mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
inst✝ : CompleteSpace E
this : (↑(adjoint T * T) - ↑(T * adjoint T)).IsSymmetric
⊢ (∀ (x : E), ⟪(adjoint T * T) x, x⟫_𝕜 - ⟪(T * adjoint T) x, x⟫_𝕜 = 0) ↔ ∀ (v : E), ‖T v‖ = ‖(adjoint T) v‖

Tactic:
mul_apply_eq_comp,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
inst✝ : CompleteSpace E
this : (↑(adjoint T * T) - ↑(T * adjoint T)).IsSymmetric
⊢ (∀ (x : E), ⟪(adjoint T) (T x), x⟫_𝕜 - ⟪T ((adjoint T) x), x⟫_𝕜 = 0) ↔ ∀ (v : E), ‖T v‖ = ‖(adjoint T) v‖
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.centroid = (↑n * (1 / (↑n + 1))) • (s.faceOppositeCentroid i -ᵥ s.points i) +ᵥ s.points i

Tactic:
rw [eq_vadd_iff_vsub_eq, centroid_vsub_point_eq_smul_vsub,
      faceOppositeCentroid_vsub_point_eq_smul_vsub, smul_smul, one_div, mul_assoc,
      inv_mul_cancel₀ (by norm_cast), mul_one]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: case mp
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : FiniteDimensional k V
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
h_tot : affineSpan k (Set.range p) = ⊤
n : ℕ := Fintype.card ι - 1
hn : Fintype.card ι = n + 1
⊢ n + 1 = finrank k V + 1

Tactic:
← finrank_top,

Goal state after:
case mp
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : FiniteDimensional k V
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
h_tot : affineSpan k (Set.range p) = ⊤
n : ℕ := Fintype.card ι - 1
hn : Fintype.card ι = n + 1
⊢ n + 1 = finrank k ↥⊤ + 1
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: case mpr
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : FiniteDimensional k V
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
hc : Fintype.card ι = finrank k V + 1
⊢ affineSpan k (Set.range p) = ⊤

Tactic:
← finrank_top,

Goal state after:
case mpr
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : FiniteDimensional k V
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
hc : Fintype.card ι = finrank k ↥⊤ + 1
⊢ affineSpan k (Set.range p) = ⊤
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
⊢ tsupport ↑f = closedBall c f.rOut

Tactic:
tsupport,

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (support ↑f) = closedBall c f.rOut
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: α : Type u_3
E : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
G : Type u_7
inst✝⁴ : Group G
inst✝³ : MeasurableSpace α
inst✝² : MulAction G α
inst✝¹ : MeasurableConstSMul G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
f : α → E
g : G
⊢ ∫ (x : α), f (g • x) ∂μ = ∫ (x : α), f x ∂μ

Tactic:
by
  have h : MeasurableEmbedding fun x : α => g • x := (MeasurableEquiv.smul g).measurableEmbedding
  rw [← h.integral_map, MeasureTheory.map_smul]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
hp : 0 < p.toReal
⊢ 0 ≤ (∑' (i : α), ‖↑f i‖ ^ p.toReal) ^ (1 / p.toReal)

Tactic:
exact Real.rpow_nonneg (tsum_nonneg fun i ↦ by positivity) _

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : ↥(lp E p)
s : Finset α
⊢ ∑ i ∈ s, ‖↑f i‖ ^ p.toReal ≤ ‖f‖ ^ p.toReal

Tactic:
lp.norm_rpow_eq_tsum hp f

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : ↥(lp E p)
s : Finset α
⊢ ∑ i ∈ s, ‖↑f i‖ ^ p.toReal ≤ ∑' (i : α), ‖↑f i‖ ^ p.toReal
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
hp : 0 < p.toReal
f : ↥(lp E p)
s : Finset α
F : α → ℝ := fun i ↦ ‖↑f i‖ ^ p.toReal - ‖↑(f - ∑ i ∈ s, lp.single p i (↑f i)) i‖ ^ p.toReal
i : α
hi : i ∉ s
this : ‖↑f i‖ ^ p.toReal - ‖↑f i - if i ∈ s then ↑f i else 0‖ ^ p.toReal = 0
⊢ F i = 0

Tactic:
by
      simpa only [coeFn_sub, coeFn_sum, lp.coeFn_single, Pi.sub_apply, Finset.sum_apply,
        Finset.sum_pi_single, F] using this

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case h₂
α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
g : α → ι → ℝ
K : ℝ≥0
hg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i
a₀ a : α
M : ℝ
hM : M ∈ upperBounds (Set.range fun i ↦ ‖g a₀ i‖)
i : ι
⊢ |g a₀ i| ≤ M

Tactic:
· exact hM ⟨i, rfl⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case inr.inr.convert_2
p : ℝ[X]
x : ℝ
hx : 0 < x
this :
  HasDerivAt (((fun x ↦ Polynomial.eval x p) * fun x ↦ rexp (-x)) ∘ Inv.inv)
    ((Polynomial.eval x⁻¹ (derivative p) * rexp (-x⁻¹) + Polynomial.eval x⁻¹ p * (rexp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x
⊢ (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x]
    ((fun x ↦ Polynomial.eval x p) * fun x ↦ rexp (-x)) ∘ Inv.inv

Tactic:
· filter_upwards [lt_mem_nhds hx] with y hy
      simp [expNegInvGlue, hy.not_ge]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case h₂
x y : ℝ
hxy : x ≤ y
⊢ expNegInvGlue (1 - y) ≤ expNegInvGlue (1 - x)

Tactic:
apply expNegInvGlue.monotone (by linarith)

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

Goal state before:
pp: case neg
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
c : E
r : ℝ
h : ¬0 < r
⊢ ContDiffOn ℝ (↑n) (↑(IsometryEquiv.vaddConst c).toHomeomorph.toOpenPartialHomeomorph.symm) (ball c r)

Tactic:
· exact contDiffOn_id.sub contDiffOn_const

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

Goal state before:
pp: 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✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
ε : ℝ
this : ∀ᶠ (x_1 : ι) in l, ∀ᶠ (y : E) in 𝓝 x, dist (‖y - x‖⁻¹ * ‖g y - g x - (g' x) (y - x)‖) 0 < ε
hε : ε > 0
⊢ ∀ (x_1 : E),
    dist (‖x_1 - x‖⁻¹ * ‖g x_1 - g x - (g' x) (x_1 - x)‖) 0 < ε →
      dist (‖x_1 - x‖⁻¹ * ‖g x_1 - g x - (g' x) (x_1 - x)‖) 0 < ε

Tactic:
by simp only [imp_self, forall_const]

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

Goal state before:
pp: 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)
⊢ Tendsto (fun n ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)

Tactic:
rw [Metric.tendstoUniformlyOnFilter_iff] at hf'

Goal state after:
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 : ∀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
hf' : ∀ ε > 0, ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
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)
⊢ Tendsto (fun n ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x ↦ ‖g x‖)
hf : Integrable f μ
hg : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f ⋆[L, μ] g)

Tactic:
by
  refine continuous_iff_continuousAt.mpr fun x₀ => ?_
  have : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t : G ∂μ, ‖L (f t) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖ := by
    filter_upwards with x; filter_upwards with t
    apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl, le_ciSup hbg (x - t)]
  refine continuousAt_of_dominated ?_ this (by fun_prop) ?_
  · exact Eventually.of_forall fun x =>
      hf.aestronglyMeasurable.convolution_integrand_snd' L hg.aestronglyMeasurable
  · filter_upwards with t; fun_prop

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

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf'✝ : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z ↦ ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z ↦ ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.toSpanSingleton 𝕜 (f' n.1 n.2)) n.2
hf' : TendstoUniformlyOnFilter F' G' l (𝓝 x)
⊢ HasFDerivAt g (ContinuousLinearMap.toSpanSingleton 𝕜 (g' x)) x

Tactic:
exact hasFDerivAt_of_tendstoUniformlyOnFilter hf' hf hfg

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
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
⊢ tsupport f ⊆ Euclidean.closedBall x d

Tactic:
tsupport,

Goal state after:
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
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (support f) ⊆ Euclidean.closedBall x d
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: case a
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)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
ε C : ℝ
εpos : 0 < ε
h₀ε : ball q₀.1 ε ⊆ s
hε : ∀ (p : P) (x : G), ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C
I1 : ∀ᶠ (x : P × G) in 𝓝 q₀, AEStronglyMeasurable (fun a ↦ (L (f a)) (g x.1 (x.2 - a))) μ
I2 : Integrable (fun a ↦ (L (f a)) (g q₀.1 (q₀.2 - a))) μ
I3 : AEStronglyMeasurable (fun a ↦ L (f a) ∘SL g' (q₀.1, q₀.2 - a)) μ
K' : Set G := ⋯
K'_def : K' = -k + {q₀.2}
hK' : IsCompact K'
U : Set G
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
K'U : K' ⊆ U
hU : IntegrableOn f U μ
δ : ℝ
δpos : 0 < δ
δε : δ ≤ ε
hδ : K' + ball 0 δ ⊆ U
this : Norm (E →L[𝕜] (P × G →L[𝕜] E') →L[𝕜] P × G →L[𝕜] F) := ⋯
bound : G → ℝ := ⋯
a : G
x : P × G
hx : max ‖x.1 - q₀.1‖ ‖x.2 - q₀.2‖ < δ
⊢ ball q₀.2 δ ⊆ {q₀.2} + ball 0 δ

Tactic:
· simp only [add_ball, thickening_singleton, zero_vadd, subset_rfl]

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: E : Type uE
E' : Type uE'
F : Type uF
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedSpace ℝ E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : CompleteSpace E
inst✝⁵ : CompleteSpace E'
inst✝⁴ : CompleteSpace F
μ ν : Measure ℝ
inst✝³ : SFinite μ
inst✝² : SFinite ν
inst✝¹ : μ.IsAddRightInvariant
inst✝ : NullSingletonClass ν
f : ℝ → E
g : ℝ → E'
hf : Integrable ((Ioi 0).indicator f) ν
hg : Integrable ((Ioi 0).indicator g) μ
L : E →L[ℝ] E' →L[ℝ] F
⊢ ∫ (x : ℝ) in Ioi 0, ∫ (t : ℝ) in 0..x, (L (f t)) (g (x - t)) ∂ν ∂μ =
    (L (∫ (x : ℝ) in Ioi 0, f x ∂ν)) (∫ (x : ℝ) in Ioi 0, g x ∂μ)

Tactic:
simp_rw [← integral_indicator measurableSet_Ioi]

Goal state after:
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedAddCommGroup E'
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedSpace ℝ E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : CompleteSpace E
inst✝⁵ : CompleteSpace E'
inst✝⁴ : CompleteSpace F
μ ν : Measure ℝ
inst✝³ : SFinite μ
inst✝² : SFinite ν
inst✝¹ : μ.IsAddRightInvariant
inst✝ : NullSingletonClass ν
f : ℝ → E
g : ℝ → E'
hf : Integrable ((Ioi 0).indicator f) ν
hg : Integrable ((Ioi 0).indicator g) μ
L : E →L[ℝ] E' →L[ℝ] F
⊢ ∫ (x : ℝ), (Ioi 0).indicator (fun x ↦ ∫ (t : ℝ) in 0..x, (L (f t)) (g (x - t)) ∂ν) x ∂μ =
    (L (∫ (x : ℝ), (Ioi 0).indicator f x ∂ν)) (∫ (x : ℝ), (Ioi 0).indicator g x ∂μ)
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ tsupport (f.normed μ) = closedBall c f.rOut

Tactic:
tsupport,

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (support (f.normed μ)) = closedBall c f.rOut
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: case hac
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
x : E
⊢ ↑f x ≤ 1

Tactic:
exact f.le_one

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

Goal state before:
pp: case hab
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
x : E
K_pos : 0 < K
⊢ ↑f x ≤ 1

Tactic:
exact f.le_one

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

Goal state before:
pp: case zero
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
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 : ContinuousOn (↿g) (s ×ˢ univ)
⊢ ContinuousOn (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
exact continuousOn_convolution_right_with_param L hk hgs hf hg

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

Goal state before:
pp: case pos
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
hG : LocallyCompactSpace G
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
int_g_ne_zero : ∫ (x : G), g x ∂μ ≠ 0
⊢ ↑(μ.haarScalarFactor μ) = ↑1

Tactic:
calc
    haarScalarFactor μ μ = (∫ x, g x ∂μ) / ∫ x, g x ∂μ :=
      haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg
    _ = 1 := div_self int_g_ne_zero

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
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D - ‖x‖) / (1 + D))
⊢ 1 + D - ‖x‖ + ‖x‖ ≤ 1 + D

Tactic:
linarith only

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
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D - ‖x‖) / (1 + D))
⊢ D - (1 + D) ≤ (1 + D) * 0

Tactic:
linarith only

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
h :
  ∀ (δ : ℝ),
    0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun i j ↦ 1 - δ ≤ ‖f i - f j‖

Tactic:
rcases lt_or_ge δ 1 with (hδ' | hδ')

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
hδ' : δ < 1
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun i j ↦ 1 - δ ≤ ‖f i - f j‖

case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
h :
  ∀ (δ : ℝ),
    0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card
N : ℕ := multiplicity E + 1
hN : N = multiplicity E + 1
δ : ℝ
hδ : 0 < δ
hδ' : 1 ≤ δ
⊢ ∃ f, (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun i j ↦ 1 - δ ≤ ‖f i - f j‖
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : Group G
inst✝¹⁰ : IsTopologicalGroup G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : BorelSpace G
H : Type u_2
inst✝⁷ : Group H
inst✝⁶ : TopologicalSpace H
inst✝⁵ : IsTopologicalGroup H
inst✝⁴ : CompactSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
ν : Measure H
inst✝ : ν.IsHaarMeasure
f : G →* H
hcont : Continuous[inst✝¹², inst✝⁶] ⇑f
hsurj : Surjective ⇑f
huniv : μ univ = ν univ
this✝ : IsFiniteMeasure μ
this : (map (⇑f) μ).IsHaarMeasure
C : ℝ≥0 := (map (⇑f) μ).haarScalarFactor ν
hC : map (⇑f) μ = C • ν
⊢ ↑C • ν univ = ν univ

Tactic:
← ENNReal.smul_def,

Goal state after:
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : Group G
inst✝¹⁰ : IsTopologicalGroup G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : BorelSpace G
H : Type u_2
inst✝⁷ : Group H
inst✝⁶ : TopologicalSpace H
inst✝⁵ : IsTopologicalGroup H
inst✝⁴ : CompactSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
ν : Measure H
inst✝ : ν.IsHaarMeasure
f : G →* H
hcont : Continuous[inst✝¹², inst✝⁶] ⇑f
hsurj : Surjective ⇑f
huniv : μ univ = ν univ
this✝ : IsFiniteMeasure μ
this : (map (⇑f) μ).IsHaarMeasure
C : ℝ≥0 := (map (⇑f) μ).haarScalarFactor ν
hC : map (⇑f) μ = C • ν
⊢ C • ν univ = ν univ