mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
G' : Type wG'
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E G
g : ContinuousMultilinearMap 𝕜 E G'
x✝ : ℝ≥0
⊢ ‖f.prod g‖₊ ≤ x✝ ↔ max ‖f‖₊ ‖g‖₊ ≤ x✝

Tactic:
by

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
E' : ι' → Type wE'
inst✝⁶ : Fintype ι'
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : Fintype ι
inst✝¹ : (i' : ι') → SeminormedAddCommGroup (E' i')
inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i')
f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')
x✝ : ℝ≥0
⊢ ‖pi f‖₊ ≤ x✝ ↔ ‖f‖₊ ≤ x✝

Tactic:
by

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

Goal state before:
pp: G : ℕ → ℝ → ℝ
f : ℝ → ℝ
μ : Measure ℝ
hGf : ∀ᵐ (x : ℝ) ∂μ, Tendsto (fun n ↦ G n x) atTop (𝓝 (f x))
hG : ∀ (n : ℕ), AEMeasurable (fun x ↦ ‖G n x‖ₑ) μ
⊢ ?m.51 =ᵐ[?m.50] ?m.52

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
G : Type wG
G' : Type wG'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
inst✝ : Subsingleton ι
i : ι
f : G →L[𝕜] G'
⊢ ‖(ofSubsingleton 𝕜 G G' i) f‖ = ‖f‖

Tactic:
by

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:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
⊢ Continuous[inst✝⁵, Pi.topologicalSpace] ⇑ξ.equivFun

Tactic:
by

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
ε' : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : X → ε
μ : Measure X
s : Set X
hf : LocallyIntegrableOn f s μ
g : X → ε'
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ₑ ≤ ‖f x‖ₑ
⊢ LocallyIntegrableOn g s μ

Tactic:
by

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

Goal state before:
pp: X : Type u_1
E : Type u_6
F : Type u_7
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : Measure X
s : Set X
f : X → E
hf : LocallyIntegrableOn f s μ
g : X → F
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ ≤ ‖f x‖
⊢ LocallyIntegrableOn g s μ

Tactic:
by

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 ν : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrableOn f s μ
h : ν.restrict s ≤ μ.restrict s
⊢ LocallyIntegrableOn f s ν

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] E
g : E →L[𝕜] E
⊢ toContinuousLinearMap f = g ↔ f = ↑g

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
g : E →L[𝕜] E
f : E →ₗ[𝕜] E
⊢ ↑g = f ↔ g = toContinuousLinearMap f

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : IsTopologicalAddGroup E
inst✝⁹ : ContinuousSMul 𝕜 E
F : Type w
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousSMul 𝕜 F
inst✝³ : CompleteSpace 𝕜
inst✝² : T2Space E
inst✝¹ : T2Space F
inst✝ : FiniteDimensional 𝕜 E
e : E ≃ₗ[𝕜] F
⊢ ↑e.toContinuousLinearEquiv = e

Tactic:
by

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✝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
⊢ ‖constOfIsEmpty 𝕜 E x‖ = ‖x‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹³ : AddCommGroup E
inst✝¹² : Module 𝕜 E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : IsTopologicalAddGroup E
inst✝⁹ : ContinuousSMul 𝕜 E
F : Type w
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜 F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousSMul 𝕜 F
inst✝³ : CompleteSpace 𝕜
inst✝² : T2Space E
inst✝¹ : T2Space F
inst✝ : FiniteDimensional 𝕜 E
e : E ≃ₗ[𝕜] F
⊢ ↑e.toContinuousLinearEquiv.symm = e.symm

Tactic:
by

Goal state after:
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 (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ

Tactic:
by

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f g : X → ε
μ : Measure X
s : Set X
h : f =ᵐ[μ.restrict s] g
hf : LocallyIntegrableOn f s μ
⊢ LocallyIntegrableOn g s μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
⊢ 0 ≤ ∏ i, ‖m i‖

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
G : Type wG
G' : Type wG'
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
k n : ℕ
f : G [×n]→L[𝕜] G'
s : Finset (Fin n)
hk : #s = k
z : G
⊢ ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k)

Tactic:
by

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
⊢ ∀ (x : ↑s), ∃ u, IsOpen[inst✝³] u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s) μ

Tactic:
by

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

Goal state before:
pp: 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✝ : Nonempty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : Module 𝕜 E
inst✝¹⁵ : TopologicalSpace E
inst✝¹⁴ : IsTopologicalAddGroup E
inst✝¹³ : ContinuousSMul 𝕜 E
F : Type w
inst✝¹² : AddCommGroup F
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : IsTopologicalAddGroup F
inst✝⁸ : ContinuousSMul 𝕜 F
F' : Type x
inst✝⁷ : AddCommGroup F'
inst✝⁶ : Module 𝕜 F'
inst✝⁵ : TopologicalSpace F'
inst✝⁴ : IsTopologicalAddGroup F'
inst✝³ : ContinuousSMul 𝕜 F'
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : T2Space E
v : Basis ι 𝕜 E
⊢ Continuous[Pi.topologicalSpace, inst✝¹⁵] v.equivFun.invFun

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
f : E →L[𝕜] E
hf : f.det ≠ 0
⊢ ↑(f.toContinuousLinearEquivOfDetNeZero hf) = f

Tactic:
by

Goal state after:
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

Goal state after:
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
⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖

Tactic:
convert! ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => 1
    simp

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

Goal state before:
pp: 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:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : UniformSpace E
inst✝⁴ : T2Space E
inst✝³ : IsUniformAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
⊢ CompleteSpace E

Tactic:
by

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

Goal state before:
pp: 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✝ : NormOneClass A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
s t : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 (E ⧸ s)
s_closed : IsClosed[inst✝⁴] ↑s
s_le_t : s ≤ t
⊢ IsClosed[inst✝⁴] ↑t

Tactic:
by

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
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
this : s ⊆ ⋃ x, u x
T : Set ↑s
hT_count : T.Countable
hT_un : ⋃ i ∈ T, u i = ⋃ i, u i
⊢ s ⊆ ⋃ i ∈ T, u i

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n.succ A‖ ≤ 1

Tactic:
by

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

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

Tactic:
by

Goal state after:
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

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
s t : Submodule 𝕜 E
hs : IsClosed[inst✝³] ↑s
ht : FiniteDimensional 𝕜 ↥t
⊢ IsClosed[inst✝³] ↑(s ⊔ t)

Tactic:
by

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
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 ∧ s ⊆ ⋃ i ∈ T, u i

Tactic:
by

Goal state after:
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 ≠ ∞
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
by

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

Goal state before:
pp: 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‖

Tactic:
by

Goal state after:
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
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ

Tactic:
by

Goal state after:
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
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ max 1 ‖1‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
ha : ‖f a‖ₑ ≠ ∞
⊢ μ {a} ≠ ∞

Tactic:
by

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
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 : Set ↑s
T_count : T.Countable
hT : s ⊆ ⋃ i ∈ T, u i
⊢ s ⊆ ⋃ u_1 ∈ u '' T, u_1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ μ {b} ≠ ∞

Tactic:
by

Goal state after:
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)
⊢ IsClosed[inst✝⁶] ↑f.range

Tactic:
by

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

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

Tactic:
by

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✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : CompleteSpace 𝕜
inst✝¹² : AddCommGroup E
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : IsTopologicalAddGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : ContinuousSMul 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
inst✝² : T2Space E
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : T2Space F
f : E →ₗ[𝕜] F
hf : f.ker = ⊥
g : E ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofInjective f ⋯
⊢ IsClosed[inst✝⁶] (Set.range ⇑f)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
inst✝ : NullSingletonClass μ
ha : ‖f a‖ₑ ≠ ∞
hb : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁵ : PartialOrder α
inst✝⁴ : MeasurableSingletonClass α
inst✝³ : TopologicalSpace ε'
inst✝² : ESeminormedAddMonoid ε'
inst✝¹ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
inst✝ : NullSingletonClass μ
hb : ‖f b‖ₑ ≠ ∞
⊢ μ {b} ≠ ∞

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
c : E
⊢ IsClosedMap fun x ↦ x • c

Tactic:
by

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
⊢ (↑f.rangeRestrict).range = ⊤

Tactic:
by

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:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
inst✝ : IsCompletelyPseudoMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
⊢ MeasurableSet {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}

Tactic:
by

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
⊢ ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen[inst✝³] u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, IntegrableOn f (u ∩ s) μ

Tactic:
by

Goal state after:
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
x : Eᵤ
hx : x ∈ U
⊢ x ∈ ↑M + c • U

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
e : E := Classical.choice hE
conv : Set X := {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
mconv : MeasurableSet conv
⊢ StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x

Tactic:
by

Goal state after: