mathlib-initiative/mathlib-tactics huggingface dataset:
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 ν : Measure G
inst✝⁵ : AddGroup G
inst✝⁴ : MeasurableAdd₂ G
inst✝³ : MeasurableNeg G
inst✝² : SFinite μ
inst✝¹ : μ.IsAddRightInvariant
inst✝ : SFinite ν
hf : Integrable f ν
hg : Integrable g μ
h_meas : AEStronglyMeasurable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
⊢ (∀ᵐ (y : G) ∂ν, Integrable (fun x ↦ (L (f y)) (g (x - y))) μ) ∧
    Integrable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν

Tactic:
refine ⟨Eventually.of_forall fun t => (L (f t)).integrable_comp (hg.comp_sub_right t), ?_⟩

Goal state after:
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 ν : Measure G
inst✝⁵ : AddGroup G
inst✝⁴ : MeasurableAdd₂ G
inst✝³ : MeasurableNeg G
inst✝² : SFinite μ
inst✝¹ : μ.IsAddRightInvariant
inst✝ : SFinite ν
hf : Integrable f ν
hg : Integrable g μ
h_meas : AEStronglyMeasurable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
⊢ Integrable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
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)
q : P × G
hq : q.1 ∈ s
⊢ s ×ˢ univ ∈ 𝓝 q

Tactic:
apply (hs.prod isOpen_univ).mem_nhds

Goal state after:
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' := ⋯
A :
  ∀ p ∈ s,
    Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
      (g p)
q : P × G
hq : q.1 ∈ s
⊢ q ∈ s ×ˢ univ
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos.h_lim
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' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := ⋯
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
⊢ ∀ᵐ (a : α) ∂μ,
    Tendsto (fun n ↦ ‖n - x₀‖⁻¹ • (F n a - F x₀ a - (F' a) (n - x₀))) (𝓝 x₀)
      (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀))))

Tactic:
apply h_diff.mono
    intro a ha
    suffices Tendsto (fun x ↦ ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0) by simpa
    rw [tendsto_zero_iff_norm_tendsto_zero]
    have : (fun x ↦ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - F' a (x - x₀)‖) = fun x ↦
        ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ := by
      ext x
      rw [norm_smul_of_nonneg (nneg _)]
    rwa [hasFDerivAt_iff_tendsto, this] at ha

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

Goal state before:
pp: case neg
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
h1 : Tendsto (fun n ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (fun n ↦ g' x - f' n x) l (𝓝 0)
n : ι × E
hx : ¬x = n.2
hnx : 0 < ‖n.2 - x‖
⊢ ‖(f' n.1 x - g' x) (n.2 - x)‖ ≤ ‖(Prod.fst ∘ Prod.map (fun n ↦ g' x - f' n x) id) n‖ * ‖n.2 - x‖

Tactic:
simp only [Function.comp_apply, Prod.map_apply']

Goal state after:
case neg
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
h1 : Tendsto (fun n ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (fun n ↦ g' x - f' n x) l (𝓝 0)
n : ι × E
hx : ¬x = n.2
hnx : 0 < ‖n.2 - x‖
⊢ ‖(f' n.1 x - g' x) (n.2 - x)‖ ≤ ‖g' x - f' n.1 x‖ * ‖n.2 - x‖
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 : ℕ
hn : 0 < haar (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n)
⊢ Bornology.IsBounded (⇑f '' (x • (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n / (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n))))

Tactic:
have : Bornology.IsBounded (f '' (interior K ∩ f ⁻¹' ball 1 n)) :=
    isBounded_ball.subset <| (image_mono inter_subset_right).trans <| image_preimage_subset _ _

Goal state after:
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 : ℕ
hn : 0 < haar (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n)
this : Bornology.IsBounded (⇑f '' (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n))
⊢ Bornology.IsBounded (⇑f '' (x • (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n / (interior ↑K ∩ ⇑f ⁻¹' ball 1 ↑n))))
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
v : ℕ → G
hv : ∀ (i : ℕ), v i ∈ V i
y : ℕ → G
hy : ∀ (i : ℕ), y i ∈ k
hvy : ∀ (i : ℕ), (fun x1 x2 ↦ x1 * x2) (v i) (y i) = x
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
n : ℕ
this : W n * {z} ∈ 𝓝 z
i : ℕ
hi : y i ∈ W n * {z}
ni : n < i
I : V i ⊆ W n
⊢ x * (y i)⁻¹ ∈ V i

Tactic:
simpa [← hvy i] using hv i

Goal state after:
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
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := ⋯
D_cont : Continuous[inst✝⁹, _] D
x : G
C : Continuous[inst✝⁹, _] fun y ↦ g (y⁻¹ * x)
⊢ 0 < ν (support fun x_1 ↦ g (x_1⁻¹ * x))

Tactic:
apply C.isOpen_support.measure_pos ν

Goal state after:
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := ⋯
D_cont : Continuous[inst✝⁹, _] D
x : G
C : Continuous[inst✝⁹, _] fun y ↦ g (y⁻¹ * x)
⊢ (support fun y ↦ g (y⁻¹ * x)).Nonempty
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case pos.e_f.inr.inl
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : NormedSpace ℝ F
f : ℝ → E
g : ℝ → E'
L : E →L[ℝ] E' →L[ℝ] F
ν : Measure ℝ
inst✝ : NullSingletonClass ν
x : ℝ
h : x ∈ Ioi 0
t : ℝ
ht : t ∈ Ioo 0 x
⊢ (Ioo 0 x).indicator (fun t ↦ (L (f t)) (g (x - t))) t = (L ((Ioi 0).indicator f t)) ((Ioi 0).indicator g (x - t))

Tactic:
rw [indicator_of_mem ht, indicator_of_mem (mem_Ioi.mpr ht.1),
          indicator_of_mem (mem_Ioi.mpr <| sub_pos.mpr ht.2)]

Goal state after:
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
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := fun x ↦ ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous[inst✝⁹, _] D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
L : Set G := tsupport g
L_comp : IsCompact L
M : Set G := (fun p ↦ p.1 * p.2⁻¹) '' K ×ˢ L
M_comp : IsCompact M
M'_comp : IsCompact (closure[inst✝⁹] M)
⊢ HasCompactSupport (uncurry fun x y ↦ f (y * x) * (D (y * x))⁻¹ * g x)

Tactic:
have : ∀ (p : G × G), p ∉ L ×ˢ closure M →
            f (p.2 * p.1) * (D (p.2 * p.1))⁻¹ * g p.1 = 0 := by
          rintro ⟨x, y⟩ hxy
          by_cases H : x ∈ L; swap
          · simp [image_eq_zero_of_notMem_tsupport H]
          have : f (y * x) = 0 := by
            apply image_eq_zero_of_notMem_tsupport
            contrapose hxy
            simp only [mem_prod, H, true_and]
            apply subset_closure
            simp only [M, mem_image, mem_prod, Prod.exists]
            exact ⟨y * x, x, ⟨hxy, H⟩, by group⟩
          simp [this]

Goal state after:
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := fun x ↦ ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous[inst✝⁹, _] D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
L : Set G := tsupport g
L_comp : IsCompact L
M : Set G := (fun p ↦ p.1 * p.2⁻¹) '' K ×ˢ L
M_comp : IsCompact M
M'_comp : IsCompact (closure[inst✝⁹] M)
this : ∀ p ∉ L ×ˢ closure[inst✝⁹] M, f (p.2 * p.1) * (D (p.2 * p.1))⁻¹ * g p.1 = 0
⊢ HasCompactSupport (uncurry fun x y ↦ f (y * x) * (D (y * x))⁻¹ * g x)
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case inr.refine_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
n : ℕ∞
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n ↦ ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ∞ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
r : ℕ → ℝ
rpos : ∀ (n : ℕ), 0 < r n
hr : ∀ (n i : ℕ), i ≤ n → ∀ (x : E), ‖iteratedFDeriv ℝ i (r n • g n) x‖ ≤ ↑(δ n)
S : ∀ (x : E), Summable fun n ↦ (r n • g n) x
⊢ (range fun x ↦ ∑' (n : ℕ), (r n • g n) x) ⊆ Icc 0 1

Tactic:
rintro - ⟨y, rfl⟩
    refine ⟨tsum_nonneg fun n => mul_nonneg (rpos n).le (g_nonneg n y), le_trans ?_ c_lt.le⟩
    have A : HasSum (fun n => (δ n : ℝ)) c := NNReal.hasSum_coe.2 δc
    simp only [Pi.smul_apply, smul_eq_mul, NNReal.val_eq_coe, ← A.tsum_eq]
    apply Summable.tsum_le_tsum _ (S y) A.summable
    intro n
    apply (le_abs_self _).trans
    simpa only [norm_iteratedFDeriv_zero] using! hr n 0 zero_le y

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

Goal state before:
pp: case inr
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
Hf : LocallyCompactSpace G
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂μ'.haarScalarFactor μ • μ

Tactic:
simp only [haarScalarFactor, Hf, not_true_eq_false, ite_false]
    exact (exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ).choose_spec
      f hf h'f

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:
simpa [H] using haarScalarFactor_eq_mul μ' μ μ'

Goal state after:
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
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
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)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑isoG.symm) μ
eg : eP → eG → eE' := fun ep ex ↦ isoE'.symm (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF := ↑(isoE'.arrowCongr isoF).symm ∘SL L
R : eP × eG → eF := fun q ↦ (ef ⋆[eL, eμ] eg q.1) q.2
R_contdiff : ContDiffOn 𝕜 (↑n) R ((⇑isoP ⁻¹' s) ×ˢ univ)
A : ContDiffOn 𝕜 (↑n) (⇑isoF ∘ R ∘ ⇑(isoP.prodCongr isoG).symm) (s ×ˢ univ)
⊢ ⇑isoF ∘ R ∘ ⇑(isoP.prodCongr isoG).symm = fun q ↦ (f ⋆[L, μ] g q.1) q.2

Tactic:
apply funext

Goal state after:
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
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
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)
eG : Type (max uG uE' uF uP) := ⋯
this✝¹ : MeasurableSpace eG := ⋯
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ⋯
eF : Type (max uF uG uE' uP) := ⋯
eP : Type (max uP uG uE' uF) := ⋯
isoG : eG ≃L[𝕜] G := ⋯
isoE' : eE' ≃L[𝕜] E' := ⋯
isoF : eF ≃L[𝕜] F := ⋯
isoP : eP ≃L[𝕜] P := ⋯
ef : eG → E := ⋯
eμ : Measure eG := ⋯
eg : eP → eG → eE' := ⋯
eL : E →L[𝕜] eE' →L[𝕜] eF := ⋯
R : eP × eG → eF := ⋯
R_contdiff : ContDiffOn 𝕜 (↑n) R ((⇑isoP ⁻¹' s) ×ˢ univ)
A : ContDiffOn 𝕜 (↑n) (⇑isoF ∘ R ∘ ⇑(isoP.prodCongr isoG).symm) (s ×ˢ univ)
⊢ ∀ (x : P × G), (⇑isoF ∘ R ∘ ⇑(isoP.prodCongr isoG).symm) x = (f ⋆[L, μ] g x.1) x.2
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case st
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
A : 0 < (R - 1) / (R + 1)
⊢ (fun p ↦ ((p.1 - 1) / (p.1 + 1), ((p.1 + 1) / 2)⁻¹ • p.2)) (R, x) ∈ Ioo 0 1 ×ˢ univ

Tactic:
have B : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith

Goal state after:
case st
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
A : 0 < (R - 1) / (R + 1)
B : (R - 1) / (R + 1) < 1
⊢ (fun p ↦ ((p.1 - 1) / (p.1 + 1), ((p.1 + 1) / 2)⁻¹ • p.2)) (R, x) ∈ Ioo 0 1 ×ˢ univ
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
i j : Fin N.succ
hij : i ≠ j
H₁ : a.r i ≤ dist (a.c i) (a.c j)
H₂ : a.r j ≤ τ * a.r i
⊢ (a.r (last N))⁻¹ * a.r i ≤ (a.r (last N))⁻¹ * dist (a.c i) (a.c j) ∧
      (a.r (last N))⁻¹ * a.r j ≤ (a.r (last N))⁻¹ * (τ * a.r i) ∨
    (a.r (last N))⁻¹ * a.r j ≤ (a.r (last N))⁻¹ * dist (a.c j) (a.c i) ∧
      (a.r (last N))⁻¹ * a.r i ≤ (a.r (last N))⁻¹ * (τ * a.r j)

Tactic:
left

Goal state after:
case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
i j : Fin N.succ
hij : i ≠ j
H₁ : a.r i ≤ dist (a.c i) (a.c j)
H₂ : a.r j ≤ τ * a.r i
⊢ (a.r (last N))⁻¹ * a.r i ≤ (a.r (last N))⁻¹ * dist (a.c i) (a.c j) ∧
    (a.r (last N))⁻¹ * a.r j ≤ (a.r (last N))⁻¹ * (τ * a.r i)
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
i j : Fin N.succ
inej : i ≠ j
ah : Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
τpos : 0 < τ
I : (1 - δ / 4) * τ ≤ 1
J : 1 - δ ≤ 1 - δ / 4
K : 1 - δ / 4 ≤ τ⁻¹
k : Fin N.succ
⊢ a.r (last N) ≤ τ * a.r k

Tactic:
exact a.hlast' k hτ

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

Goal state before:
pp: case ha
α : Type u_1
inst✝⁴ : MetricSpace α
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < μ s
h✝ : Nonempty α
Npos : N ≠ 0
o : Set α
so : s ⊆ o
omeas : MeasurableSet o
μo : μ o = μ s
a : BallPackage (↑s) α := { c := fun x ↦ ↑x, r := fun x ↦ r ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
u : Fin N → Set ↑s
hu : ∀ (i : Fin N), (u i).PairwiseDisjoint fun j ↦ closedBall (a.c j) (a.r j)
hu' : range a.c ⊆ ⋃ i, ⋃ j ∈ u i, ball (a.c j) (a.r j)
u_count : ∀ (i : Fin N), (u i).Countable
v : Fin N → Set α := fun i ↦ ⋃ x ∈ u i, closedBall (↑x) (r ↑x)
A : s = ⋃ i, s ∩ v i
⊢ ↑N ≠ ∞

Tactic:
finiteness

Goal state after: