).toReal + (μ (s \ t)).toReal = (μ s).toReal | case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s ∩ t) ≠ ⊤
case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ (μ (s ∩ t)).toReal + (μ (s \ t)).toReal = (μ s).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff | [244, 1] | [250, 49] | exact measure_ne_top_of_subset inter_subset_left h | case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s ∩ t) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s ∩ t) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_inter_add_diff | [244, 1] | [250, 49] | exact measure_ne_top_of_subset diff_subset h | case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
ht : MeasurableSet t
h : autoParam (μ s ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_eq | [267, 1] | [273, 50] | simp only [Measure.real] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∆ t) = μ.real (s \ t) + μ.real (t \ s) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ (μ (s ∆ t)).toReal = (μ (s \ t)).toReal + (μ (t \ s)).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∆ t) = μ.real (s \ t) + μ.real (t \ s)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_eq | [267, 1] | [273, 50] | rw [← ENNReal.toReal_add, measure_symmDiff_eq hs ht] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ (μ (s ∆ t)).toReal = (μ (s \ t)).toReal + (μ (t \ s)).toReal | case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤
case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ (t \ s) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ (μ (s ∆ t)).toReal = (μ (s \ t)).toReal + (μ (t \ s)).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_eq | [267, 1] | [273, 50] | exact measure_ne_top_of_subset diff_subset h₁ | case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ (s \ t) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_eq | [267, 1] | [273, 50] | exact measure_ne_top_of_subset diff_subset h₂ | case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ (t \ s) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ (t \ s) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | rcases eq_top_or_lt_top (μ u) with hu|hu | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u) | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u)
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | have : μ (s ∆ u) = ∞ := measure_symmDiff_eq_top h₁ hu | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u) | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
this : μ (s ∆ u) = ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | simp only [measureReal_def, this, ENNReal.top_toReal] | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
this : μ (s ∆ u) = ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u) | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
this : μ (s ∆ u) = ⊤
⊢ 0 ≤ (μ (s ∆ t)).toReal + (μ (t ∆ u)).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
this : μ (s ∆ u) = ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
this : μ (s ∆ u) = ⊤
⊢ 0 ≤ (μ (s ∆ t)).toReal + (μ (t ∆ u)).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u = ⊤
this : μ (s ∆ u) = ⊤
⊢ 0 ≤ (μ (s ∆ t)).toReal + (μ (t ∆ u)).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | apply le_trans _ (measureReal_union_le (s ∆ t) (t ∆ u)) | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t ∪ t ∆ u) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t) + μ.real (t ∆ u)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | apply measureReal_mono (symmDiff_triangle s t u) ?_ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t ∪ t ∆ u) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ (s ∆ t ⊔ t ∆ u) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ.real (s ∆ u) ≤ μ.real (s ∆ t ∪ t ∆ u)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_symmDiff_le | [275, 1] | [284, 98] | exact measure_union_ne_top (measure_symmDiff_ne_top h₁ h₂) (measure_symmDiff_ne_top h₂ hu.ne) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ (s ∆ t ⊔ t ∆ u) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t u : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
hu : μ u < ⊤
⊢ μ (s ∆ t ⊔ t ∆ u) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_biUnion_finset₀ | [290, 1] | [294, 83] | simp only [measureReal_def, measure_biUnion_finset₀ hd hm, ENNReal.toReal_sum h] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
s : Finset ι
f : ι → Set α
hd : (↑s).Pairwise (AEDisjoint μ on f)
hm : ∀ b ∈ s, NullMeasurableSet (f b) μ
h : autoParam (∀ b ∈ s, μ (f b) ≠ ⊤) _auto✝
⊢ μ.real (⋃ b ∈ s, f b) = ∑ p ∈ s, μ.real (f p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
s : Finset ι
f : ι → Set α
hd : (↑s).Pairwise (AEDisjoint μ on f)
hm : ∀ b ∈ s, NullMeasurableSet (f b) μ
h : autoParam (∀ b ∈ s, μ (f b) ≠ ⊤) _auto✝
⊢ μ.real (⋃ b ∈ s, f b) = ∑ p ∈ s, μ.real (f p)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.sum_measureReal_preimage_singleton | [303, 1] | [306, 91] | simp only [measureReal_def, ← sum_measure_preimage_singleton s hf, ENNReal.toReal_sum h] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
s : Finset β
f : α → β
hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})
h : autoParam (∀ a ∈ s, μ (f ⁻¹' {a}) ≠ ⊤) _auto✝
⊢ ∑ b ∈ s, μ.real (f ⁻¹' {b}) = μ.real (f ⁻¹' ↑s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
s : Finset β
f : α → β
hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})
h : autoParam (∀ a ∈ s, μ (f ⁻¹' {a}) ≠ ⊤) _auto✝
⊢ ∑ b ∈ s, μ.real (f ⁻¹' {b}) = μ.real (f ⁻¹' ↑s)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null' | [315, 1] | [320, 54] | simp only [measureReal_def] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ.real (s₁ \ s₂) = μ.real s₁ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ (μ (s₁ \ s₂)).toReal = (μ s₁).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ.real (s₁ \ s₂) = μ.real s₁
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null' | [315, 1] | [320, 54] | rw [measure_diff_null'] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ (μ (s₁ \ s₂)).toReal = (μ s₁).toReal | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ (s₁ ∩ s₂) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ (μ (s₁ \ s₂)).toReal = (μ s₁).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null' | [315, 1] | [320, 54] | apply (measureReal_eq_zero_iff _).1 h | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ (s₁ ∩ s₂) = 0 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ (s₁ ∩ s₂) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ (s₁ ∩ s₂) = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null' | [315, 1] | [320, 54] | exact measure_ne_top_of_subset inter_subset_left h' | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ (s₁ ∩ s₂) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ (s₁ ∩ s₂) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null | [322, 1] | [326, 86] | rcases eq_top_or_lt_top (μ s₁) with H|H | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real (s₁ \ s₂) = μ.real s₁ | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
H : μ s₁ = ⊤
⊢ μ.real (s₁ \ s₂) = μ.real s₁
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
H : μ s₁ < ⊤
⊢ μ.real (s₁ \ s₂) = μ.real s₁ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real (s₁ \ s₂) = μ.real s₁
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null | [322, 1] | [326, 86] | simp [measureReal_def, H, measure_diff_eq_top H h'] | case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
H : μ s₁ = ⊤
⊢ μ.real (s₁ \ s₂) = μ.real s₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
H : μ s₁ = ⊤
⊢ μ.real (s₁ \ s₂) = μ.real s₁
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_null | [322, 1] | [326, 86] | exact measureReal_diff_null' (measureReal_mono_null inter_subset_right h h') H.ne | case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
H : μ s₁ < ⊤
⊢ μ.real (s₁ \ s₂) = μ.real s₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : μ.real s₂ = 0
h' : autoParam (μ s₂ ≠ ⊤) _auto✝
H : μ s₁ < ⊤
⊢ μ.real (s₁ \ s₂) = μ.real s₁
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_add_diff | [328, 1] | [332, 64] | rw [← measureReal_union' (@disjoint_sdiff_right _ s t) hs h₁
(measure_ne_top_of_subset diff_subset h₂), union_diff_self] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t✝ : Set α
hs : MeasurableSet s
t : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real s + μ.real (t \ s) = μ.real (s ∪ t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t✝ : Set α
hs : MeasurableSet s
t : Set α
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real s + μ.real (t \ s) = μ.real (s ∪ t)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff' | [334, 1] | [338, 7] | rw [union_comm, ← measureReal_add_diff hm s h₂ h₁] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
hm : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s \ t) = μ.real (s ∪ t) - μ.real t | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
hm : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s \ t) = μ.real t + μ.real (s \ t) - μ.real t | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
hm : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s \ t) = μ.real (s ∪ t) - μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff' | [334, 1] | [338, 7] | ring | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
hm : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s \ t) = μ.real t + μ.real (s \ t) - μ.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t s : Set α
hm : MeasurableSet t
h₁ : autoParam (μ s ≠ ⊤) _auto✝
h₂ : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (s \ t) = μ.real t + μ.real (s \ t) - μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff | [340, 1] | [343, 98] | rw [measureReal_diff' _ h₂ h₁ (measure_ne_top_of_subset h h₁), union_eq_self_of_subset_right h] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₂ ⊆ s₁
h₂ : MeasurableSet s₂
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ.real (s₁ \ s₂) = μ.real s₁ - μ.real s₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : s₂ ⊆ s₁
h₂ : MeasurableSet s₂
h₁ : autoParam (μ s₁ ≠ ⊤) _auto✝
⊢ μ.real (s₁ \ s₂) = μ.real s₁ - μ.real s₂
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.le_measureReal_diff | [345, 1] | [351, 65] | simp only [tsub_le_iff_left] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ - μ.real s₂ ≤ μ.real (s₁ \ s₂) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ ≤ μ.real s₂ + μ.real (s₁ \ s₂) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ - μ.real s₂ ≤ μ.real (s₁ \ s₂)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.le_measureReal_diff | [345, 1] | [351, 65] | calc
μ.real s₁ ≤ μ.real (s₂ ∪ s₁) := measureReal_le_measureReal_union_right h
_ = μ.real (s₂ ∪ s₁ \ s₂) := congr_arg μ.real union_diff_self.symm
_ ≤ μ.real s₂ + μ.real (s₁ \ s₂) := measureReal_union_le _ _ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ ≤ μ.real s₂ + μ.real (s₁ \ s₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
h : autoParam (μ s₂ ≠ ⊤) _auto✝
⊢ μ.real s₁ ≤ μ.real s₂ + μ.real (s₁ \ s₂)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_lt_of_lt_add | [353, 1] | [356, 45] | rw [measureReal_diff hst hs ht'] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
h : μ.real t < μ.real s + ε
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (t \ s) < ε | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
h : μ.real t < μ.real s + ε
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real t - μ.real s < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
h : μ.real t < μ.real s + ε
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (t \ s) < ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_lt_of_lt_add | [353, 1] | [356, 45] | linarith | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
h : μ.real t < μ.real s + ε
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real t - μ.real s < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
h : μ.real t < μ.real s + ε
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real t - μ.real s < ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_diff_le_iff_le_add | [358, 1] | [361, 53] | rw [measureReal_diff hst hs ht', tsub_le_iff_left] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (t \ s) ≤ ε ↔ μ.real t ≤ μ.real s + ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
ht' : autoParam (μ t ≠ ⊤) _auto✝
⊢ μ.real (t \ s) ≤ ε ↔ μ.real t ≤ μ.real s + ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_eq_measureReal_of_null_diff | [363, 1] | [367, 73] | rw [measureReal_eq_zero_iff h] at h_nulldiff | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t : Set α
hst : s ⊆ t
h_nulldiff : μ.real (t \ s) = 0
h : autoParam (μ (t \ s) ≠ ⊤) _auto✝
⊢ μ.real s = μ.real t | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t : Set α
hst : s ⊆ t
h_nulldiff : μ (t \ s) = 0
h : autoParam (μ (t \ s) ≠ ⊤) _auto✝
⊢ μ.real s = μ.real t | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t : Set α
hst : s ⊆ t
h_nulldiff : μ.real (t \ s) = 0
h : autoParam (μ (t \ s) ≠ ⊤) _auto✝
⊢ μ.real s = μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_eq_measureReal_of_null_diff | [363, 1] | [367, 73] | simp [measureReal_def, measure_eq_measure_of_null_diff hst h_nulldiff] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t : Set α
hst : s ⊆ t
h_nulldiff : μ (t \ s) = 0
h : autoParam (μ (t \ s) ≠ ⊤) _auto✝
⊢ μ.real s = μ.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ s t : Set α
hst : s ⊆ t
h_nulldiff : μ (t \ s) = 0
h : autoParam (μ (t \ s) ≠ ⊤) _auto✝
⊢ μ.real s = μ.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_eq_measureReal_of_between_null_diff | [369, 1] | [375, 35] | have A : μ s₁ = μ s₂ ∧ μ s₂ = μ s₃ :=
measure_eq_measure_of_between_null_diff h12 h23 ((measureReal_eq_zero_iff h').1 h_nulldiff) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : μ.real (s₃ \ s₁) = 0
h' : autoParam (μ (s₃ \ s₁) ≠ ⊤) _auto✝
⊢ μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : μ.real (s₃ \ s₁) = 0
h' : autoParam (μ (s₃ \ s₁) ≠ ⊤) _auto✝
A : μ s₁ = μ s₂ ∧ μ s₂ = μ s₃
⊢ μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : μ.real (s₃ \ s₁) = 0
h' : autoParam (μ (s₃ \ s₁) ≠ ⊤) _auto✝
⊢ μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_eq_measureReal_of_between_null_diff | [369, 1] | [375, 35] | simp [measureReal_def, A.1, A.2] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : μ.real (s₃ \ s₁) = 0
h' : autoParam (μ (s₃ \ s₁) ≠ ⊤) _auto✝
A : μ s₁ = μ s₂ ∧ μ s₂ = μ s₃
⊢ μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁✝ s₂✝ t s₁ s₂ s₃ : Set α
h12 : s₁ ⊆ s₂
h23 : s₂ ⊆ s₃
h_nulldiff : μ.real (s₃ \ s₁) = 0
h' : autoParam (μ (s₃ \ s₁) ≠ ⊤) _auto✝
A : μ s₁ = μ s₂ ∧ μ s₂ = μ s₃
⊢ μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_compl | [388, 1] | [391, 44] | rw [compl_eq_univ_diff] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
h₁ : MeasurableSet s
⊢ μ.real sᶜ = μ.real univ - μ.real s | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
h₁ : MeasurableSet s
⊢ μ.real (univ \ s) = μ.real univ - μ.real s | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
h₁ : MeasurableSet s
⊢ μ.real sᶜ = μ.real univ - μ.real s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_compl | [388, 1] | [391, 44] | exact measureReal_diff (subset_univ s) h₁ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
h₁ : MeasurableSet s
⊢ μ.real (univ \ s) = μ.real univ - μ.real s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
inst✝ : IsFiniteMeasure μ
h₁ : MeasurableSet s
⊢ μ.real (univ \ s) = μ.real univ - μ.real s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_congr_of_subset | [393, 1] | [400, 79] | simp [measureReal_def] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ.real (s₁ ∪ t₁) = μ.real (s₂ ∪ t₂) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ (μ (s₁ ∪ t₁)).toReal = (μ (s₂ ∪ t₂)).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ.real (s₁ ∪ t₁) = μ.real (s₂ ∪ t₂)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_congr_of_subset | [393, 1] | [400, 79] | rw [measure_union_congr_of_subset hs _ ht] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ (μ (s₁ ∪ t₁)).toReal = (μ (s₂ ∪ t₂)).toReal | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ t₂ ≤ μ t₁
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ s₂ ≤ μ s₁ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ (μ (s₁ ∪ t₁)).toReal = (μ (s₂ ∪ t₂)).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_congr_of_subset | [393, 1] | [400, 79] | exact (ENNReal.toReal_le_toReal h₂ (measure_ne_top_of_subset ht h₂)).1 htμ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ t₂ ≤ μ t₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ t₂ ≤ μ t₁
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_union_congr_of_subset | [393, 1] | [400, 79] | exact (ENNReal.toReal_le_toReal h₁ (measure_ne_top_of_subset hs h₁)).1 hsμ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ s₂ ≤ μ s₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t t₁ t₂ : Set α
hs : s₁ ⊆ s₂
hsμ : μ.real s₂ ≤ μ.real s₁
ht : t₁ ⊆ t₂
htμ : μ.real t₂ ≤ μ.real t₁
h₁ : autoParam (μ s₂ ≠ ⊤) _auto✝
h₂ : autoParam (μ t₂ ≠ ⊤) _auto✝
⊢ μ s₂ ≤ μ s₁
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.sum_measureReal_le_measureReal_univ | [402, 1] | [408, 40] | simp only [measureReal_def] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ i ∈ s, μ.real (t i) ≤ μ.real univ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ x ∈ s, (μ (t x)).toReal ≤ (μ univ).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ i ∈ s, μ.real (t i) ≤ μ.real univ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.sum_measureReal_le_measureReal_univ | [402, 1] | [408, 40] | rw [← ENNReal.toReal_sum (fun i hi ↦ measure_ne_top _ _)] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ x ∈ s, (μ (t x)).toReal ≤ (μ univ).toReal | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ (∑ a ∈ s, μ (t a)).toReal ≤ (μ univ).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ x ∈ s, (μ (t x)).toReal ≤ (μ univ).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.sum_measureReal_le_measureReal_univ | [402, 1] | [408, 40] | apply ENNReal.toReal_mono (measure_ne_top _ _) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ (∑ a ∈ s, μ (t a)).toReal ≤ (μ univ).toReal | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ a ∈ s, μ (t a) ≤ μ univ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ (∑ a ∈ s, μ (t a)).toReal ≤ (μ univ).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.sum_measureReal_le_measureReal_univ | [402, 1] | [408, 40] | exact sum_measure_le_measure_univ h H | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ a ∈ s, μ (t a) ≤ μ univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t✝ : Set α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (↑s).PairwiseDisjoint t
⊢ ∑ a ∈ s, μ (t a) ≤ μ univ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal | [412, 1] | [422, 66] | apply exists_nonempty_inter_of_measure_univ_lt_sum_measure μ h | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), (t i ∩ t j).Nonempty | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ μ univ < ∑ i ∈ s, μ (t i) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), (t i ∩ t j).Nonempty
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal | [412, 1] | [422, 66] | apply (ENNReal.toReal_lt_toReal (measure_ne_top _ _) _).1 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ μ univ < ∑ i ∈ s, μ (t i) | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ (μ univ).toReal < (∑ i ∈ s, μ (t i)).toReal
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∑ i ∈ s, μ (t i) ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ μ univ < ∑ i ∈ s, μ (t i)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal | [412, 1] | [422, 66] | convert H | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ (μ univ).toReal < (∑ i ∈ s, μ (t i)).toReal | case h.e'_4
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ (∑ i ∈ s, μ (t i)).toReal = ∑ i ∈ s, μ.real (t i) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ (μ univ).toReal < (∑ i ∈ s, μ (t i)).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal | [412, 1] | [422, 66] | rw [ENNReal.toReal_sum (fun i hi ↦ measure_ne_top _ _)] | case h.e'_4
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ (∑ i ∈ s, μ (t i)).toReal = ∑ i ∈ s, μ.real (t i) | case h.e'_4
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∑ a ∈ s, (μ (t a)).toReal = ∑ i ∈ s, μ.real (t i) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ (∑ i ∈ s, μ (t i)).toReal = ∑ i ∈ s, μ.real (t i)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal | [412, 1] | [422, 66] | rfl | case h.e'_4
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∑ a ∈ s, (μ (t a)).toReal = ∑ i ∈ s, μ.real (t i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∑ a ∈ s, (μ (t a)).toReal = ∑ i ∈ s, μ.real (t i)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal | [412, 1] | [422, 66] | exact (ENNReal.sum_lt_top (fun i hi ↦ measure_ne_top _ _)).ne | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∑ i ∈ s, μ (t i) ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∑ i ∈ s, μ (t i) ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add | [427, 1] | [435, 98] | apply nonempty_inter_of_measure_lt_add μ ht h's h't | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ μ u < μ s + μ t | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add | [427, 1] | [435, 98] | apply (ENNReal.toReal_lt_toReal hu _).1 | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ μ u < μ s + μ t | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (μ u).toReal < (μ s + μ t).toReal
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ μ s + μ t ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ μ u < μ s + μ t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add | [427, 1] | [435, 98] | rw [ENNReal.toReal_add (measure_ne_top_of_subset h's hu) (measure_ne_top_of_subset h't hu)] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (μ u).toReal < (μ s + μ t).toReal | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (μ u).toReal < (μ s).toReal + (μ t).toReal | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (μ u).toReal < (μ s + μ t).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add | [427, 1] | [435, 98] | exact h | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (μ u).toReal < (μ s).toReal + (μ t).toReal | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (μ u).toReal < (μ s).toReal + (μ t).toReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add | [427, 1] | [435, 98] | exact ENNReal.add_ne_top.2 ⟨measure_ne_top_of_subset h's hu, measure_ne_top_of_subset h't hu⟩ | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ μ s + μ t ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
ht : MeasurableSet t
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ μ s + μ t ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add' | [440, 1] | [446, 63] | rw [add_comm] at h | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real s + μ.real t
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add' | [440, 1] | [446, 63] | rw [inter_comm] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (t ∩ s).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (s ∩ t).Nonempty
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.nonempty_inter_of_measureReal_lt_add' | [440, 1] | [446, 63] | exact nonempty_inter_of_measureReal_lt_add μ hs h't h's h hu | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (t ∩ s).Nonempty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
m : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : autoParam (μ u ≠ ⊤) _auto✝
⊢ (t ∩ s).Nonempty
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.measureReal_prod_prod | [448, 1] | [451, 61] | simp only [measureReal_def, prod_prod, ENNReal.toReal_mul] | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
μ : Measure α
ν : Measure β
inst✝ : SigmaFinite ν
s : Set α
t : Set β
⊢ (μ.prod ν).real (s ×ˢ t) = μ.real s * ν.real t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ✝ : Measure α
s✝ s₁ s₂ t✝ : Set α
μ : Measure α
ν : Measure β
inst✝ : SigmaFinite ν
s : Set α
t : Set β
⊢ (μ.prod ν).real (s ×ˢ t) = μ.real s * ν.real t
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | constructor | ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ μ1 = μ2 ↔ ∀ (x : S), μ1 {x} = μ2 {x} | case mp
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ μ1 = μ2 → ∀ (x : S), μ1 {x} = μ2 {x}
case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ (∀ (x : S), μ1 {x} = μ2 {x}) → μ1 = μ2 | 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✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ μ1 = μ2 ↔ ∀ (x : S), μ1 {x} = μ2 {x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | rintro rfl | case mp
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ μ1 = μ2 → ∀ (x : S), μ1 {x} = μ2 {x} | case mp
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 : Measure S
⊢ ∀ (x : S), μ1 {x} = μ1 {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ μ1 = μ2 → ∀ (x : S), μ1 {x} = μ2 {x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | simp | case mp
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 : Measure S
⊢ ∀ (x : S), μ1 {x} = μ1 {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 : Measure S
⊢ ∀ (x : S), μ1 {x} = μ1 {x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | intro h | case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ (∀ (x : S), μ1 {x} = μ2 {x}) → μ1 = μ2 | case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
⊢ μ1 = μ2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
⊢ (∀ (x : S), μ1 {x} = μ2 {x}) → μ1 = μ2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | ext s | case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
⊢ μ1 = μ2 | case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
⊢ μ1 s = μ2 s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
⊢ μ1 = μ2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | have hs : Set.Finite s := Set.toFinite s | case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
⊢ μ1 s = μ2 s | case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
hs : s.Finite
⊢ μ1 s = μ2 s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
⊢ μ1 s = μ2 s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | rw [← hs.coe_toFinset, ← Finset.sum_measure_singleton μ1, ← Finset.sum_measure_singleton μ2] | case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
hs : s.Finite
⊢ μ1 s = μ2 s | case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
hs : s.Finite
⊢ ∑ x ∈ hs.toFinset, μ1 {x} = ∑ x ∈ hs.toFinset, μ2 {x} | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
hs : s.Finite
⊢ μ1 s = μ2 s
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.Measure.ext_iff_singleton | [454, 1] | [465, 16] | simp_rw [h] | case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
hs : s.Finite
⊢ ∑ x ∈ hs.toFinset, μ1 {x} = ∑ x ∈ hs.toFinset, μ2 {x} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h
ι : Type u_1
α : Type u_2
β : Type u_3
x✝ : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
s✝ s₁ s₂ t : Set α
S : Type u_4
inst✝² : Fintype S
inst✝¹ : MeasurableSpace S
inst✝ : MeasurableSingletonClass S
μ1 μ2 : Measure S
h : ∀ (x : S), μ1 {x} = μ2 {x}
s : Set S
a✝ : MeasurableSet s
hs : s.Finite
⊢ ∑ x ∈ hs.toFinset, μ1 {x} = ∑ x ∈ hs.toFinset, μ2 {x}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/ToMathlib/MeasureReal.lean | MeasureTheory.ext_iff_measureReal_singleton | [467, 1] | [476, 16] | rw [Measure.ext_iff_singleton] | 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
⊢ μ1 = μ2 ↔ ∀ (x : S), μ1.real {x} = μ2.real {x} | 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} = μ2 {x}) ↔ ∀ (x : S), μ1.real {x} = μ2.real {x} | 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
⊢ μ1 = μ2 ↔ ∀ (x : S), 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] | congr! with x | 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} = μ2 {x}) ↔ ∀ (x : S), 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
⊢ μ1 {x} = μ2 {x} ↔ μ1.real {x} = μ2.real {x} | 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} = μ2 {x}) ↔ ∀ (x : S), 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] | have h1 : μ1 {x} ≠ ⊤ := by finiteness | 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
⊢ 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} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1.real {x} = μ2.real {x} | 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
⊢ 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] | have h2 : μ2 {x} ≠ ⊤ := by finiteness | 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} ≠ ⊤
⊢ 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.real {x} = μ2.real {x} | 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} ≠ ⊤
⊢ μ1 {x} = μ2 {x} ↔ μ1.real {x} = μ2.real {x}
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.