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
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
⊢ Integrable f μ

Tactic:
by
  have : ∀ᵐ x ∂μ, ‖f x‖ ≤ max ‖g₁ x‖ ‖g₂ x‖ := by
    filter_upwards [h_le₁, h_le₂] with x hx1 hx2
    simp only [Real.norm_eq_abs]
    exact abs_le_max_abs_abs hx1 hx2
  have h_le_add : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖‖g₁ x‖ + ‖g₂ x‖‖ := by
    filter_upwards [this] with x hx
    refine hx.trans ?_
    conv_rhs => rw [Real.norm_of_nonneg (by positivity)]
    exact max_le_add_of_nonneg (norm_nonneg _) (norm_nonneg _)
  exact Integrable.mono (by fun_prop) hf h_le_add

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : Subsingleton ι
i : ι
⊢ ‖(ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ ≤ 1

Tactic:
norm_ofSubsingleton

Goal state after:
𝕜 : Type u
ι : Type v
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : Subsingleton ι
i : ι
⊢ ‖ContinuousLinearMap.id 𝕜 G‖ ≤ 1
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
⊢ IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
by_cases hab : a < b

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ

case neg
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : ¬a < b
⊢ IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case a
𝕜 : Type u
ι : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Fintype ι
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ ‖1‖

Tactic:
apply opNorm_le_bound

Goal state after:
case a.hMp
𝕜 : Type u
ι : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Fintype ι
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ 0 ≤ ‖1‖

case a.hM
𝕜 : Type u
ι : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Fintype ι
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ∀ (m : ι → A), ‖(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) m‖ ≤ ‖1‖ * ∏ i, ‖m i‖
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
by_cases hab : a < b

Goal state after:
case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ

case neg
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : ¬a < b
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
t : Submodule 𝕜 E
s_compl_t : IsCompl s t
this : FiniteDimensional 𝕜 ↥t
⊢ IsClosed[inst✝⁶] ↑(Submodule.map f (s ⊔ t))

Tactic:
Submodule.map_sup

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
t : Submodule 𝕜 E
s_compl_t : IsCompl s t
this : FiniteDimensional 𝕜 ↥t
⊢ IsClosed[inst✝⁶] ↑(Submodule.map f s ⊔ Submodule.map f t)
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✝ g✝ : α →ₘ[μ] β
f : α → β
hf : AEStronglyMeasurable f μ
g : α → β
hg : AEStronglyMeasurable g μ
hfi : MeasureTheory.Integrable f μ
hgi : MeasureTheory.Integrable g μ
⊢ MeasureTheory.Integrable (f + g) μ

Tactic:
exact hfi.add hgi

Goal state after:
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
inst✝ : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
f₁ f₂ : E i →L[𝕜] E₁ i
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
⊢ ((compContinuousLinearMapL (update f i (f₁ + f₂))) g) x =
    ((compContinuousLinearMapL (update f i f₁) + compContinuousLinearMapL (update f i f₂)) g) x

Tactic:
change (g fun j ↦ update f i (f₁ + f₂) j <| x j) =
        (g fun j ↦ update f i f₁ j <| x j) + g fun j ↦ update f i f₂ j (x j)

Goal state after:
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
inst✝ : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
f₁ f₂ : E i →L[𝕜] E₁ i
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
⊢ (g fun j ↦ (update f i (f₁ + f₂) j) (x j)) = (g fun j ↦ (update f i f₁ j) (x j)) + g fun j ↦ (update f i f₂ j) (x j)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ Integrable (fun a ↦ K.indicator f a • g a) μ

Tactic:
apply Integrable.smul_of_top_left

Goal state after:
case hφ
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := ⋯
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ Integrable (K.indicator f) μ

case hf
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := ⋯
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ MemLp g ∞ μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
F : Type u_5
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup F
p : ℝ
f : α →ₛ F
μ : Measure α
⊢ eLpNorm' (⇑f) p μ = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (⇑f ⁻¹' {y})) ^ (1 / p)

Tactic:
have h_map : (‖f ·‖ₑ ^ p) = f.map (‖·‖ₑ ^ p) := by simp; rfl

Goal state after:
α : Type u_1
F : Type u_5
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup F
p : ℝ
f : α →ₛ F
μ : Measure α
h_map : (fun x ↦ ‖f x‖ₑ ^ p) = ⇑(map (fun x ↦ ‖x‖ₑ ^ p) f)
⊢ eLpNorm' (⇑f) p μ = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (⇑f ⁻¹' {y})) ^ (1 / p)
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
h' : ∀ (s : Set α), μ s = ∞ → μ' s = ∞
s : Set α
hs : MeasurableSet s
hμ's : μ' s < ∞
hμs : μ s < ∞
⊢ C * μ.real s ≤ C * μ'.real s

Tactic:
simp only [measureReal_def]
      gcongr
      exact hμ's.ne

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

Goal state before:
pp: case pos
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : g (f a) = 0
⊢ (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = ∑ j ∈ f.range with g j = g (f a), (T (⇑f ⁻¹' {j})) (g j)

Tactic:
simp_rw [h0]
    rw [map_zero, Finset.sum_eq_zero fun x hx => ?_]
    rw [mem_filter] at hx
    rw [hx.2, map_zero]

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

Goal state before:
pp: case hb
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx✝ : 0 ≤ f x
hx : 0 < f x
⊢ ν (⇑f ⁻¹' {f x}) ≠ ∞

Tactic:
exact integrable_iff.mp hfν (f x) hx.ne' |>.ne

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

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
h_add : FinMeasAdditive μ T
c : ℝ
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, c • (T (⇑f ⁻¹' {x})) x = c • setToSimpleFunc T f

Tactic:
simp only [setToSimpleFunc, smul_sum]

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

Goal state before:
pp: α : Type u_1
F : Type u_3
𝕜 : Type u_6
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : SMulZeroClass 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : DistribSMul 𝕜 F
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, c • (T (⇑f ⁻¹' {x})) x = c • setToSimpleFunc T f

Tactic:
simp only [setToSimpleFunc, smul_sum]

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

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
T : Set α → F' →L[ℝ] F
f : α →ₛ F'
⊢ ∑ x ∈ f.range, ‖(T (⇑f ⁻¹' {x})) x‖ ≤ ∑ x ∈ f.range, ‖T (⇑f ⁻¹' {x})‖ * ‖x‖

Tactic:
gcongr with b

Goal state after:
case h
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
T : Set α → F' →L[ℝ] F
f : α →ₛ F'
b : F'
a✝ : b ∈ f.range
⊢ ‖(T (⇑f ⁻¹' {b})) b‖ ≤ ‖T (⇑f ⁻¹' {b})‖ * ‖b‖
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
⊢ u ~[l] u

Tactic:
IsEquivalent,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
⊢ (u - u) =o[l] u
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
⊢ u ~[l] 0 ↔ u =ᶠ[l] 0

Tactic:
IsEquivalent,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
⊢ (u - 0) =o[l] 0 ↔ u =ᶠ[l] 0
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

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

Tactic:
IsEquivalent,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (u - v) =o[l] v ↔ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v
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
hz : ∀ᶠ (x : α) in l, v x ≠ 0
hequiv : u ~[l] v
this : Tendsto (fun x ↦ u x / v x - v x / v x) l (𝓝 0)
⊢ Tendsto (u / v) l (𝓝 1)

Tactic:
have key : Tendsto (fun x ↦ v x / v x) l (𝓝 1) :=
      (tendsto_congr' <| hz.mono fun x hnz ↦ @div_self _ _ (v x) hnz).mpr tendsto_const_nhds

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
hequiv : u ~[l] v
this : Tendsto (fun x ↦ u x / v x - v x / v x) l (𝓝 0)
key : Tendsto (fun x ↦ v x / v x) l (𝓝 1)
⊢ Tendsto (u / v) l (𝓝 1)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case inr.const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
c : E
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
⊢ ε ≠ 0 →
    MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ →
      ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g

Tactic:
εpos

Goal state after:
case inr.const
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
c : E
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
εpos : ε ≠ 0
⊢ MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ →
    ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
b : ℝ≥0
h : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b
hf : ¬Integrable (fun a ↦ ↑(f a)) μ
⊢ 0 ≤ ↑b

Tactic:
exact b.2

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

Goal state before:
pp: case neg
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
z₁ : E := (↑r * (↑‖z‖)⁻¹) • z
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
eq : f z = ↑‖z‖ / ↑r * f z₁
⊢ ‖↑‖z‖‖ / ‖↑r‖ * ‖f z₁‖ ≤ c / r * ‖z‖

Tactic:
RCLike.norm_coe_norm,

Goal state after:
case neg
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
z₁ : E := (↑r * (↑‖z‖)⁻¹) • z
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
eq : f z = ↑‖z‖ / ↑r * f z₁
⊢ ‖z‖ / ‖↑r‖ * ‖f z₁‖ ≤ c / r * ‖z‖
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
r' : ℝ := max r 2⁻¹
hr' : r' < 1
hlt : 0 < r'
hdlt : d < d / r'
y₀ : E
hy₀F : y₀ ∈ F
hxy₀ : dist x y₀ < d / r'
x_ne_y₀ : x - y₀ ∉ F
y : E
hy : y ∈ F
hy₀y : y₀ + y ∈ F
⊢ r' * dist x y₀ < d

Tactic:
exact (lt_div_iff₀' hlt).1 hxy₀

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
H : x = 0
⊢ False

Tactic:
by simp [H] at xF

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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : T = T'
f : ↥(Lp E 1 μ)
⊢ (setToL1 hT) f = (setToL1 hT') f

Tactic:
apply setToL1_unique hT (A := setToL1 hT') _ f

Goal state after:
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : T = T'
f : ↥(Lp E 1 μ)
⊢ ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = (setToL1 hT') ↑f
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : ↥(Lp E 1 μ)
⊢ (setToL1 hT) f = (setToL1 hT') f

Tactic:
apply setToL1_unique hT (A := setToL1 hT') _ f

Goal state after:
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = T' s
f : ↥(Lp E 1 μ)
⊢ ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = (setToL1 hT') ↑f
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
⊢ HasSum (fun i ↦ ∫ (x : X), f x ∂μ i) (∫ (x : X), f x ∂Measure.sum μ)

Tactic:
simp only [HasSum, ← integral_finsetSum_measure fun i _ ↦ hfi i]

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
⊢ Tendsto (fun s ↦ ∫ (x : X), f x ∂∑ i ∈ s, μ i) (SummationFilter.unconditional ι).filter
    (𝓝 (∫ (x : X), f x ∂Measure.sum μ))
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
this : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
by
    filter_upwards [this] with a ha_tendsto
    convert! ha_tendsto.neg
    · simp [f']
    · simp [F']

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
s : Set α
inst✝ : One β
hs : ∀ a ∈ frontier s, f a = 1
hf : ContinuousOn f (closure[inst✝²] s)
⊢ Continuous[inst✝², inst✝¹] (s.mulIndicator f)

Tactic:
exact continuous_piecewise hs hf continuousOn_const

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C

Tactic:
rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩

Goal state after:
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
hC : ∀ ⦃x : β⦄, x ∈ range ⇑f ∪ range ⇑g → ∀ ⦃y : β⦄, y ∈ range ⇑f ∪ range ⇑g → dist x y ≤ C
⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C} =
    sInf {x | ∃ (h : 0 ≤ x), NNReal.mk x h ∈ {C | ∀ (x : α), nndist (f x) (g x) ≤ C}}

Tactic:
simp_rw [mem_ofPred_eq, ← NNReal.coe_le_coe, NNReal.coe_mk, exists_prop, coe_nndist]

Goal state after: