mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁸ : NormedField 𝕜
inst✝⁷ : (i : ι) → TopologicalSpace (E i)
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
S : Set (ContinuousMultilinearMap 𝕜 E F)
hS : IsVonNBounded 𝕜 S
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
⊢ IsVonNBounded 𝕜 (image2 (fun f x ↦ f x) S s)

Tactic:
intro U hU

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁸ : NormedField 𝕜
inst✝⁷ : (i : ι) → TopologicalSpace (E i)
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
S : Set (ContinuousMultilinearMap 𝕜 E F)
hS : IsVonNBounded 𝕜 S
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
U : Set F
hU : U ∈ 𝓝 0
⊢ Absorbs 𝕜 U (image2 (fun f x ↦ f x) S s)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : s₂ ⊆ s₁
h₂ : MeasurableSet s₂
h₁ : μ s₁ ≠ ∞
⊢ μ.real (s₁ \ s₂) = μ.real s₁ - μ.real s₂

Tactic:
by
  rw [measureReal_sdiff' h₂ h₁ (measure_ne_top_of_subset h h₁), union_eq_self_of_subset_right h]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g₁ g₂ : α → ℝ
hf : AEStronglyMeasurable f μ
h_le₁ : g₁ ≤ᵐ[μ] f
h_le₂ : f ≤ᵐ[μ] g₂
h_int₁ : Integrable g₁ μ
h_int₂ : Integrable g₂ μ
x : α
hx1 : g₁ x ≤ f x
⊢ f x ≤ g₂ x → ‖f x‖ ≤ max ‖g₁ x‖ ‖g₂ x‖

Tactic:
hx2

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g₁ g₂ : α → ℝ
hf : AEStronglyMeasurable f μ
h_le₁ : g₁ ≤ᵐ[μ] f
h_le₂ : f ≤ᵐ[μ] g₂
h_int₁ : Integrable g₁ μ
h_int₂ : Integrable g₂ μ
x : α
hx1 : g₁ x ≤ f x
hx2 : f x ≤ g₂ x
⊢ ‖f x‖ ≤ max ‖g₁ x‖ ‖g₂ x‖
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
x : α
hx : f x < ∞
⊢ ↑(f x).toNNReal = f x

Tactic:
simp only [hx.ne, Ne, not_false_iff, coe_toNNReal]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
s : Set α
μ : Measure α
ε' : Type u_7
inst✝² : TopologicalSpace ε'
inst✝¹ : ENormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
h1s : support f ⊆ s
⊢ IntegrableOn f s μ ↔ Integrable f μ

Tactic:
by
  refine ⟨fun h => ?_, fun h => h.integrableOn⟩
  refine h.integrable_of_forall_notMem_eq_zero fun x hx => ?_
  contrapose! hx
  exact h1s (mem_support.2 hx)

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f : α → E
⊢ IntegrableAtFilter (-f) l μ ↔ IntegrableAtFilter f l μ

Tactic:
by
  refine ⟨fun h ↦ ?_, fun h ↦ h.neg⟩
  convert! h.neg; simp

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:
rw [sum_const, card_univ, nsmul_eq_mul]
      ring

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✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : Fintype ι
𝕜' : Type u_1
inst✝³ : SeminormedRing 𝕜'
inst✝² : Module 𝕜' G
inst✝¹ : IsBoundedSMul 𝕜' G
inst✝ : SMulCommClass 𝕜 𝕜' G
c : 𝕜'
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
⊢ ‖c • f m‖ ≤ ‖c‖ * ‖f‖ * ∏ i, ‖m i‖

Tactic:
norm_smul_le,

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : Fintype ι
𝕜' : Type u_1
inst✝³ : SeminormedRing 𝕜'
inst✝² : Module 𝕜' G
inst✝¹ : IsBoundedSMul 𝕜' G
inst✝ : SMulCommClass 𝕜 𝕜' G
c : 𝕜'
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
⊢ ‖c‖ * ‖f m‖ ≤ ‖c‖ * ‖f‖ * ∏ i, ‖m i‖
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: case neg
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
hm : Multipliable (fun i ↦ f i) L
h✝¹ : ¬(L.HasSupport ∧ ((Function.mulSupport fun i ↦ f i) ∩ L.support).Finite)
h✝ : ¬HasProd (fun i ↦ f i) 1 L
⊢ AEStronglyMeasurable (Exists.choose hm) μ

Tactic:
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.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε'' : Type u_5
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace ε''
inst✝³ : ESeminormedAddMonoid ε''
μ : Measure X
inst✝² : PseudoMetrizableSpace ε''
f : X → ε''
inst✝¹ : LinearOrder X
inst✝ : CompactIccSpace X
⊢ Integrable f μ ↔ (IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f atTop μ) ∧ LocallyIntegrable f μ

Tactic:
constructor
  · exact fun hf ↦ ⟨⟨hf.integrableAtFilter _, hf.integrableAtFilter _⟩, hf.locallyIntegrable⟩
  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
    exact (IntegrableAtFilter.sup_iff.mpr h.1).filter_mono cocompact_le_atBot_atTop

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

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : LocallyIntegrableOn f s μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ LocallyIntegrableOn (fun x ↦ f x * g x) s μ

Tactic:
rw [MeasureTheory.locallyIntegrableOn_iff hs] at hf ⊢
  exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEEqFun.mk ⇑(toSimpleFunc (f.toLp hfi)) ?m.37 = AEEqFun.mk ⇑f ?m.38

pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑(toSimpleFunc (f.toLp hfi))) μ

pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑f) μ

Tactic:
exact Classical.choose_spec (toLp f hfi).2

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
⊢ ⇑(toSimpleFunc 0) =ᵐ[μ] 0

Tactic:
filter_upwards [toSimpleFunc_eq_toFun (0 : Lp.simpleFunc E p μ),
    Lp.coeFn_zero E 1 μ] with _ h₁ _

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
a✝¹ : α
h₁ : (toSimpleFunc 0) a✝¹ = ↑↑↑0 a✝¹
a✝ : ↑↑0 a✝¹ = 0 a✝¹
⊢ (toSimpleFunc 0) a✝¹ = 0 a✝¹
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
⊢ DominatedFinMeasAdditive μ' T C

Tactic:
have h' : ∀ s, μ s = ∞ → μ' s = ∞ := fun s hs ↦ top_unique <| hs.symm.trans_le (h _)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
h' : ∀ (s : Set α), μ s = ∞ → μ' s = ∞
⊢ DominatedFinMeasAdditive μ' T C
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁵ : NormedDivisionRing 𝕜
inst✝¹⁴ : NormedDivisionRing 𝕜₂
inst✝¹³ : AddCommGroup E
inst✝¹² : NormedAddCommGroup Eₗ
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : NormedAddCommGroup Fₗ
inst✝⁹ : Module 𝕜 E
inst✝⁸ : Module 𝕜 Eₗ
inst✝⁷ : IsBoundedSMul 𝕜 Eₗ
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : Module 𝕜₂ Fₗ
inst✝⁴ : IsBoundedSMul 𝕜₂ Fₗ
inst✝³ : CompleteSpace Eₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖
h_dense₂ : DenseRange ⇑e₂
h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖
⊢ ∀ x ∈ Set.range ⇑e₁, ((e₁ ∘ₛₗ ↑f.symm).extendOfNorm e₂) ((↑((e₂ ∘ₛₗ ↑f).extendOfNorm e₁)).toFun x) = x

Tactic:
rintro _ ⟨_, rfl⟩

Goal state after:
case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁵ : NormedDivisionRing 𝕜
inst✝¹⁴ : NormedDivisionRing 𝕜₂
inst✝¹³ : AddCommGroup E
inst✝¹² : NormedAddCommGroup Eₗ
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : NormedAddCommGroup Fₗ
inst✝⁹ : Module 𝕜 E
inst✝⁸ : Module 𝕜 Eₗ
inst✝⁷ : IsBoundedSMul 𝕜 Eₗ
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : Module 𝕜₂ Fₗ
inst✝⁴ : IsBoundedSMul 𝕜₂ Fₗ
inst✝³ : CompleteSpace Eₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖
h_dense₂ : DenseRange ⇑e₂
h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖
w✝ : E
⊢ ((e₁ ∘ₛₗ ↑f.symm).extendOfNorm e₂) ((↑((e₂ ∘ₛₗ ↑f).extendOfNorm e₁)).toFun (e₁ w✝)) = e₁ w✝
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁵ : NormedDivisionRing 𝕜
inst✝¹⁴ : NormedDivisionRing 𝕜₂
inst✝¹³ : AddCommGroup E
inst✝¹² : NormedAddCommGroup Eₗ
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : NormedAddCommGroup Fₗ
inst✝⁹ : Module 𝕜 E
inst✝⁸ : Module 𝕜 Eₗ
inst✝⁷ : IsBoundedSMul 𝕜 Eₗ
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : Module 𝕜₂ Fₗ
inst✝⁴ : IsBoundedSMul 𝕜₂ Fₗ
inst✝³ : CompleteSpace Eₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖
h_dense₂ : DenseRange ⇑e₂
h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖
⊢ ∀ x ∈ Set.range ⇑e₂, (↑((e₂ ∘ₛₗ ↑f).extendOfNorm e₁)).toFun (((e₁ ∘ₛₗ ↑f.symm).extendOfNorm e₂) x) = x

Tactic:
rintro _ ⟨_, rfl⟩

Goal state after:
case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁵ : NormedDivisionRing 𝕜
inst✝¹⁴ : NormedDivisionRing 𝕜₂
inst✝¹³ : AddCommGroup E
inst✝¹² : NormedAddCommGroup Eₗ
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : NormedAddCommGroup Fₗ
inst✝⁹ : Module 𝕜 E
inst✝⁸ : Module 𝕜 Eₗ
inst✝⁷ : IsBoundedSMul 𝕜 Eₗ
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : Module 𝕜₂ Fₗ
inst✝⁴ : IsBoundedSMul 𝕜₂ Fₗ
inst✝³ : CompleteSpace Eₗ
inst✝² : CompleteSpace Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : E ≃ₛₗ[σ₁₂] F
e₁ : E →ₗ[𝕜] Eₗ
e₂ : F →ₗ[𝕜₂] Fₗ
h_dense₁ : DenseRange ⇑e₁
h_norm₁ : ∃ C, ∀ (x : E), ‖e₂ (f x)‖ ≤ C * ‖e₁ x‖
h_dense₂ : DenseRange ⇑e₂
h_norm₂ : ∃ C, ∀ (x : F), ‖e₁ (f.symm x)‖ ≤ C * ‖e₂ x‖
w✝ : F
⊢ (↑((e₂ ∘ₛₗ ↑f).extendOfNorm e₁)).toFun (((e₁ ∘ₛₗ ↑f.symm).extendOfNorm e₂) (e₂ w✝)) = e₂ w✝
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case refine_2
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
y : F
hy : y ∈ {x ∈ f.range | x ≠ 0}
⊢ μ.real (s.piecewise (⇑f) 0 ⁻¹' {y}) • y = (μ.restrict s).real (⇑f ⁻¹' {y}) • y

Tactic:
Set.piecewise_eq_indicator,

Goal state after:
case refine_2
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
y : F
hy : y ∈ {x ∈ f.range | x ≠ 0}
⊢ μ.real (s.indicator ⇑f ⁻¹' {y}) • y = (μ.restrict s).real (⇑f ⁻¹' {y}) • y
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
⊢ ‖∫ (a : α), f a ∂μ‖ ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
simp only [integral_eq_setToFun]
  exact (norm_setToFun_le_toReal _ (by simp)).trans (by simp)

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : a ~[l] b
huv : u ~[l] v
φ : α → 𝕜
hφ : Tendsto φ l (𝓝 1)
habφ : a =ᶠ[l] φ * b
⊢ (fun x ↦ a x • u x) ~[l] fun x ↦ b x • v x

Tactic:
have : ((fun x ↦ a x • u x) - (fun x ↦ b x • v x)) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x) := by
    convert!
      (habφ.comp₂ (· • ·) <| EventuallyEq.refl _ u).fun_sub
        (EventuallyEq.refl _ fun x ↦ b x • v x) using 1
    ext
    rw [Pi.mul_apply, mul_comm, mul_smul, ← smul_sub]

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : a ~[l] b
huv : u ~[l] v
φ : α → 𝕜
hφ : Tendsto φ l (𝓝 1)
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
⊢ (fun x ↦ a x • u x) ~[l] fun x ↦ b x • v x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : ClosedIciTopology E
f g : α → E
hf : 0 ≤ᵐ[μ] f
hgi : Integrable g μ
h : f ≤ᵐ[μ] g
⊢ ∫ (a : α), f a ∂μ ≤ ∫ (a : α), g a ∂μ

Tactic:
by_cases hfi : Integrable f μ
  · exact integral_mono_ae hfi hgi h
  · exact integral_undef hfi ▸ integral_nonneg_of_ae (hf.trans h)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
a : α
ha : Monotone fun n ↦ f n a
n : ℕ
⊢ 0 ≤ f' n a

Tactic:
simp [f', ha zero_le]

Goal state after:
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹⁵ : Ring R
inst✝¹⁴ : AddCommGroup V
inst✝¹³ : Module R V
inst✝¹² : TopologicalSpace P
inst✝¹¹ : AddTorsor V P
inst✝¹⁰ : AddCommGroup W
inst✝⁹ : Module R W
inst✝⁸ : TopologicalSpace Q
inst✝⁷ : AddTorsor W Q
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace W
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : ContinuousVAdd V P
inst✝ : ContinuousVAdd W Q
f : P →ᴬ[R] Q
p₀ p₁ : P
c : R
⊢ f ((lineMap p₀ p₁) c) = (lineMap (f p₀) (f p₁)) c

Tactic:
by
  simp_rw [coe_lineMap_eq, apply_lineMap]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] G)
h_eq : ∀ (a : α), (SimpleFunc.map (fun x ↦ ‖x‖ₑ) (toSimpleFunc f)) a = ‖(toSimpleFunc f) a‖ₑ
x : G
a✝ : x ∈ (toSimpleFunc f).range
hx0 : x = 0
⊢ ‖x‖ₑ * μ (⇑(toSimpleFunc f) ⁻¹' {x}) ≠ ∞

Tactic:
rw [hx0]; simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
a : α
ha_mono : Monotone fun i ↦ f i a
i j : ℕ
⊢ i ≤ j → (fun i ↦ f' i a) i ≤ (fun i ↦ f' i a) j

Tactic:
hij

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
a : α
ha_mono : Monotone fun i ↦ f i a
i j : ℕ
hij : i ≤ j
⊢ (fun i ↦ f' i a) i ≤ (fun i ↦ f' i a) j
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
a : α
ha : Antitone fun i ↦ f i a
i j : ℕ
⊢ i ≤ j → (fun i ↦ f' i a) i ≤ (fun i ↦ f' i a) j

Tactic:
hij

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
f' : ℕ → α → ℝ := fun i a ↦ -f i a
F' : α → ℝ := fun a ↦ -F a
a : α
ha : Antitone fun i ↦ f i a
i j : ℕ
hij : i ≤ j
⊢ (fun i ↦ f' i a) i ≤ (fun i ↦ f' i a) j
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
by
  /- This result is already known for spaces `ι → ℝ`. We use a continuous linear equiv between
    `E'` and such a space to transfer the result to `E'`. -/
  let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E'
  let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
  have LA : LipschitzWith ‖A.toContinuousLinearMap‖₊ A := by apply A.lipschitz
  have L : LipschitzOnWith (‖A.toContinuousLinearMap‖₊ * K) (A ∘ f) s :=
    LA.comp_lipschitzOnWith hf
  obtain ⟨g, hg, gs⟩ :
    ∃ g : α → ι → ℝ, LipschitzWith (‖A.toContinuousLinearMap‖₊ * K) g ∧ EqOn (A ∘ f) g s :=
    L.extend_pi
  refine ⟨A.symm ∘ g, ?_, ?_⟩
  · have LAsymm : LipschitzWith ‖A.symm.toContinuousLinearMap‖₊ A.symm := by
      apply A.symm.lipschitz
    apply (LAsymm.comp hg).weaken
    rw [lipschitzExtensionConstant, ← mul_assoc]
    exact mul_le_mul' (le_max_left _ _) le_rfl
  · intro x hx
    have : A (f x) = g x := gs hx
    simp only [(· ∘ ·), ← this, A.symm_apply_apply]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
inst✝ : IsEmpty α
⊢ dist f g = 0

Tactic:
by
  rw [(ext isEmptyElim : f = g), dist_self]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
x : ↑(range ⇑f)ᶜ
⊢ dist (h₁ ↑x) (h₂ ↑x) = dist ((extend f g₁ h₁) ↑x) ((extend f g₂ h₂) ↑x)

Tactic:
rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : FiniteDimensional 𝕜 E
f : X → E →L[𝕜] F
s : Set X
x : X
h : ∀ (y : E), ContinuousWithinAt (fun q ↦ (f q) y) s x
e : (E →L[𝕜] F) ≃L[𝕜] Fin (finrank 𝕜 E) → F :=
  ((ContinuousLinearEquiv.ofFinrankEq ⋯).arrowCongr 1).trans (ContinuousLinearEquiv.piRing (Fin (finrank 𝕜 E)))
⊢ ContinuousWithinAt f s x

Tactic:
e.toHomeomorph.isInducing.continuousWithinAt_iff

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : FiniteDimensional 𝕜 E
f : X → E →L[𝕜] F
s : Set X
x : X
h : ∀ (y : E), ContinuousWithinAt (fun q ↦ (f q) y) s x
e : (E →L[𝕜] F) ≃L[𝕜] Fin (finrank 𝕜 E) → F :=
  ((ContinuousLinearEquiv.ofFinrankEq ⋯).arrowCongr 1).trans (ContinuousLinearEquiv.piRing (Fin (finrank 𝕜 E)))
⊢ ContinuousWithinAt (⇑e.toHomeomorph ∘ f) s x
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
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
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝¹ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : StronglyMeasurable f
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
hF : CompleteSpace F
hfi' : Integrable f μ'
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range f ∪ {0})
i : ℕ
y : E
x✝ : y ∈ (SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i).range
hy : ⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {y} = ∅
⊢ (T (⇑(SimpleFunc.approxOn f ⋯ (insert 0 (range f)) 0 ⋯ i) ∘ φ ⁻¹' {y})) y = 0

Tactic:
simp [hy, hT.1.map_empty_eq_zero]

Goal state after: