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

Goal state before:
pp: case inr
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
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
hx : f x ≠ 0
⊢ f x • ↑↑(u + v) x = f x • ↑↑u x + f x • ↑↑v x

Tactic:
rw [h'' _, Pi.add_apply, smul_add]

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
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
hx : f x ≠ 0
⊢ ↑(f x) ≠ 0
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
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
⊢ Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f

Tactic:
let D := max C 1

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
D : ℝ := max C 1
⊢ Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝³ : Fintype ι
this✝² : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝¹ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this✝ : FiniteDimensional 𝕜 E
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
f : E →ₗ[𝕜] 𝕜
H : ¬finrank 𝕜 ↥f.range = 0
Z : 1 + finrank 𝕜 ↥f.ker = n + 1
this : finrank 𝕜 ↥f.range = 1
⊢ finrank 𝕜 ↥f.ker = n

Tactic:
add_comm,

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝³ : Fintype ι
this✝² : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝¹ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this✝ : FiniteDimensional 𝕜 E
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
f : E →ₗ[𝕜] 𝕜
H : ¬finrank 𝕜 ↥f.range = 0
Z : finrank 𝕜 ↥f.ker + 1 = n + 1
this : finrank 𝕜 ↥f.range = 1
⊢ finrank 𝕜 ↥f.ker = n
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ Integrable (fun a ↦ max (f a) 0) μ

Tactic:
constructor <;> fun_prop

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_8
inst✝² : NormedAddCommGroup 𝕜
inst✝¹ : SMulZeroClass 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : α → β
hf : Integrable f μ
⊢ Integrable (c • f) μ

Tactic:
constructor <;> fun_prop

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
𝕜 : Type u_8
ε : Type u_9
inst✝⁵ : TopologicalSpace ε
inst✝⁴ : ESeminormedAddMonoid ε
inst✝³ : NormedAddCommGroup 𝕜
inst✝² : SMul 𝕜 ε
inst✝¹ : ContinuousConstSMul 𝕜 ε
inst✝ : ENormSMulClass 𝕜 ε
c : 𝕜
f : α → ε
hf : Integrable f μ
⊢ Integrable (c • f) μ

Tactic:
constructor <;> fun_prop

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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
z z' : G
⊢ ContinuousMultilinearMap.mkPiRing 𝕜 ι (z + z') =
    ContinuousMultilinearMap.mkPiRing 𝕜 ι z + ContinuousMultilinearMap.mkPiRing 𝕜 ι z'

Tactic:
ext
    simp [smul_add]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
A B : Submodule 𝕜 E
B_cofg : FiniteDimensional 𝕜 (E ⧸ B)
hB : IsClosed[inst✝³] ↑B
hAB : Disjoint A B
C : Submodule 𝕜 E
B_le_C : B ≤ C
C_compl_A : IsCompl C A
C_cofg : FiniteDimensional 𝕜 (E ⧸ C)
⊢ A.ClosedComplemented

Tactic:
have hC : IsClosed (C : Set E) := isClosed_mono_of_finiteDimensional_quotient hB B_le_C

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
A B : Submodule 𝕜 E
B_cofg : FiniteDimensional 𝕜 (E ⧸ B)
hB : IsClosed[inst✝³] ↑B
hAB : Disjoint A B
C : Submodule 𝕜 E
B_le_C : B ≤ C
C_compl_A : IsCompl C A
C_cofg : FiniteDimensional 𝕜 (E ⧸ C)
hC : IsClosed[inst✝³] ↑C
⊢ A.ClosedComplemented
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : AddCommGroup F
inst✝³ : TopologicalSpace F
inst✝² : Module 𝕜 F
inst✝¹ : T2Space F
f : E →L[𝕜] F
inst✝ : FiniteDimensional 𝕜 ↥(↑f).range
⊢ (↑f).ker.ClosedComplemented

Tactic:
by
  suffices FiniteDimensional 𝕜 (E ⧸ f.ker) from .of_finiteDimensional_quotient f.isClosed_ker
  exact f.toLinearMap.quotKerEquivRange.symm.finiteDimensional

Goal state after:
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 : ↥(Lp β 1 μ)
⊢ edist f g = ∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ

Tactic:
simp only [Lp.edist_def, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_enorm, Pi.sub_apply,
    toReal_one, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
  simp [edist_eq_enorm_sub]

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)
⊢ LocallyIntegrableOn f s μ

Tactic:
intro x _

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✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f (μ.restrict s)
x : X
a✝ : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x) μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case refine_4
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
inst✝ : PseudoMetrizableSpace ε
hf : LocallyIntegrable f μ
k : Set X
hk : IsCompact k
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, ∃ u, IsOpen[inst✝³] u ∧ t ⊆ u ∧ IntegrableOn f u μ

Tactic:
intro x _

Goal state after:
case refine_4
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
inst✝ : PseudoMetrizableSpace ε
hf : LocallyIntegrable f μ
k : Set X
hk : IsCompact k
x : X
a✝ : x ∈ k
⊢ ∃ t ∈ 𝓝[k] x, ∃ u, IsOpen[inst✝³] u ∧ t ⊆ u ∧ IntegrableOn f u μ
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✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : ContinuousMultilinearMap 𝕜 E₁ G
g dg : (i : ι) → E i →L[𝕜] E₁ i
v : (i : ι) → E i
⊢ ((f.fderivCompContinuousLinearMap g) dg) v = ∑ i, f fun j ↦ (update g i (dg i) j) (v j)

Tactic:
simp [fderivCompContinuousLinearMap]

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ MemLp (⇑f) p μ

Tactic:
refine ⟨f.aestronglyMeasurable, ?_⟩

Goal state after:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm (⇑f) p μ < ∞
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: 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)
hSp : ∀ i ∈ sι, μ (S i) ≠ ∞
h_disj : ∀ i ∈ sι, ∀ j ∈ sι, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i)

Tactic:
classical
  revert hSp h_disj
  refine Finset.induction_on sι ?_ ?_
  · simp only [Finset.notMem_empty, IsEmpty.forall_iff, iUnion_false, iUnion_empty, sum_empty,
      imp_true_iff, T_empty]
  intro a s has h hps h_disj
  rw [Finset.sum_insert has, ← h]
  swap; · exact fun i hi => hps i (Finset.mem_insert_of_mem hi)
  swap
  · exact fun i hi j hj hij =>
      h_disj i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj) hij
  rw [←
    h_add (S a) (⋃ i ∈ s, S i) (hS_meas a) (measurableSet_biUnion _ fun i _ => hS_meas i)
      (hps a (Finset.mem_insert_self a s))]
  · congr; convert! Finset.iSup_insert a s S
  · exact (measure_biUnion_lt_top s.finite_toSet fun i hi ↦
      (hps i <| Finset.mem_insert_of_mem hi).lt_top).ne
  · simp_rw [Set.disjoint_iUnion_right]
    intro i hi
    refine h_disj a (Finset.mem_insert_self a s) i (Finset.mem_insert_of_mem hi) fun hai ↦ ?_
    rw [← hai] at hi
    exact has hi

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
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) (-x) = ∑ x ∈ f.range, -(T (⇑f ⁻¹' {x})) x

Tactic:
exact Finset.sum_congr rfl fun x _ => map_neg _ _

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

Goal state before:
pp: α : Type u_1
β : Type u_3
inst✝ : NormedField β
u v : α → β
l : Filter α
φ : α → β
hφ : Tendsto φ l (𝓝 1)
h : u =ᶠ[l] φ * v
⊢ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u⁻¹ =ᶠ[l] φ * v⁻¹

Tactic:
← inv_one

Goal state after:
α : Type u_1
β : Type u_3
inst✝ : NormedField β
u v : α → β
l : Filter α
φ : α → β
hφ : Tendsto φ l (𝓝 1)
h : u =ᶠ[l] φ * v
⊢ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1⁻¹)), u⁻¹ =ᶠ[l] φ * v⁻¹
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
u v : α → β
l : Filter α
inst✝ : ClosedIicTopology β
h : u ~[l] v
⊢ ∃ φ, (∀ᶠ (x : α) in l, 0 < φ x) ∧ u =ᶠ[l] φ * v

Tactic:
obtain ⟨φ, hφ, h_eq⟩ := h.exists_eq_mul

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
u v : α → β
l : Filter α
inst✝ : ClosedIicTopology β
h : u ~[l] v
φ : α → β
hφ : Tendsto φ l (𝓝 1)
h_eq : u =ᶠ[l] φ * v
⊢ ∃ φ, (∀ᶠ (x : α) in l, 0 < φ x) ∧ u =ᶠ[l] φ * v
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
u v t w : α → ℝ
l : Filter α
hu : 0 ≤ v
hw : 0 ≤ w
htu : u ~[l] v
hvw : t ~[l] w
⊢ u + t ~[l] v + w

Tactic:
by
  simp only [IsEquivalent, add_sub_add_comm]
  change (fun x ↦ (u - v) x + (t - w) x) =o[l] (fun x ↦ v x + w x)
  conv => enter [3, x]; rw [← abs_eq_self.mpr (hu x), ← abs_eq_self.mpr (hw x)]
  simpa [← Real.norm_eq_abs] using .add_add htu hvw

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

Goal state before:
pp: 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 : α), ‖(toSimpleFunc f) x‖ₑ ∂μ).toReal = ∑ x ∈ (toSimpleFunc f).range, μ.real (⇑(toSimpleFunc f) ⁻¹' {x}) * ‖x‖

Tactic:
simp_rw [← h_eq, measureReal_def]

Goal state after:
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 : α), (SimpleFunc.map (fun x ↦ ‖x‖ₑ) (toSimpleFunc f)) x ∂μ).toReal =
    ∑ x ∈ (toSimpleFunc f).range, (μ (⇑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: 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
⊢ ∀ a ∈ (toSimpleFunc f).range, ‖a‖ₑ * μ (⇑(toSimpleFunc f) ⁻¹' {a}) ≠ ∞

Tactic:
intro x _

Goal state after:
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
⊢ ‖x‖ₑ * μ (⇑(toSimpleFunc f) ⁻¹' {x}) ≠ ∞
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: case inr
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x : F
s : Set F
hx : x ∈ s
h₀ : infDist x sᶜ ≠ 0
⊢ closedBall x (infDist x sᶜ) ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s

Tactic:
rw [← closure_ball x h₀]

Goal state after:
case inr
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x : F
s : Set F
hx : x ∈ s
h₀ : infDist x sᶜ ≠ 0
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball x (infDist x sᶜ)) ⊆
    closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
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
le_ae : ∀ᵐ (a : α) ∂μ, 0 ≤ ‖f a‖
h : ¬AEStronglyMeasurable f μ
⊢ ‖∫ (a : α), f a ∂μ‖ ≤ ∫ (a : α), ‖f a‖ ∂μ

Tactic:
rw [integral_non_aestronglyMeasurable h, norm_zero]

Goal state after:
case neg
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
le_ae : ∀ᵐ (a : α) ∂μ, 0 ≤ ‖f a‖
h : ¬AEStronglyMeasurable f μ
⊢ 0 ≤ ∫ (a : α), ‖f a‖ ∂μ
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
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ setToFun μ T hT (s.indicator fun x_1 ↦ x) = (T s) x

Tactic:
rw [setToFun_congr_ae hT (@indicatorConstLp_coeFn _ _ _ 1 _ _ _ hs hμs x).symm]
  rw [L1.setToFun_eq_setToL1 hT]
  exact L1.setToL1_indicatorConstLp hT hs hμs x

Goal state after:
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 ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν

Tactic:
rw [integral_add_measure h₁ h₂, nndist_comm, nndist_eq_nnnorm, add_sub_cancel_left]
  exact enorm_integral_le_lintegral_enorm _

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

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
hf : Function.Injective ⇑f
⊢ ∃ K, AntilipschitzWith K ⇑f

Tactic:
by
  obtain ⟨K, -, hK⟩ := f.linear.injective_iff_antilipschitz.mp (f.linear_injective_iff.mpr hf)
  refine ⟨K, AntilipschitzWith.of_le_mul_dist fun x y ↦ ?_⟩
  rw [dist_eq_norm_vsub E, dist_eq_norm_vsub F, ← f.linearMap_vsub]
  exact ZeroHomClass.bound_of_antilipschitz f.linear hK (x -ᵥ y)

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

Goal state before:

Tactic:
‖u e‖₊ = ‖u (∑ i, v.equivFun e i • v i)‖₊ := by rw [v.sum_equivFun]
      _ = ‖∑ i, v.equivFun e i • (u <| v i)‖₊ := by simp only [equivFun_apply, map_sum, map_smul]
      _ ≤ ∑ i, ‖v.equivFun e i • (u <| v i)‖₊ := nnnorm_sum_le _ _
      _ = ∑ i, ‖v.equivFun e i‖₊ * ‖u (v i)‖₊ := by simp only [nnnorm_smul]
      _ ≤ ∑ i, ‖v.equivFun e i‖₊ * M := by gcongr; apply hu
      _ = (∑ i, ‖v.equivFun e i‖₊) * M := by rw [Finset.sum_mul]
      _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) * M := by
        gcongr
        calc
          ∑ i, ‖v.equivFun e i‖₊ ≤ Fintype.card ι • ‖φ e‖₊ := Pi.sum_nnnorm_apply_le_nnnorm _
          _ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) := nsmul_le_nsmul_right (φ.le_opNNNorm e) _
      _ = Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊ := by simp only [smul_mul_assoc, mul_right_comm]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
hx : x ∈ s
hs : s ≠ univ
y : E
hys : y ∈ (interior s)ᶜ
hyd : infDist x sᶜ = dist x y
⊢ ∃ y ∈ frontier s, infDist x sᶜ = dist x y

Tactic:
refine ⟨y, ⟨Metric.closedBall_infDist_compl_subset_closure hx <|
    Metric.mem_closedBall.2 <| ge_of_eq ?_, hys⟩, hyd⟩

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
s : Set E
hx : x ∈ s
hs : s ≠ univ
y : E
hys : y ∈ (interior s)ᶜ
hyd : infDist x sᶜ = dist x y
⊢ infDist x sᶜ = dist y x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
hxf : 0 x ≤ f x
⊢ 0 x ≤ g x → f x * g x = ‖f x‖ * ‖g x‖

Tactic:
hxg

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
x : α
hxf : 0 x ≤ f x
hxg : 0 x ≤ g x
⊢ f x * g x = ‖f x‖ * ‖g x‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫ (a : α), f a * g a ∂μ = ∫ (a : α), ‖f a‖ * ‖g a‖ ∂μ
h_right_f : ∫ (a : α), f a ^ p ∂μ = ∫ (a : α), ‖f a‖ ^ p ∂μ
x : α
⊢ 0 x ≤ g x → g x ^ q = ‖g x‖ ^ q

Tactic:
hxg

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hg_nonneg : 0 ≤ᵐ[μ] g
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫ (a : α), f a * g a ∂μ = ∫ (a : α), ‖f a‖ * ‖g a‖ ∂μ
h_right_f : ∫ (a : α), f a ^ p ∂μ = ∫ (a : α), ‖f a‖ ^ p ∂μ
x : α
hxg : 0 x ≤ g x
⊢ g x ^ q = ‖g x‖ ^ q
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ Tendsto (fun n ↦ ⇑(thickenedIndicator ⋯ E)) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))

Tactic:
by
  have key := thickenedIndicatorAux_tendsto_indicator_closure δseq_lim E
  rw [tendsto_pi_nhds] at *
  intro x
  rw [show indicator (closure E) (fun _ => (1 : ℝ≥0)) x =
        (indicator (closure E) (fun _ => (1 : ℝ≥0∞)) x).toNNReal
      by refine (congr_fun (comp_indicator_const 1 ENNReal.toNNReal toNNReal_zero) x).symm]
  refine Tendsto.comp (tendsto_toNNReal ?_) (key x)
  by_cases x_mem : x ∈ closure E <;> simp [x_mem]

Goal state after: