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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
this : (c • μ).IsHaarMeasure
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ ↑c * ↑(μ'.haarScalarFactor (c • μ)) = ↑c * ((∫ (x : G), g x ∂μ') / ∫ (x : G), g x ∂c • μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
⊢ ∀ k < N,
    ∃ j < i,
      (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color j

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
G P : Type uP
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : NormedSpace 𝕜 G
inst✝⁶ : NormedAddCommGroup P
inst✝⁵ : NormedSpace 𝕜 P
f : G → E
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hf : LocallyIntegrable f μ
n : ℕ
ih :
  ∀ {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 E']
    [inst_3 : NormedSpace ℝ F] [inst_4 : NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'},
    (∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) →
      ContDiffOn 𝕜 (↑↑n) (↿g) (s ×ˢ univ) → ContDiffOn 𝕜 (↑↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
E' F : Type uP
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hg :
  DifferentiableOn 𝕜 (↿g) (s ×ˢ univ) ∧
    (↑n = ⊤ → AnalyticOn 𝕜 (↿g) (s ×ˢ univ)) ∧ ContDiffOn 𝕜 (↑n) (fderiv 𝕜 ↿g) (s ×ˢ univ)
f' : P → G → P × G →L[𝕜] F := fun p a ↦ (f ⋆[precompR (P × G) L, μ] fun x ↦ fderiv 𝕜 (uncurry g) (p, x)) a
A : ∀ (q₀ : P × G), q₀.1 ∈ s → HasFDerivAt (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (f' q₀.1 q₀.2) q₀
H : ContDiffOn 𝕜 (↑n) (↿f') (s ×ˢ univ)
⊢ ContDiffOn 𝕜 (↑n) (fderiv 𝕜 fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
this : (c • μ).IsHaarMeasure
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ ↑c * ((∫ (x : G), g x ∂μ') / ∫ (x : G), g x ∂c • μ) = ↑c * ((∫ (x : G), g x ∂μ') / c • ∫ (x : G), g x ∂μ)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
this : (c • μ).IsHaarMeasure
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ ↑c ≠ 0

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
this : (c • μ).IsHaarMeasure
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ ↑c * ((∫ (x : G), g x ∂μ') / c • ∫ (x : G), g x ∂μ) = (∫ (x : G), g x ∂μ') / ∫ (x : G), g x ∂μ

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
⊢ c * μ'.haarScalarFactor (c • μ) = μ'.haarScalarFactor μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ∞ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∈ ball 0 1
⊢ 0 < f x

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
⊢ (c • μ').haarScalarFactor (c • μ) = μ'.haarScalarFactor μ

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁵ : TopologicalSpace G
inst✝⁴ : Group G
inst✝³ : IsTopologicalGroup G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
μ : Measure G
inst✝ : μ.IsHaarMeasure
⊢ μ.haarScalarFactor μ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ∞ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
⊢ x ∉ support f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
A : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball 0 1)
f : E → ℝ
f_support : support f = ball 0 1
f_smooth : ContDiff ℝ ∞ f
f_range : range f ⊆ Icc 0 1
B : ∀ (x : E), f x ∈ Icc 0 1
x : E
hx : x ∉ ball 0 1
I1 : x ∉ support f
⊢ -x ∉ support f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
⊢ ∀ n ≤ N, p.color (G n) = n

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ ν : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : ν.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
hG : LocallyCompactSpace G
g : G → ℝ
g_cont : Continuous[inst✝⁸, _] g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
Z : μ'.haarScalarFactor ν • ∫ (x : G), g x ∂ν = (μ'.haarScalarFactor μ * μ.haarScalarFactor ν) • ∫ (x : G), g x ∂ν
⊢ 0 < ∫ (x : G), g x ∂ν

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ ν : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : ν.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
⊢ μ'.haarScalarFactor ν = μ'.haarScalarFactor μ * μ.haarScalarFactor ν

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ ∃ u, ContDiff ℝ ∞ u ∧ (∀ (x : E), u x ∈ Icc 0 1) ∧ support u = ball 0 1 ∧ ∀ (x : E), u (-x) = u x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
⊢ HasCompactSupport u

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
⊢ 0 < ∫ (x : E), u x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
⊢ w D = fun x ↦ ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ 0 ≤ w D x

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ AEMeasurable (⇑φ) μ'

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x y : E
⊢ ∀ a ∈ closedBall 0 1, 0 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
G P : Type uP
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : NormedSpace 𝕜 G
inst✝⁶ : NormedAddCommGroup P
inst✝⁵ : NormedSpace 𝕜 P
f : G → E
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hf : LocallyIntegrable f μ
n : ℕ
ih :
  ∀ {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 E']
    [inst_3 : NormedSpace ℝ F] [inst_4 : NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'},
    (∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) →
      ContDiffOn 𝕜 (↑↑n) (↿g) (s ×ˢ univ) → ContDiffOn 𝕜 (↑↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
E' F : Type uP
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E'
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hg :
  DifferentiableOn 𝕜 (↿g) (s ×ˢ univ) ∧
    (↑n = ⊤ → AnalyticOn 𝕜 (↿g) (s ×ˢ univ)) ∧ ContDiffOn 𝕜 (↑n) (fderiv 𝕜 ↿g) (s ×ˢ univ)
f' : P → G → P × G →L[𝕜] F := fun p a ↦ (f ⋆[precompR (P × G) L, μ] fun x ↦ fderiv 𝕜 (uncurry g) (p, x)) a
A : ∀ (q₀ : P × G), q₀.1 ∈ s → HasFDerivAt (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (f' q₀.1 q₀.2) q₀
⊢ ∀ (p : P) (x : G), p ∈ s → x ∉ k → fderiv 𝕜 (uncurry g) (p, x) = 0

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ AEStronglyMeasurable (⇑{ toFun := f, continuous_toFun := f_cont }) (map (⇑φ) μ')

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ AEMeasurable (⇑φ) μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ ∫ (x : E), w D x ∂μ = 1

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ AEStronglyMeasurable (⇑{ toFun := f, continuous_toFun := f_cont }) (map (⇑φ) μ)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ D • ball 0 1 = ball 0 D

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
B : D • ball 0 1 = ball 0 D
⊢ D ^ finrank ℝ E ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
⊢ ∀ n ≤ N, G n < p.lastStep

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ support (w D) = ball 0 D

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ HasCompactSupport (w D)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
⊢ y D (-x) = y D x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ ball x D ⊆ ball 0 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ ∀ y ∈ ball x D, φ y = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
B : ∀ y ∈ ball x D, φ y = 1
Bx : φ x = 1
⊢ ∀ y ∈ ball x D, φ y = φ x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
G E' F P : Type uP
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
μ : Measure G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
f : G → E
n : ℕ∞
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 (↑n) (↿g) (s ×ˢ univ)
⊢ ContDiffOn 𝕜 (↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ p.index (G n) = Classical.epsilon fun t ↦ p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∈ closedBall 0 (1 - D)
⊢ y D x = 1

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
⊢ (map (⇑φ) μ').haarScalarFactor (map (⇑φ) μ) = μ'.haarScalarFactor μ

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
H : μ'.haarScalarFactor μ = 0
⊢ False

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous[inst✝⁷, _] f
h'f : HasCompactSupport f
u : ℕ → ℝ
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → ℝ → ℝ := fun n x ↦ ↑((thickenedIndicator ⋯ {1}) x)
n : ℕ
⊢ Continuous[inst✝⁷, _] (v n ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
y : E
hy : y ∈ ball x D
h'y : ‖y‖ ≤ 1
⊢ ball y D ⊆ ball 0 (1 + D)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
hx : x ∉ ball 0 (1 + D)
⊢ ∀ y ∈ ball x D, φ y = 0

Tactic:
by

Goal state after: