mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
𝕜✝ : Type u_2
𝕜'✝ : Type u_3
E : Type u_4
F✝ : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F✝
g : E → F✝
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜' F✝
σ : 𝕜 →+* 𝕜'
inst✝⁵ : RingHomIsometric σ
F : Type u_8
G : Type u_9
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fact (1 ≤ p)
B : G →L[𝕜] E →L[𝕜] F
c : G
f : ↥(Lp E p μ)
⊢ ‖((compLpₗ₂ p μ B) c) f‖ ≤ ‖B‖ * ‖c‖ * ‖f‖

Tactic:
simp only [compLpₗ₂_apply, compLpₗ_apply]

Goal state after:
α : Type u_1
𝕜✝ : Type u_2
𝕜'✝ : Type u_3
E : Type u_4
F✝ : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F✝
g : E → F✝
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜' F✝
σ : 𝕜 →+* 𝕜'
inst✝⁵ : RingHomIsometric σ
F : Type u_8
G : Type u_9
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fact (1 ≤ p)
B : G →L[𝕜] E →L[𝕜] F
c : G
f : ↥(Lp E p μ)
⊢ ‖(B c).compLp f‖ ≤ ‖B‖ * ‖c‖ * ‖f‖
Module: Mathlib.MeasureTheory.Function.LpOrder

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
⊢ AddLeftMono ↥(Lp E p μ)

Tactic:
refine ⟨fun f g₁ g₂ hg₁₂ => ?_⟩

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
f g₁ g₂ : ↥(Lp E p μ)
hg₁₂ : g₁ ≤ g₂
⊢ f + g₁ ≤ f + g₂
Module: Mathlib.MeasureTheory.Function.LpOrder

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : Fact (1 ≤ p)
f g : ↥(Lp E p μ)
hfg : ↑↑|f| ≤ᵐ[μ] ↑↑|g|
x : α
hx : ↑↑|f| x ≤ ↑↑|g| x
hxf : ↑↑|f| x = |↑↑f x|
hxg : ↑↑|g| x = |↑↑g x|
⊢ ‖↑↑f x‖ ≤ ‖↑↑g x‖

Tactic:
rw [hxf, hxg] at hx

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝⁴ : NormedAddCommGroup E
inst✝³ : Lattice E
inst✝² : HasSolidNorm E
inst✝¹ : IsOrderedAddMonoid E
inst✝ : Fact (1 ≤ p)
f g : ↥(Lp E p μ)
hfg : ↑↑|f| ≤ᵐ[μ] ↑↑|g|
x : α
hx : |↑↑f x| ≤ |↑↑g x|
hxf : ↑↑|f| x = |↑↑f x|
hxg : ↑↑|g| x = |↑↑g x|
⊢ ‖↑↑f x‖ ≤ ‖↑↑g x‖
Module: Mathlib.Topology.Algebra.Module.Multilinear.Basic

Goal state before:
pp: R : Type u
ι : Type v
M : Type u_1
inst✝⁷ : Fintype ι
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace M
inst✝¹ : ContinuousMul R
inst✝ : ContinuousSMul R M
⊢ 0 = 0 fun x ↦ 1

Tactic:
zero_apply

Goal state after:
R : Type u
ι : Type v
M : Type u_1
inst✝⁷ : Fintype ι
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace M
inst✝¹ : ContinuousMul R
inst✝ : ContinuousSMul R M
⊢ 0 = 0
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ.real s₂ = 0
h' : μ s₂ ≠ ∞
⊢ μ.real (s₁ \ s₂) = μ.real s₁

Tactic:
rcases eq_top_or_lt_top (μ s₁) with H | H

Goal state after:
case inl
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ.real s₂ = 0
h' : μ s₂ ≠ ∞
H : μ s₁ = ∞
⊢ μ.real (s₁ \ s₂) = μ.real s₁

case inr
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ.real s₂ = 0
h' : μ s₂ ≠ ∞
H : μ s₁ < ∞
⊢ μ.real (s₁ \ s₂) = μ.real s₁
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: E : Type u_2
p : ℝ≥0∞
inst✝³ : NormedAddCommGroup E
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasureOnCompacts μ
f : X → E
hf : HasCompactSupport f
h2f : AEStronglyMeasurable f μ
C : ℝ
hfC : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
this : MemLp f ∞ μ
⊢ MemLp f p μ

Tactic:
exact this.mono_exponent_of_measure_support_ne_top
    (fun x ↦ image_eq_zero_of_notMem_tsupport) (hf.measure_lt_top.ne) le_top

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c c' : E
⊢ indicatorConstLp p hs hμs c + indicatorConstLp p hs hμs c' = indicatorConstLp p hs hμs (c + c')

Tactic:
indicatorConstLp,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c c' : E
⊢ MemLp.toLp (s.indicator fun x ↦ c) ⋯ + MemLp.toLp (s.indicator fun x ↦ c') ⋯ =
    MemLp.toLp (s.indicator fun x ↦ c + c') ⋯
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c c' : E
⊢ indicatorConstLp p hs hμs c - indicatorConstLp p hs hμs c' = indicatorConstLp p hs hμs (c - c')

Tactic:
indicatorConstLp,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c c' : E
⊢ MemLp.toLp (s.indicator fun x ↦ c) ⋯ - MemLp.toLp (s.indicator fun x ↦ c') ⋯ =
    MemLp.toLp (s.indicator fun x ↦ c - c') ⋯
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
⊢ ‖indicatorConstLp p hs hμs c‖ ≤ ‖c‖ * μ.real s ^ (1 / p.toReal)

Tactic:
indicatorConstLp,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
⊢ ‖MemLp.toLp (s.indicator fun x ↦ c) ⋯‖ ≤ ‖c‖ * μ.real s ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s t : Set α
hs : MeasurableSet s
hsμ : μ s ≠ ∞
ht : MeasurableSet t
htμ : μ t ≠ ∞
c : E
hc : c ≠ 0
⊢ indicatorConstLp p hs hsμ c = indicatorConstLp p ht htμ c ↔ (s.indicator fun x ↦ c) =ᵐ[μ] t.indicator fun x ↦ c

Tactic:
indicatorConstLp,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s t : Set α
hs : MeasurableSet s
hsμ : μ s ≠ ∞
ht : MeasurableSet t
htμ : μ t ≠ ∞
c : E
hc : c ≠ 0
⊢ MemLp.toLp (s.indicator fun x ↦ c) ⋯ = MemLp.toLp (t.indicator fun x ↦ c) ⋯ ↔
    (s.indicator fun x ↦ c) =ᵐ[μ] t.indicator fun x ↦ c
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : AEStronglyMeasurable f μ
⊢ HasFiniteIntegral (fun x ↦ ‖f x‖ₑ) μ ↔ HasFiniteIntegral f μ

Tactic:
hasFiniteIntegral_enorm_iff

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
s : Set α
f : ↥(Lp E p μ)
hp : 1 ≤ p
hμs : μ s ≠ ∞
hμ_restrict_univ : (μ.restrict s) univ < ∞
hμ_finite : IsFiniteMeasure (μ.restrict s)
⊢ MemLp (↑↑f) 1 (μ.restrict s)

Tactic:
exact ((Lp.memLp _).restrict s).mono_exponent hp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
A :
  ∀ (i : ι),
    (∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)
⊢ C * ∑ _i, ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) =
    C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖

Tactic:
sum_const,

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
A :
  ∀ (i : ι),
    (∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)
⊢ C * #univ • (‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)) =
    C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
A :
  ∀ (i : ι),
    (∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)
⊢ C * Fintype.card ι • (‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)) =
    C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖

Tactic:
nsmul_eq_mul

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
A :
  ∀ (i : ι),
    (∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤ ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)
⊢ C * (↑(Fintype.card ι) * (‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1))) =
    C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetrizableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
h's : IsSeparable s
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
⊢ AEStronglyMeasurable f (μ.restrict s)

Tactic:
borelize β

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetrizableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
h's : IsSeparable s
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (μ.restrict s)
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
⊢ AEStronglyMeasurable f (μ.restrict s)

Tactic:
borelize β

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (μ.restrict s)
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G
⊢ ‖- -f.1 + -f.2‖ = ‖f.1 - f.2‖

Tactic:
neg_neg,

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G
⊢ ‖f.1 + -f.2‖ = ‖f.1 - f.2‖
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s t : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
ht : MeasurableSet t
hts : t ⊆ s
⊢ AEStronglyMeasurable f (μ.restrict t)

Tactic:
borelize β

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s t : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
ht : MeasurableSet t
hts : t ⊆ s
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (μ.restrict t)
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
p q : Submodule 𝕜 E
h : IsCompl p q
hp : IsClosed[inst✝⁴] ↑p
inst✝ : FiniteDimensional 𝕜 (E ⧸ p)
φ : E ⧸ p →L[𝕜] ↥q := LinearMap.toContinuousLinearMap ↑(p.quotientEquivOfIsCompl q h)
⊢ IsTopCompl p q

Tactic:
have := (φ ∘L p.mkQL).isTopCompl_of_proj fun x ↦ by simp [φ]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
p q : Submodule 𝕜 E
h : IsCompl p q
hp : IsClosed[inst✝⁴] ↑p
inst✝ : FiniteDimensional 𝕜 (E ⧸ p)
φ : E ⧸ p →L[𝕜] ↥q := LinearMap.toContinuousLinearMap ↑(p.quotientEquivOfIsCompl q h)
this : IsTopCompl q (↑(φ ∘SL p.mkQL)).ker
⊢ IsTopCompl p q
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁶ : MeasurableSpace X
inst✝⁵ : CommMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : PseudoMetrizableSpace E
inst✝² : ContinuousMul E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
⊢ AEStronglyMeasurable (∏'[L] (i : ι), f i) μ

Tactic:
rw [tprod_def, finprod_def']
  split_ifs with hm
  any_goals exact aestronglyMeasurable_one
  · refine Finset.aestronglyMeasurable_prod _ (fun _ _ ↦ ?_)
    rw [Set.mulIndicator]
    split_ifs <;> fun_prop
  · apply aestronglyMeasurable_of_tendsto_ae L.filter (f := fun s => ∏ i ∈ s, f i) (by fun_prop)
    exact .of_forall fun x ↦ hm.choose_spec.apply_nhds x

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → E i →L[𝕜] E₁ i
m : (i : ι) → E i
⊢ ‖g‖ * ∏ i, ‖(f i) (m i)‖ ≤ ‖g‖ * ∏ i, ‖f i‖ * ‖m i‖

Tactic:
by gcongr with i; exact (f i).le_opNorm (m i)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ 0 x = 0

Tactic:
zero_apply

Goal state after:
case neg
α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ 0 = 0
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f : ↥(Lp E p μ)
⊢ f ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Set.range Subtype.val)

Tactic:
mem_closure_iff_seq_limit

Goal state after:
α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f : ↥(Lp E p μ)
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range Subtype.val) ∧ Tendsto x atTop (𝓝 f)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
s : Set α
h_zero : μ s = 0
⊢ weightedSMul μ s = 0

Tactic:
by
  ext1 x; rw [weightedSMul_apply, measureReal_def, h_zero]; simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
⊢ g ∈ closure[instTopologicalSpaceSubtype] (Set.range (coeSimpleFuncNonnegToLpNonneg p μ G))

Tactic:
mem_closure_iff_seq_limit

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
ν : Measure α
f : α →ₛ E
hf : Integrable (⇑f) (μ + ν)
s : Set α
x✝ : MeasurableSet s
hμνs : μ s ≠ ∞ ∧ ν s ≠ ∞
⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s

Tactic:
rw [weightedSMul_add_measure _ _ hμνs.1 hμνs.2]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal

Tactic:
rw [integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi]

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal - (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal =
    (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v) → ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v

Tactic:
rintro ⟨φ, hφ, h⟩

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
φ : α → β
hφ : Tendsto φ l (𝓝 0)
h : u - v =ᶠ[l] φ * v
⊢ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v) → ∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v

Tactic:
rintro ⟨φ, hφ, h⟩

Goal state after:
case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
φ : α → β
hφ : Tendsto φ l (𝓝 1)
h : u =ᶠ[l] φ * v
⊢ ∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬Integrable f μ
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal

Tactic:
· rw [integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi]
    have h_min : ∫⁻ a, ENNReal.ofReal (-f a) ∂μ = 0 := by
      rw [lintegral_eq_zero_iff']
      · refine hf.mono ?_
        simp only [Pi.zero_apply]
        intro a h
        simp only [h, neg_nonpos, ofReal_eq_zero]
      · exact measurable_ofReal.comp_aemeasurable hfm.aemeasurable.neg
    rw [h_min, toReal_zero, _root_.sub_zero]

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : ¬Integrable f μ
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q✝ : Type u_5
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : PseudoMetricSpace Q✝
inst✝³ : NormedAddTorsor W Q✝
Q : Type u_6
inst✝² : VAdd V Q
inst✝¹ : VSub V Q
inst✝ : PseudoMetricSpace Q
f : P → Q
hf : Surjective f
vadd : ∀ (c : V) (x : P), f (c +ᵥ x) = c +ᵥ f x
vsub : ∀ (x y : P), x -ᵥ y = f x -ᵥ f y
norm : ∀ (x y : P), dist x y = dist (f x) (f y)
⊢ ∀ (x y : Q), dist x y = ‖x -ᵥ y‖

Tactic:
by simp [hf.forall, ← norm, NormedAddTorsor.dist_eq_norm', ← vsub]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : ∀ ⦃x : β⦄, x ∈ s → ∀ ⦃y : β⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ p • a + q • b ∈ s

Tactic:
exact hs ha hb hp hq hpq

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hfi : Integrable (fun x ↦ ↑(f x)) μ
b : ℝ≥0
⊢ (∫ (a : α), ↑(f a) ∂μ).toNNReal ≤ b ↔ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b

Tactic:
Real.toNNReal_le_iff_le_coe

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hfi : Integrable (fun x ↦ ↑(f x)) μ
b : ℝ≥0
⊢ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b ↔ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b