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

Goal state before:
pp: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : H → α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
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, HasFDerivAt (fun x ↦ F x a) (F' x a) x
this : NormedSpace ℝ H := ⋯
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' x₀ a) x₀
⊢ ∀ (x : α),
    ((∀ x_1 ∈ s, HasFDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1) ∧ ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x) →
      LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) (ball x₀ ε)

Tactic:
rintro a ⟨ha_deriv, ha_bound⟩

Goal state after:
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : H → α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
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, HasFDerivAt (fun x ↦ F x a) (F' x a) x
this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' x₀ a) x₀
a : α
ha_deriv : ∀ x ∈ s, HasFDerivAt (fun x ↦ F x a) (F' x a) x
ha_bound : ∀ x ∈ s, ‖F' x a‖ ≤ bound a
⊢ LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) (ball x₀ ε)
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
⊢ ∀ (x : α),
    ((∀ x_1 ∈ s, HasDerivAt (fun x_2 ↦ F x_2 x) (F' x_1 x) x_1) ∧ ∀ x_1 ∈ s, ‖F' x_1 x‖ ≤ bound x) →
      LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) (ball x₀ ε)

Tactic:
rintro a ⟨ha_deriv, ha_bound⟩

Goal state after:
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
a : α
ha_deriv : ∀ x ∈ s, HasDerivAt (fun x ↦ F x a) (F' x a) x
ha_bound : ∀ x ∈ s, ‖F' x a‖ ≤ bound a
⊢ LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) (ball x₀ ε)
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x ↦ ‖g x‖) (fun x ↦ ‖k x‖) x (mul ℝ ℝ) μ
hfgk : ConvolutionExistsAt (fun x ↦ ‖f x‖) ((fun x ↦ ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x ↦ ‖k x‖) x₀ (mul ℝ ℝ) ν
⊢ AEStronglyMeasurable (fun x ↦ k (x₀ - x.2 - x.1)) (μ.prod ν)

Tactic:
refine (hk.mono_ac ?_).comp_measurable (by fun_prop)

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x ↦ ‖g x‖) (fun x ↦ ‖k x‖) x (mul ℝ ℝ) μ
hfgk : ConvolutionExistsAt (fun x ↦ ‖f x‖) ((fun x ↦ ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x ↦ ‖k x‖) x₀ (mul ℝ ℝ) ν
⊢ Measure.map (fun x ↦ x₀ - x.2 - x.1) (μ.prod ν) ≪ μ
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x ↦ ‖g x‖) (fun x ↦ ‖k x‖) x (mul ℝ ℝ) μ
hfgk : ConvolutionExistsAt (fun x ↦ ‖f x‖) ((fun x ↦ ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x ↦ ‖k x‖) x₀ (mul ℝ ℝ) ν
y : G
⊢ QuasiMeasurePreserving (fun x ↦ x₀ - (x, y).2 - (x, y).1) μ μ

Tactic:
dsimp only

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hf : AEStronglyMeasurable f ν
hg : AEStronglyMeasurable g μ
hk : AEStronglyMeasurable k μ
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x ↦ ‖g x‖) (fun x ↦ ‖k x‖) x (mul ℝ ℝ) μ
hfgk : ConvolutionExistsAt (fun x ↦ ‖f x‖) ((fun x ↦ ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x ↦ ‖k x‖) x₀ (mul ℝ ℝ) ν
y : G
⊢ QuasiMeasurePreserving (fun x ↦ x₀ - y - x) μ μ
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
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ ∃ f, tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ (↑n) f ∧ range f ⊆ Icc 0 1 ∧ f x = 1

Tactic:
refine ⟨f, f_tsupp.trans hd, ?_, ?_, ?_, ?_⟩

Goal state after:
case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ HasCompactSupport f

case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ ContDiff ℝ (↑n) f

case refine_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ range f ⊆ Icc 0 1

case refine_4
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ f x = 1
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
x : β
A : ∀ (z : β), p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep
h : ¬p.c x ∈ p.iUnionUpTo p.lastStep
H : p.τ * p.r x < p.R p.lastStep
⊢ 0 < p.R p.lastStep

Tactic:
apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H

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
⊢ ∃ c, ∀ (f : G → ℝ), Continuous[inst✝⁷, _] f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ

Tactic:
by_cases H : LocallyCompactSpace G; swap
  · refine ⟨0, fun f f_cont f_comp ↦ ?_⟩
    rcases f_comp.eq_zero_or_locallyCompactSpace_of_group f_cont with hf | hf
    · simp [hf]
    · exact (H hf).elim
  -- Fix some nonzero continuous function with compact support `g`.
  obtain ⟨⟨g, g_cont⟩, g_comp, g_nonneg, g_one⟩ :
    ∃ (g : C(G, ℝ)), HasCompactSupport g ∧ 0 ≤ g ∧ g 1 ≠ 0 := exists_continuous_nonneg_pos 1
  have int_g_pos : 0 < ∫ x, g x ∂μ :=
    g_cont.integral_pos_of_hasCompactSupport_nonneg_nonzero g_comp g_nonneg g_one
  -- The proportionality constant we are looking for will be the ratio of the integrals of `g`
  -- with respect to `μ'` and `μ`.
  let c : ℝ := (∫ x, g x ∂μ)⁻¹ * (∫ x, g x ∂μ')
  have c_nonneg : 0 ≤ c :=
    mul_nonneg (inv_nonneg.2 (integral_nonneg g_nonneg)) (integral_nonneg g_nonneg)
  refine ⟨.mk c c_nonneg, fun f f_cont f_comp ↦ ?_⟩
  /- use the lemma `integral_mulLeftInvariant_mulRightInvariant_combo` for `μ` and then `μ'`
  to reexpress the integral of `f` as the integral of `g` times a factor which only depends
  on a right-invariant measure `ν`. We use `ν = μ.inv` for convenience. -/
  let ν := μ.inv
  have A : ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ :=
    integral_isMulLeftInvariant_isMulRightInvariant_combo f_cont f_comp g_cont g_comp g_nonneg g_one
  rw [← mul_inv_eq_iff_eq_mul₀ int_g_pos.ne'] at A
  have B : ∫ x, f x ∂μ' = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ' :=
    integral_isMulLeftInvariant_isMulRightInvariant_combo f_cont f_comp g_cont g_comp g_nonneg g_one
  /- Since the `ν`-factor is the same for `μ` and `μ'`, this gives the result. -/
  rw [← A, mul_assoc, mul_comm] at B
  simp [B, integral_smul_nnreal_measure, c, NNReal.smul_def]

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

Goal state before:
pp: case hf.hf
E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
x : E
hR : 1 < R
right✝ : (R, x).2 ∈ univ
⊢ 0 < (R, x).1 + 1

Tactic:
dsimp only

Goal state after:
case hf.hf
E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
x : E
hR : 1 < R
right✝ : (R, x).2 ∈ univ
⊢ 0 < R + 1
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
x : E
hR : 1 < R
right✝ : (R, x).2 ∈ univ
⊢ 0 < ((R, x).1 + 1) / 2

Tactic:
dsimp only

Goal state after:
E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
x : E
hR : 1 < R
right✝ : (R, x).2 ∈ univ
⊢ 0 < (R + 1) / 2
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure[inst✝⁹] s)
⊢ μ'.haarScalarFactor μ • μ s ≤ μ' s

Tactic:
by
  apply le_of_forall_lt (fun r hr ↦ ?_)
  let ν := haarScalarFactor μ' μ • μ
  have : ν s ≠ ∞ := ((measure_mono subset_closure).trans_lt h's.measure_lt_top).ne
  obtain ⟨-, hf, ⟨f, f_cont, f_comp, rfl⟩, νf⟩ :
      ∃ K ⊆ s, (∃ f, Continuous f ∧ HasCompactSupport f ∧ K = f ⁻¹' {1}) ∧ r < ν K :=
    innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group ⟨hs, this⟩ r
      (by convert! hr)
  calc
  r < ν (f ⁻¹' {1}) := νf
  _ = μ' (f ⁻¹' {1}) :=
    (measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport _ _ f_cont f_comp).symm
  _ ≤ μ' s := measure_mono hf

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Finset E
hs : ∀ c ∈ s, ‖c‖ ≤ 2
h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
μ : Measure E := Measure.addHaar
δ : ℝ := 1 / 2
ρ : ℝ := 5 / 2
ρpos : 0 < ρ
A : Set E := ⋃ c ∈ s, ball c δ
hA : A = ⋃ c ∈ s, ball c δ
D : (↑s).Pairwise (Disjoint on fun c ↦ ball c δ)
A_subset : A ⊆ ball 0 ρ
⊢ ↑s.card * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) = μ A

Tactic:
rw [hA, measure_biUnion_finset D fun c _ => measurableSet_ball]
        have I : 0 < δ := by norm_num
        simp only [μ.addHaar_ball_of_pos _ I]
        simp only [Finset.sum_const, nsmul_eq_mul, mul_assoc]

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

Goal state before:
pp: case inr
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy_lt : jx < p.lastStep
jyi : p.color jx = ↑i
hy : p.index jx ∈ s i
x_ne_y : p.index jx ≠ p.index jx
jxy : jx ≤ jx
⊢ jx < jx

Tactic:
exact (x_ne_y rfl).elim

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
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
hm : (range f).Nonempty
h'm : (range f).Countable
M : ∀ (i : ℕ), MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
sm : s ⊆ ⋃ y, f y • (k * k⁻¹)
s_eq : s = ⋃ n, s ∩ f n • (k * k⁻¹)
⊢ μ' s = ∑' (n : ℕ), μ' (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)

Tactic:
← measure_iUnion (disjoint_disjointed _) M,

Goal state after:
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
hm : (range f).Nonempty
h'm : (range f).Countable
M : ∀ (i : ℕ), MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
sm : s ⊆ ⋃ y, f y • (k * k⁻¹)
s_eq : s = ⋃ n, s ∩ f n • (k * k⁻¹)
⊢ μ' s = μ' (⋃ i, disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
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
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
hm : (range f).Nonempty
h'm : (range f).Countable
M : ∀ (i : ℕ), MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
sm : s ⊆ ⋃ y, f y • (k * k⁻¹)
s_eq : s = ⋃ n, s ∩ f n • (k * k⁻¹)
I : μ' s = ∑' (n : ℕ), μ' (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)
⊢ ν s = ∑' (n : ℕ), ν (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)

Tactic:
← measure_iUnion (disjoint_disjointed _) M,

Goal state after:
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
hm : (range f).Nonempty
h'm : (range f).Countable
M : ∀ (i : ℕ), MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
sm : s ⊆ ⋃ y, f y • (k * k⁻¹)
s_eq : s = ⋃ n, s ∩ f n • (k * k⁻¹)
I : μ' s = ∑' (n : ℕ), μ' (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)
⊢ ν s = ν (⋃ i, disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
⊢ Convex ℝ≥0 {r | HolderWith C r f}

Tactic:
← holderOnWith_univ

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
f : X → Y
C : ℝ≥0
⊢ Convex ℝ≥0 {r | HolderOnWith C r f univ}
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case pos
f : ℝ → ℝ
a b : ℝ
h : ContinuousAt f a
hd₀ : DifferentiableOn ℝ f (Ioo a b)
g₀ : a < b
⊢ ContinuousOn f (Ico a b)

Tactic:
exact Ioo_union_left g₀ ▸ hd₀.continuousOn.union_continuousAt isOpen_Ioo (by simp_all)

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

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ < 0
hx : f x₀ = 0
⊢ deriv (fun x ↦ -f x) x₀ > 0

Tactic:
simpa [deriv.neg]

Goal state after: