mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
⊢ Function.Injective fun x ↦
    match x with
    | Sum.inl a => f a
    | Sum.inr b => g b

Tactic:
by
      rintro (a₁ | b₁) (a₂ | b₂) f_eq <;>
        simp only at f_eq
      · rw [f.injective f_eq]
      · exfalso
        exact disj.le_bot ⟨⟨a₁, f_eq⟩, ⟨b₂, by simp⟩⟩
      · exfalso
        exact disj.le_bot ⟨⟨a₂, rfl⟩, ⟨b₁, f_eq⟩⟩
      · rw [g.injective f_eq]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : SeminormedRing A
inst✝⁵ : Algebra 𝕜 A
E : Type u_3
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module A E
inst✝¹ : IsBoundedSMul A E
inst✝ : IsScalarTower 𝕜 A E
x : A
y : E
⊢ ‖(x, y).1 • (x, y).2‖ ≤ 1 * ‖x‖ * ‖y‖

Tactic:
one_mul,

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : SeminormedRing A
inst✝⁵ : Algebra 𝕜 A
E : Type u_3
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module A E
inst✝¹ : IsBoundedSMul A E
inst✝ : IsScalarTower 𝕜 A E
x : A
y : E
⊢ ‖(x, y).1 • (x, y).2‖ ≤ ‖x‖ * ‖y‖
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 (s₁ ∩ s₂) = 0
h' : μ s₁ ≠ ∞
⊢ (μ (s₁ \ s₂)).toReal = (μ s₁).toReal

Tactic:
measure_sdiff_null'

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

α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : μ s₁ ≠ ∞
⊢ μ (s₁ ∩ s₂) = 0
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
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) = ‖c‖ₑ * ENNReal.ofReal (μ s ^ (1 / p.toReal)).toReal

Tactic:
ENNReal.ofReal_toReal

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
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) = ‖c‖ₑ * μ s ^ (1 / p.toReal)

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
⊢ μ s ^ (1 / p.toReal) ≠ ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case pos
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : IsFiniteMeasure μ
f : α → ε
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : ¬p = 0
h_top : p = ∞
⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ

Tactic:
simp [h_top]

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : ¬p = 0
h_top : p = ∞
⊢ Integrable (fun x ↦ ‖f x‖ ^ p.toReal) μ

Tactic:
simp [h_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✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
⊢ indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) c

Tactic:
rw [← MemLp.toLp_const, indicatorConstLp]

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
⊢ MemLp.toLp (Set.univ.indicator fun x ↦ c) ⋯ = MemLp.toLp (fun x ↦ c) ⋯
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, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖) ≤
    C * ∑ _i, ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)

Tactic:
by gcongr; apply A

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 g : ContinuousMultilinearMap 𝕜 E G
x : (i : ι) → E i
⊢ ‖(f + g) x‖ ≤ ‖f‖ * ∏ i, ‖x i‖ + ‖g‖ * ∏ i, ‖x i‖

Tactic:
exact norm_add_le_of_le (le_opNorm _ _) (le_opNorm _ _)

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
this : ∀ (x : ↑s), ∃ u, IsOpen[inst✝³] u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s) μ
⊢ ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen[inst✝³] u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, IntegrableOn f (u ∩ s) μ

Tactic:
choose u u_open xu hu using this

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen[inst✝³] (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) μ
⊢ ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen[inst✝³] u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case refine_1
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖

pp: case refine_2
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖

Tactic:
· refine opNorm_le_bound (norm_nonneg (1 : A)) ?_
    simp

Goal state after:
case refine_2
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case refine_1.inl
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen[inst✝³] (u n)

Tactic:
rw [h]
      exact isOpen_empty

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

Goal state before:
pp: case refine_2
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
⊢ s ⊆ ⋃ n, u n

pp: case refine_3
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
⊢ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ

Tactic:
· intro x hx
    obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
    have : v ∈ range u := by rw [← hu]; exact subset_insert ∅ T hv
    obtain ⟨n, rfl⟩ : ∃ n, u n = v := by simpa only [mem_range] using this
    exact mem_iUnion_of_mem _ h'v

Goal state after:
case refine_3
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
⊢ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ↥(Lp β 1 μ)
⊢ ENNReal.ofReal (∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ).toReal = ∫⁻ (x : α), ‖↑↑f x‖ₑ ∂μ

Tactic:
ENNReal.ofReal_toReal

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ↥(Lp β 1 μ)
⊢ ∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ = ∫⁻ (x : α), ‖↑↑f x‖ₑ ∂μ

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ↥(Lp β 1 μ)
⊢ ∫⁻ (a : α), ‖↑↑f a‖ₑ ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case mpr
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
h : (IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f atTop μ) ∧ LocallyIntegrable f μ
⊢ IntegrableAtFilter f (cocompact X) μ

Tactic:
exact (IntegrableAtFilter.sup_iff.mpr h.1).filter_mono cocompact_le_atBot_atTop

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

Goal state before:
pp: case refine_2.a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
i : ι
hi : a ∈ s
hai : a = i
⊢ False

Tactic:
exact has hi

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
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
c : E
hc : c ≠ 0
s : Set α
hs : MeasurableSet s
hcs : MemLp (⇑(piecewise s hs (const α c) (const α 0))) p μ
⊢ μ s < ∞

Tactic:
by
  have : Function.support (const α c) = Set.univ := Function.support_const hc
  simpa only [memLp_iff_finMeasSupp hp_pos hp_ne_top, finMeasSupp_iff_support,
    support_indicator, Set.inter_univ, this] using hcs

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
⊢ k • ↑f ∈ simpleFunc E p μ

Tactic:
by
      rcases f with ⟨f, ⟨s, hs⟩⟩
      use k • s
      apply Eq.trans (AEEqFun.smul_mk k s s.aestronglyMeasurable).symm _
      rw [hs]
      rfl

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 : ↥(simpleFunc E p μ)
⊢ ⇑(toSimpleFunc (-f)) =ᵐ[μ] -⇑(toSimpleFunc f)

Tactic:
filter_upwards [toSimpleFunc_eq_toFun (-f), toSimpleFunc_eq_toFun f,
    Lp.coeFn_neg (f : Lp E p μ)] with _

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : ↥(simpleFunc E p μ)
a✝ : α
⊢ (toSimpleFunc (-f)) a✝ = ↑↑↑(-f) a✝ →
    (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑↑(-↑f) a✝ = (-↑↑↑f) a✝ → (toSimpleFunc (-f)) a✝ = (-⇑(toSimpleFunc f)) a✝
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
c : ℝ≥0∞
s : Set α
⊢ weightedSMul (c • μ) s = c.toReal • weightedSMul μ s

Tactic:
ext1 x
  simp [weightedSMul_apply, smul_smul]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
b : E
x✝ : b ∈ f.range
⊢ μ.real (⇑f ⁻¹' {b}) * (ENNReal.toReal ∘ g) b = (g b).toReal * (μ (⇑f ⁻¹' {b})).toReal

Tactic:
mul_comm,

Goal state after:
α : Type u_1
E : Type u_2
inst✝ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
f : α →ₛ E
g : E → ℝ≥0∞
hf : Integrable (⇑f) μ
hg0 : g 0 = 0
ht : ∀ (b : E), g b ≠ ∞
hf' : f.FinMeasSupp μ
b : E
x✝ : b ∈ f.range
⊢ (ENNReal.toReal ∘ g) b * μ.real (⇑f ⁻¹' {b}) = (g b).toReal * (μ (⇑f ⁻¹' {b})).toReal
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
f : ↥(α →₁ₛ[μ] E)
⊢ ‖integral f‖ ≤ ‖f‖

Tactic:
rw [integral, norm_eq_integral]
  exact (toSimpleFunc f).norm_integral_le_integral_norm (SimpleFunc.integrable f)

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

Goal state before:
pp: case const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
⊢ ∀ (c : E) {s : Set α} (hs : MeasurableSet s),
    MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ →
      motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))

Tactic:
intro c s hs h

Goal state after:
case const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
c : E
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
⊢ motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: 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 = 0

Tactic:
rw [lintegral_eq_zero_iff']

Goal state after:
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
⊢ (fun a ↦ ENNReal.ofReal (-f a)) =ᵐ[μ] 0

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
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (-f a)) μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: 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
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (-f a)) μ

Tactic:
· exact measurable_ofReal.comp_aemeasurable hfm.aemeasurable.neg

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
P : Type u_7
inst✝ : NormedAddCommGroup P
f : α → P
hf : Integrable f μ
⊢ ENNReal.ofReal (∫⁻ (x : α), ‖f x‖ₑ ∂μ).toReal = ∫⁻ (x : α), ‖f x‖ₑ ∂μ

Tactic:
ENNReal.ofReal_toReal

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
P : Type u_7
inst✝ : NormedAddCommGroup P
f : α → P
hf : Integrable f μ
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ = ∫⁻ (x : α), ‖f x‖ₑ ∂μ

α : Type u_1
m : MeasurableSpace α
μ : Measure α
P : Type u_7
inst✝ : NormedAddCommGroup P
f : α → P
hf : Integrable f μ
⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ ≠ ∞
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case e'_5
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
x✝ : α
⊢ b x✝ • (φ x✝ • u x✝ - v x✝) = (φ x✝ * b x✝) • u x✝ - b x✝ • v x✝

Tactic:
mul_comm,

Goal state after:
case e'_5
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
x✝ : α
⊢ b x✝ • (φ x✝ • u x✝ - v x✝) = (b x✝ * φ x✝) • u x✝ - b x✝ • v x✝
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
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)
C : ℝ
hC : C > 0
hCuv : IsBigOWith C l u v
⊢ (fun x ↦ φ x • u x - v x) =o[l] v

Tactic:
rw [IsEquivalent] at *

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 - b) =o[l] b
huv : (u - v) =o[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)
C : ℝ
hC : C > 0
hCuv : IsBigOWith C l u v
⊢ (fun x ↦ φ x • u x - v x) =o[l] v
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
⊢ ∫⁻ (a : α), ↑(f a) ∂μ = ENNReal.ofReal (∫⁻ (a : α), ↑(f a) ∂μ).toReal

Tactic:
ENNReal.ofReal_toReal

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

α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hfi : Integrable (fun x ↦ ↑(f x)) μ
⊢ ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

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

Tactic:
rw [lintegral_eq_zero_iff']

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ (fun a ↦ ENNReal.ofReal (f a)) =ᵐ[μ] 0 ↔ f =ᵐ[μ] 0

α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (f a)) μ
Module: Mathlib.Topology.Algebra.AffineSubspace

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁹ : Ring R
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module R V
inst✝⁶ : TopologicalSpace P
inst✝⁵ : AddTorsor V P
inst✝⁴ : AddCommGroup W
inst✝³ : Module R W
inst✝² : TopologicalSpace Q
inst✝¹ : AddTorsor W Q
e : P ≃ᴬ[R] Q
s : AffineSubspace R P
inst✝ : Nonempty ↥s
⊢ Continuous ((↑e).affineSubspaceMap s).invFun

Tactic:
by simpa [Topology.IsEmbedding.subtypeVal.continuous_iff] using!
      (e.continuous_invFun.comp continuous_subtype_val).congr fun x ↦
        (e.eq_symm_apply.mpr
          (AffineEquiv.affineSubspaceMap_apply_symm_apply e.toAffineEquiv s x)).symm

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

Goal state before:
pp: case e_f
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
⊢ ‖x‖ₑ.toReal * (μ (⇑(toSimpleFunc f) ⁻¹' {x})).toReal = (μ (⇑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖

Tactic:
mul_comm,

Goal state after:
case e_f
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
⊢ (μ (⇑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖ₑ.toReal = (μ (⇑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖
Module: Mathlib.Analysis.Normed.Module.RCLike.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : z = 0
⊢ ‖f z‖ ≤ c / r * ‖z‖

Tactic:
rw [z_zero]
    simp only [map_zero, norm_zero, mul_zero]
    exact le_rfl

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ (setToL1 hT) (indicatorConstLp 1 hs hμs x) = (T s) x

Tactic:
rw [← Lp.simpleFunc.coe_indicatorConst hs hμs x]
  exact setToL1_simpleFunc_indicatorConst hT hs hμs.lt_top x

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

Goal state before:
pp: case hg
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : Integrable f (Measure.map Subtype.val (Measure.comap Subtype.val μ))
hi : Countable { x // x ∈ s }
⊢ AEStronglyMeasurable f (Measure.map Subtype.val (Measure.comap Subtype.val μ))

Tactic:
exact Integrable.aestronglyMeasurable hf

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

Goal state before:
pp: case hfm
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_6
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.HolderConjugate q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
⊢ AEStronglyMeasurable (fun a ↦ ‖f a‖ ^ p) μ

Tactic:
exact (hf.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable

Goal state after: