mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Haar.Disintegration

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := L.ker
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
⊢ map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
h_neg : 0 ≤ᵐ[μ.restrict t] -f
⊢ f ≤ᵐ[μ.restrict t] 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ f =ᵐ[μ.restrict t] 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := L.ker
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
⊢ ∃ c₀, c₀ ≠ 0 ∧ c₀ ≠ ∞ ∧ map (⇑M.symm) μ = c₀ • μS.prod μT

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x ↦ (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) (↑n) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∉ s
c : M
Hx : x ∈ tsupport ⇑(f c)
this : g c (↑(chartAt H c) x) = 0
⊢ (f c) x * g c (↑(chartAt H c) x) = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := L.ker
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ∞
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
⊢ ∃ c₁, c₁ ≠ 0 ∧ c₁ ≠ ∞ ∧ map (⇑L') μT = c₁ • ν

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ f =ᵐ[μ.restrict t] 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
this✝ : FiniteDimensional 𝕜 E
this : ProperSpace F
S : Submodule 𝕜 E := L.ker
T : Submodule 𝕜 E
hT : IsCompl S T
M : (↥S × ↥T) ≃ₗ[𝕜] E := S.prodEquivOfIsCompl T hT
M_cont : Continuous ⇑M.symm
P : ↥S × ↥T →ₗ[𝕜] ↥T := snd 𝕜 ↥S ↥T
P_cont : Continuous ⇑P
I✝ : Function.Bijective ⇑(L.domRestrict T)
L' : ↥T ≃ₗ[𝕜] F := LinearEquiv.ofBijective (L.domRestrict T) I✝
L'_cont : Continuous ⇑L'
A : L = ↑L' ∘ₗ P ∘ₗ ↑M.symm
I : map (⇑L) μ = map (⇑L') (map (⇑P) (map (⇑M.symm) μ))
μS : Measure ↥S := addHaar
μT : Measure ↥T := addHaar
c₀ : ℝ≥0∞
c₀_pos : c₀ ≠ 0
c₀_fin : c₀ ≠ ∞
h₀ : map (⇑M.symm) μ = c₀ • μS.prod μT
J : map (⇑P) (μS.prod μT) = μS univ • μT
c₁ : ℝ≥0∞
c₁_pos : c₁ ≠ 0
c₁_fin : c₁ ≠ ∞
h₁ : map (⇑L') μT = c₁ • ν
⊢ 0 < c₀ * c₁

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: case pos
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x ↦ (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) (↑n) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∉ s
c : M
Hx : x ∈ tsupport ⇑(f c)
⊢ x ∉ ↑(chartAt H c) ⁻¹' (chartAt H c).target ∩ ↑(chartAt H c) ⁻¹' ↑(chartAt H c).symm ⁻¹' s

Tactic:
contrapose hx

Goal state after:
case pos
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x ↦ (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) (↑n) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x c : M
Hx : x ∈ tsupport ⇑(f c)
hx : x ∈ ↑(chartAt H c) ⁻¹' (chartAt H c).target ∩ ↑(chartAt H c) ⁻¹' ↑(chartAt H c).symm ⁻¹' s
⊢ x ∈ s
Module: Mathlib.MeasureTheory.Function.AEEqOfIntegral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, (f - g) x ∂μ = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
⊢ f =ᵐ[μ.restrict t] g

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x ↦ (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) (↑n) (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∉ s
c : M
Hx : x ∉ tsupport ⇑(f c)
⊢ x ∉ support ⇑(f c)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : FiniteDimensional ℝ E
F : Type u_2
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : IsManifold I ∞ M
inst✝² : MeasurableSpace M
inst✝¹ : BorelSpace M
inst✝ : T2Space M
f : M → F
μ : Measure M
U : Set M
hU : IsOpen[inst✝⁵] U
hSig : IsSigmaCompact U
hf : LocallyIntegrableOn f U μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → tsupport g ⊆ U → ∫ (x : M), g x • f x ∂μ = 0
⊢ ∀ᵐ (x : M) ∂μ, x ∈ U → f x = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
⊢ ∃ c, 0 < c ∧ c < ∞ ∧ map (⇑L) μ = (c * addHaar univ) • ν

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
c : ℝ≥0∞
c_pos : 0 < c
hc : map (⇑L) μ = (c * addHaar univ) • ν
⊢ 0 < c * addHaar univ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
⊢ ∃ c, 0 < c ∧ map (⇑L) μ = c • ν

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SigmaFinite μ
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
⊢ f =ᵐ[μ] 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝² : μ.IsAddHaarMeasure
inst✝¹ : ν.IsAddHaarMeasure
inst✝ : LocallyCompactSpace E
h : Function.Surjective ⇑L
s : Set F
hs : MeasurableSet s
⊢ (∀ᵐ (x : E) ∂μ, L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SigmaFinite μ
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, (f - g) x ∂μ = 0

Tactic:
by

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

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 : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
this✝² : FiniteDimensional 𝕜 E
this✝¹ : FiniteDimensional 𝕜 F
this✝ : ProperSpace E
this : ProperSpace F
M : F × E →ₗ[𝕜] F := LinearMap.id.coprod L
M_cont : Continuous ⇑M
⊢ Function.Surjective ⇑M

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : SigmaFinite μ
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
⊢ f =ᵐ[μ] g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : IsManifold I ∞ M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : T2Space M
f f' : M → F
μ : Measure M
inst✝ : SigmaCompactSpace M
hf : LocallyIntegrable f μ
hf' : LocallyIntegrable f' μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = ∫ (x : M), g x • f' x ∂μ
⊢ ∀ᵐ (x : M) ∂μ, (f - f') x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : IsManifold I ∞ M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : T2Space M
f f' : M → F
μ : Measure M
inst✝ : SigmaCompactSpace M
hf : LocallyIntegrable f μ
hf' : LocallyIntegrable f' μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = ∫ (x : M), g x • f' x ∂μ
⊢ ∀ᵐ (x : M) ∂μ, f x = f' x

Tactic:
by

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

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 : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : MeasurableSpace F
inst✝⁵ : BorelSpace F
inst✝⁴ : NormedSpace 𝕜 F
L : E →ₗ[𝕜] F
μ : Measure E
ν : Measure F
inst✝³ : μ.IsAddHaarMeasure
inst✝² : ν.IsAddHaarMeasure
inst✝¹ : LocallyCompactSpace E
inst✝ : LocallyCompactSpace F
s : Set F
hs : MeasurableSet s
⊢ (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s : Set M
hs : IsOpen[inst✝⁵] s
⊢ ∃ f, support f = s ∧ ContMDiff I 𝓘(ℝ, ℝ) (↑n) f ∧ ∀ (x : M), 0 ≤ f x

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
v : E
L : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight 1 v
p : E
this : ∀ᵐ (s : ℝ), DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
s : ℝ
hs : DifferentiableAt ℝ (fun t ↦ f (p + t • v)) s
⊢ DifferentiableAt ℝ (fun t ↦ f (p + t • v)) (s + 0)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
v : E
⊢ ∀ᵐ (p : E) ∂μ, LineDifferentiableAt ℝ f p v

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : AEFinStronglyMeasurable f μ
⊢ f =ᵐ[μ] 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, (f - g) x ∂μ = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
t : ℝ
ht : 0 < t
x : E
⊢ ‖t⁻¹ • (f (x + t • v) - f x) * g x‖ = t⁻¹ * ‖f (x + t • v) - f x‖ * ‖g x‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hf : AEFinStronglyMeasurable f μ
hg : AEFinStronglyMeasurable g μ
⊢ f =ᵐ[μ] g

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
t : ℝ
ht : 0 < t
x : E
⊢ t⁻¹ * ‖f (x + t • v) - f x‖ * ‖g x‖ ≤ t⁻¹ * (↑C * ‖x + t • v - x‖) * ‖g x‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f g : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
hg : Integrable g μ
v : E
t : ℝ
ht : 0 < t
x : E
⊢ t⁻¹ * (↑C * ‖x + t • v - x‖) * ‖g x‖ = ↑C * ‖v‖ * ‖g x‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m0
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, f x ∂μ = 0
hf : FinStronglyMeasurable f (μ.trim hm)
t : Set α
ht_meas : MeasurableSet t
htf_zero : ∀ x ∈ tᶜ, f x = 0
htμ : SigmaFinite ((μ.trim hm).restrict t)
⊢ SigmaFinite ((μ.restrict t).trim hm)

Tactic:
by

Goal state after: