mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A✝ :
  ∀ p ∈ s,
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (g p)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
A : IsCompact ({q₀.1} ×ˢ k)
t : Set (P × G)
kt : {q₀.1} ×ˢ k ⊆ t
t_open : IsOpen[instTopologicalSpaceProd] t
ht : Bornology.IsBounded (g' '' t)
⊢ ∃ ε, 0 < ε ∧ thickening ε ({q₀.1} ×ˢ k) ⊆ t ∧ ball q₀.1 ε ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformly f' g' l
hf : ∀ᶠ (n : ι) in l, ∀ (x : 𝕜), HasDerivAt (f n) (f' n x) x
hfg : ∀ (x : 𝕜), Tendsto (fun n ↦ f n x) l (𝓝 (g x))
x : 𝕜
⊢ ∀ᶠ (n : ι) in l, ∀ x ∈ Set.univ, HasDerivAt (f n) (f' n x) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformly f' g' l
hf✝ : ∀ᶠ (n : ι) in l, ∀ (x : 𝕜), HasDerivAt (f n) (f' n x) x
hfg : ∀ (x : 𝕜), Tendsto (fun n ↦ f n x) l (𝓝 (g x))
x : 𝕜
hf : ∀ᶠ (n : ι) in l, ∀ x ∈ Set.univ, HasDerivAt (f n) (f' n x) x
⊢ ∀ x ∈ Set.univ, Tendsto (fun n ↦ f n x) l (𝓝 (g x))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformly f' g' l
hf✝ : ∀ᶠ (n : ι) in l, ∀ (x : 𝕜), HasDerivAt (f n) (f' n x) x
hfg✝ : ∀ (x : 𝕜), Tendsto (fun n ↦ f n x) l (𝓝 (g x))
x : 𝕜
hf : ∀ᶠ (n : ι) in l, ∀ x ∈ Set.univ, HasDerivAt (f n) (f' n x) x
hfg : ∀ x ∈ Set.univ, Tendsto (fun n ↦ f n x) l (𝓝 (g x))
⊢ TendstoUniformlyOn f' g' l Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformly f' g' l
hf : ∀ᶠ (n : ι) in l, ∀ (x : 𝕜), HasDerivAt (f n) (f' n x) x
hfg : ∀ (x : 𝕜), Tendsto (fun n ↦ f n x) l (𝓝 (g x))
x : 𝕜
⊢ HasDerivAt g (g' x) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hr : 0 < r
hs : IsClosed s
h : s ⊆ ball x r
hne : s.Nonempty
this : IsCompact s
⊢ ContinuousOn (fun y ↦ dist y x) s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hr : 0 < r
hs : IsClosed s
h : s ⊆ ball x r
⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
r : ℝ
s : Set α
hs : IsClosed s
h : s ⊆ ball x r
⊢ ∃ r' < r, s ⊆ ball x r'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
x₀ : H
s : Set H
inst✝ : NormedSpace ℝ H
μ : Measure ℝ
F : H → ℝ → E
F' : H → ℝ → H →L[ℝ] E
a b : ℝ
bound : ℝ → ℝ
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))
hF_int : IntervalIntegrable (F x₀) μ a b
hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))
h_bound : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ s, ‖F' x t‖ ≤ bound t
bound_integrable : IntervalIntegrable bound μ a b
h_diff : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ∀ x ∈ s, HasFDerivAt (fun x ↦ F x t) (F' x t) x
⊢ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : ProperSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a z : α
r : ℝ
hf : ContinuousOn f (closedBall a r)
hz : z ∈ closedBall a r
hf1 : ∀ z' ∈ sphere a r, f z < f z'
⊢ ∃ z ∈ ball a r, IsLocalMin f z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
⊢ Continuous (dist x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝ : PseudoMetricSpace α
⊢ ProperSpace α ↔ ∀ (x : α), IsProperMap (dist x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
⊢ Continuous (nndist x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
x : α
⊢ Continuous NNReal.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
r : ℝ
⊢ closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
r : ℝ
⊢ IsCompact (closedBall x r)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
r : ℝ
h : r ≠ 0
⊢ closure[inst✝⁵] (ball x r) = closedBall x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
F' : α → E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) s
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
hF'_int : Integrable (⇑L ∘ F') μ
key : HasFDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
⊢ Integrable F' μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
R : ℝ
s : Set E
x : E
hR : 0 < R
hs : IsClosed[inst✝⁵] s
h : s ⊆ ball x R
⊢ ∃ r ∈ Ioo 0 R, s ⊆ ball x r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : SeminormedAddCommGroup G
x₀ : G
R : ℝ
hf : support f ⊆ ball 0 R
hg : ∀ x ∈ ball x₀ R, g x = g x₀
t : G
⊢ (L (f t)) (g (x₀ - t)) = (L (f t)) (g x₀)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (𝓝 x).HasBasis (fun r ↦ 0 < r) (closedBall x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
⊢ (𝓝 x).HasBasis (fun r ↦ 0 < r) (ball x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : SeminormedAddCommGroup G
x₀ : G
R : ℝ
hf : support f ⊆ ball 0 R
hg : ∀ x ∈ ball x₀ R, g x = g x₀
⊢ (f ⋆[L, μ] g) x₀ = ∫ (t : G), (L (f t)) (g x₀) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.EuclideanDist

Goal state before:
pp: F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
inst✝ : FiniteDimensional ℝ G
f g : F → G
n : ℕ∞
hf : ContDiff ℝ (↑n) f
hg : ContDiff ℝ (↑n) g
h : ∀ (x : F), f x ≠ g x
⊢ ContDiff ℝ ↑n fun x ↦ Euclidean.dist (f x) (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Data.Set.Pointwise.Support

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Group α
inst✝¹ : MulAction α β
inst✝ : One γ
c : α
f : β → γ
⊢ (mulSupport fun x ↦ f (c⁻¹ • x)) = c • mulSupport f

Tactic:
by

Goal state after:
Module: Mathlib.Data.Set.Pointwise.Support

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : Group α
inst✝¹ : MulAction α β
inst✝ : Zero γ
c : α
f : β → γ
⊢ (support fun x ↦ f (c⁻¹ • x)) = c • support f

Tactic:
by

Goal state after:
Module: Mathlib.Data.Set.Pointwise.Support

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : One γ
c : α
hc : c ≠ 0
f : β → γ
⊢ (mulSupport fun x ↦ f (c⁻¹ • x)) = c • mulSupport f

Tactic:
by

Goal state after:
Module: Mathlib.Data.Set.Pointwise.Support

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Zero γ
c : α
hc : c ≠ 0
f : β → γ
⊢ (support fun x ↦ f (c⁻¹ • x)) = c • support f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : Nontrivial E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
⊢ NullSingletonClass μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace H
inst✝¹ : SeminormedGroup H
inst✝ : OpensMeasurableSpace H
f : G →* H
hf : Measurable ⇑f
x : G
K : PositiveCompacts G := Classical.arbitrary (PositiveCompacts G)
⊢ ∃ n, 0 < haar (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁸ : MeasurableSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace H
inst✝¹ : SeminormedGroup H
inst✝ : OpensMeasurableSpace H
f : G →* H
hf : Measurable ⇑f
x : G
⊢ ∃ s ∈ 𝓝 x, Bornology.IsBounded (⇑f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
F' : α → E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) s
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x ↦ F x a) (F' a) x₀
⊢ Integrable F' μ ∧ HasDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
hF : CompleteSpace F
hE : finrank ℝ E = 0
this : Subsingleton E
⊢ f = fun x ↦ f 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : SFinite μ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hif : Integrable f μ
hf : support f ⊆ ball 0 R
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
⊢ ConvolutionExistsAt f g x₀ L μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
F' : 𝕜 → α → E
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, HasDerivAt (fun x ↦ F x a) (F' x a) x
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
diff_x₀ : ∀ᵐ (a : α) ∂μ, HasDerivAt (fun x ↦ F x a) (F' x₀ a) x₀
⊢ ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) (ball x₀ ε)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
hF : CompleteSpace F
hR : R ≠ 0
⊢ ∫ (y : E), f y ∂map (fun x ↦ R • x) μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
⊢ ∫ (x : E), f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
hR : 0 ≤ R
⊢ ∫ (x : E), f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
F' : 𝕜 → α → E
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, HasDerivAt (fun x ↦ F x a) (F' x a) x
⊢ Integrable (F' x₀) μ ∧ HasDerivAt (fun n ↦ ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
⊢ ∫ (x : E), f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ (x : E), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
hR : 0 ≤ R
⊢ ∫ (x : E), f (R⁻¹ • x) ∂μ = R ^ finrank ℝ E • ∫ (x : E), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = ∫ (x : E) in ⇑e ⁻¹' ⇑e.symm ⁻¹' s, f (e x) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (y : E) in ⇑e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (y : E) in ⇑e.symm ⁻¹' s, f y ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ |(R ^ finrank ℝ E)⁻¹| • ∫ (y : E) in ⇑e.symm ⁻¹' s, f y ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : 0 < R
⊢ ∫ (x : E) in s, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ (x : E) in R • s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
g : ℝ → F
a : ℝ
⊢ ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
g : ℝ → F
a : ℝ
⊢ ∫ (x : ℝ), g (a⁻¹ * x) = |a| • ∫ (y : ℝ), g y

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
g : ℝ → F
a : ℝ
⊢ ∫ (x : ℝ), g (x * a) = |a⁻¹| • ∫ (y : ℝ), g y

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
g : ℝ → F
a : ℝ
⊢ ∫ (x : ℝ), g (x * a⁻¹) = |a| • ∫ (y : ℝ), g y

Tactic:
by

Goal state after: