mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → E
this : (fun x ↦ ↑(f x)) =ᵐ[μ] fun x ↦ ↑(AEMeasurable.mk f hf x)
⊢ Integrable g (μ.withDensity fun x ↦ ↑(f x)) ↔ Integrable g (μ.withDensity fun x ↦ ↑(AEMeasurable.mk f hf x))

Tactic:
rw [withDensity_congr_ae this]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
r : ℝ
u : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
⊢ (↑(f x) ≠ 0 → ↑↑(r • u) x = (r • ↑↑u) x) →
    ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = f x • ↑↑(r • u) x →
      ↑↑(r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = (r • ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯)) x →
        ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x →
          ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = ↑↑((RingHom.id ℝ) r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x

Tactic:
h

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
r : ℝ
u : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
h : ↑(f x) ≠ 0 → ↑↑(r • u) x = (r • ↑↑u) x
⊢ ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = f x • ↑↑(r • u) x →
    ↑↑(r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = (r • ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯)) x →
      ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x →
        ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(r • u) x) ⋯) x = ↑↑((RingHom.id ℝ) r • MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case a
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
x✝ : G
x : (i : ι) → E i
⊢ ‖(constOfIsEmpty 𝕜 E x✝) x‖ ≤ ‖x✝‖

Tactic:
constOfIsEmpty_apply

Goal state after:
case a
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
x✝ : G
x : (i : ι) → E i
⊢ ‖x✝‖ ≤ ‖x✝‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case zero
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_1
inst✝² : SeminormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = 1

Tactic:
rw [norm_mkPiAlgebraFin_zero]
    simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : ContinuousSMul 𝕜 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space F
inst✝ : FiniteDimensional 𝕜 F
f : E →L[𝕜] F
⊢ IsStrictMap ⇑f

Tactic:
isStrictMap_iff_isQuotientMap_rangeFactorization

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : CompleteSpace 𝕜
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : ContinuousSMul 𝕜 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : T2Space F
inst✝ : FiniteDimensional 𝕜 F
f : E →L[𝕜] F
⊢ IsQuotientMap (rangeFactorization ⇑f)
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:
h

Goal state after:
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] ∅
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
h_ind : ∀ (n : ℕ), U ⊆ ↑M + c ^ n • U
h_small : Tendsto (fun n ↦ c ^ n • U) atTop (𝓝 0).smallSets
x : Eᵤ
hx : x ∈ U
m : ℕ → Eᵤ
hm : ∀ (n : ℕ), m n ∈ ↑M
u : ℕ → Eᵤ
hu : ∀ (n : ℕ), u n ∈ c ^ n • U
h_eq : ∀ (n : ℕ), (fun x1 x2 ↦ x1 + x2) (m n) (u n) = x
⊢ Tendsto u atTop (𝓝 0)

Tactic:
intro W hW

Goal state after:
𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
h_ind : ∀ (n : ℕ), U ⊆ ↑M + c ^ n • U
h_small : Tendsto (fun n ↦ c ^ n • U) atTop (𝓝 0).smallSets
x : Eᵤ
hx : x ∈ U
m : ℕ → Eᵤ
hm : ∀ (n : ℕ), m n ∈ ↑M
u : ℕ → Eᵤ
hu : ∀ (n : ℕ), u n ∈ c ^ n • U
h_eq : ∀ (n : ℕ), (fun x1 x2 ↦ x1 + x2) (m n) (u n) = x
W : Set Eᵤ
hW : W ∈ 𝓝 0
⊢ W ∈ map u atTop
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'
inst✝ : DecidableEq ι
m : (i : ι) → E i
i : ι
x y : E i
⊢ { toFun := fun x_1 ↦ (f x_1) (update m i (x + y)), map_add' := ⋯, map_smul' := ⋯ }.mkContinuous
      (‖f‖ * ∏ i_1, ‖update m i (x + y) i_1‖) ⋯ =
    { toFun := fun x_1 ↦ (f x_1) (update m i x), map_add' := ⋯, map_smul' := ⋯ }.mkContinuous
        (‖f‖ * ∏ i_1, ‖update m i x i_1‖) ⋯ +
      { toFun := fun x ↦ (f x) (update m i y), map_add' := ⋯, map_smul' := ⋯ }.mkContinuous
        (‖f‖ * ∏ i_1, ‖update m i y i_1‖) ⋯

Tactic:
by
        ext1
        simp only [add_apply, ContinuousMultilinearMap.map_update_add, LinearMap.coe_mk,
          LinearMap.mkContinuous_apply, AddHom.coe_mk]

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✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f (μ.restrict s)
x : X
a✝ : x ∈ s
t : Set X
ht_mem : t ∈ 𝓝 x
ht_int : IntegrableOn f t (μ.restrict s)
u : Set X
hu_sub : u ⊆ t
hu_o : IsOpen[inst✝³] u
hu_mem : x ∈ u
⊢ IntegrableOn f (s ∩ u) μ

Tactic:
simpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using
    ht_int.mono_set hu_sub

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 ≃ₗᵢ[𝕜] E₁ i
⊢ ‖g.compContinuousLinearMap fun i ↦ ↑↑(f i)‖ = ‖g‖

Tactic:
apply le_antisymm (g.norm_compContinuous_linearIsometry_le fun i => (f i).toLinearIsometry)

Goal state after:
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 ≃ₗᵢ[𝕜] E₁ i
⊢ ‖g‖ ≤ ‖g.compContinuousLinearMap fun i ↦ (f i).toLinearIsometry.toContinuousLinearMap‖
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
μ : Measure X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A μ
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
⊢ IntegrableOn (fun x ↦ g x * g' x) A μ

Tactic:
rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  exact hg.mul_bdd ((hg'.mono hAK).aestronglyMeasurable hA)
    (ae_restrict_of_forall_mem hA fun x hx => hC x (hAK hx))

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

Goal state before:
pp: case refine_2
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
⊢ ((∀ 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)) →
    (∀ i ∈ insert a s, μ (S i) ≠ ∞) →
      (∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)) →
        T (⋃ i ∈ insert a s, S i) = ∑ i ∈ insert a s, T (S i)

Tactic:
h

Goal state after:
case refine_2
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)
⊢ (∀ i ∈ insert a s, μ (S i) ≠ ∞) →
    (∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)) →
      T (⋃ i ∈ insert a s, S i) = ∑ i ∈ insert a s, T (S i)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
𝕜 : Type u_6
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝³ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
inst✝² : SeminormedAddGroup 𝕜
inst✝¹ : DistribSMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
hT : DominatedFinMeasAdditive μ T C
c : 𝕜
⊢ DominatedFinMeasAdditive μ (fun s ↦ c • T s) (‖c‖ * C)

Tactic:
refine ⟨hT.1.smul c, fun s hs hμs => (norm_smul_le _ _).trans ?_⟩

Goal state after:
α : Type u_1
𝕜 : Type u_6
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝³ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
inst✝² : SeminormedAddGroup 𝕜
inst✝¹ : DistribSMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
hT : DominatedFinMeasAdditive μ T C
c : 𝕜
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ‖c‖ * ‖T s‖ ≤ ‖c‖ * C * μ.real s
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 g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → (toSimpleFunc (f + g)) a✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
    ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + (toSimpleFunc g) a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + (toSimpleFunc g) a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝¹ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : (toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝¹ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : (toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
⊢ ↑↑↑f a✝ + ↑↑↑g a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
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)
a✝ : α
h : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝ → (toSimpleFunc (-f)) a✝ = -(toSimpleFunc f) a✝

Tactic:
h

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
h : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -(toSimpleFunc f) a✝
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)
a✝ : α
h✝ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -(toSimpleFunc f) a✝

Tactic:
h

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
h✝ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝
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)
a✝ : α
h✝¹ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : ↑(-↑↑f) a✝ = -↑↑↑f a✝
⊢ ↑(-↑↑f) a✝ = -↑↑↑f a✝

Tactic:
h

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
h✝¹ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : ↑(-↑↑f) a✝ = -↑↑↑f a✝
⊢ -↑↑↑f a✝ = -↑↑↑f a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → (toSimpleFunc (f - g)) a✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
    ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - (toSimpleFunc g) a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - (toSimpleFunc g) a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝
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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h✝¹ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : (toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝
⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝

Tactic:
h

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 g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h✝¹ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : (toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝
⊢ ↑↑↑f a✝ - ↑↑↑g a✝ = ↑↑↑f a✝ - ↑↑↑g a✝
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
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)
a✝ : α
h : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → (toSimpleFunc (k • f)) a✝ = k • (toSimpleFunc f) a✝

Tactic:
h

Goal state after:
α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
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)
a✝ : α
h : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • (toSimpleFunc f) a✝
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
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)
a✝ : α
h✝ : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • (toSimpleFunc f) a✝

Tactic:
h

Goal state after:
α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
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)
a✝ : α
h✝ : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
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)
a✝ : α
h✝¹ : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝
⊢ ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝

Tactic:
h

Goal state after:
α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
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)
a✝ : α
h✝¹ : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝
⊢ k • ↑↑↑f a✝ = k • ↑↑↑f a✝