mathlib-initiative/mathlib-tactics huggingface dataset:
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)
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 + {q₀.2}
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) := hasOpNorm
bound : G → ℝ := U.indicator fun t ↦ ‖precompR (P × G) L‖ * ‖f t‖ * C
I4 : ∀ᵐ (a : G) ∂μ, ∀ (x : P × G), dist x q₀ < δ → ‖((precompR (P × G) L) (f a)) (g' (x.1, x.2 - a))‖ ≤ bound a
I5 : Integrable bound μ
a : G
x : P × G
hx : dist x q₀ < δ
N : s ×ˢ univ ∈ 𝓝 (x.1, x.2 - a)
Z : HasFDerivAt (↿g) (fderiv 𝕜 ↿g (x.1, x.2 - a)) (x.1, x.2 - a)
⊢ (fun x ↦ (x.1, x.2 - a)) = id - fun x ↦ (0, a)

Tactic:
ext x <;> simp only [Pi.sub_apply, _root_.id, Prod.fst_sub, sub_zero, Prod.snd_sub]

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)
⊢ ∀ x < i,
    (closedBall (p.c (p.index x)) (p.r (p.index x)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty →
      ¬N = p.color x

Tactic:
intro j

Goal state after:
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)
j : Ordinal.{u}
⊢ 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 →
      ¬N = p.color j
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
⊢ ↑n = ⊤ → AnalyticOn 𝕜 (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
simp

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
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ (∫ (x : G), g x ∂c • μ') / ∫ (x : G), g x ∂μ = (c • ∫ (x : G), g x ∂μ') / ∫ (x : G), g x ∂μ

Tactic:
simp

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:
simp

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:
rw [Bx]

Goal state after:
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 = 1
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)
B : ∀ y ∈ ball x D, φ y = 0
Bx : φ x = 0
⊢ ∀ y ∈ ball x D, φ y = φ x

Tactic:
rw [Bx]

Goal state after:
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)
B : ∀ y ∈ ball x D, φ y = 0
Bx : φ x = 0
⊢ ∀ y ∈ ball x D, φ y = 0
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case neg
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)
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
n : ℕ
x : G
hx : x ∉ tsupport f
⊢ ↑0 ≤ 0

Tactic:
simp

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

Goal state before:
pp: case refine_2
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
C : ball z (D * (1 + D - ‖x‖) / (1 + D)) ⊆ support fun y ↦ w D y * φ (x - y)
⊢ 0 < μ (support fun i ↦ w D i * φ (x - i))

Tactic:
apply lt_of_lt_of_le _ (measure_mono C)

Goal state after:
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 := ⋯
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
C : ball z (D * (1 + D - ‖x‖) / (1 + D)) ⊆ support fun y ↦ w D y * φ (x - y)
⊢ 0 < μ (ball z (D * (1 + D - ‖x‖) / (1 + D)))
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case e'_5
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)
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
x : G
T : Tendsto (fun i ↦ (thickenedIndicatorAux (u i) {1} (f x)).toNNReal) atTop (𝓝 ({1}.indicator (fun x ↦ 1) (f x)))
⊢ {1}.indicator (fun x ↦ 1) (f x) = ↑({1}.indicator (fun x ↦ 1) (f x))

Tactic:
simp

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)
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
I :
  ∀ (ν : Measure G),
    IsFiniteMeasureOnCompacts ν →
      Tendsto (fun n ↦ ∫ (x : G), v n (f x) ∂ν) atTop (𝓝 (∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂ν))
M : ∀ (n : ℕ), ∫ (x : G), v n (f x) ∂μ' = ∫ (x : G), v n (f x) ∂μ'.haarScalarFactor μ • μ
I1 :
  Tendsto (fun n ↦ ∫ (x : G), v n (f x) ∂μ'.haarScalarFactor μ • μ) atTop
    (𝓝 (∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂μ'))
J1 :
  ∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂μ' = ∫ (x : G), {1}.indicator (fun x ↦ 1) (f x) ∂μ'.haarScalarFactor μ • μ
J2 : μ'.real (f ⁻¹' {1}) = (μ'.haarScalarFactor μ • μ).real (f ⁻¹' {1})
⊢ 0 ∉ {1}

Tactic:
simp

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
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)
ν : Measure G := μ'.haarScalarFactor μ • μ
f : G → ℝ
f_cont : Continuous[inst✝⁹, _] f
hf : EqOn (⇑{ toFun := f, continuous_toFun := f_cont }) 1 (closure[inst✝⁹] s)
f_comp : HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont }
t : Set G := f ⁻¹' {1}
t_closed : IsClosed[inst✝⁹] t
⊢ 0 ∉ {1}

Tactic:
simp

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

Goal state before:
pp: case inl
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✝ : IsEmpty β
⊢ q.c '' ∅ ⊆ ⋃ i, ⋃ j ∈ (fun x ↦ ∅) i, ball (q.c j) (q.r j)

Tactic:
simp

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
⊢ ∅.card = 0 ∧ (∀ c ∈ ∅, ‖c‖ ≤ 2) ∧ ∀ c ∈ ∅, ∀ d ∈ ∅, c ≠ d → 1 ≤ ‖c - d‖

Tactic:
simp

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's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖
⊢ s.card ≤ multiplicity E

Tactic:
by
  apply le_csSup
  · refine ⟨5 ^ finrank ℝ E, ?_⟩
    rintro _ ⟨s, ⟨rfl, h⟩⟩
    exact Besicovitch.card_le_of_separated s h.1 h.2
  · simp only [mem_ofPred_eq, Ne]
    exact ⟨s, rfl, hs, h's⟩

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
hδ' : 1 ≤ δ
⊢ ∀ (i : Fin N), ‖(fun x ↦ 0) i‖ ≤ 2

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Conformal

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : NormedField R
inst✝¹ : SeminormedAddCommGroup M
inst✝ : NormedSpace R M
⊢ ContinuousLinearMap.id R M = 1 • LinearIsometry.id.toContinuousLinearMap

Tactic:
simp

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
i j : Fin N.succ
inej : i ≠ j
hi : 2 < ‖a.c i‖
hij : ‖a.c i‖ ≤ ‖a.c 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
hcrj : ‖a.c j‖ ≤ a.r j + 1
this : i < last N
⊢ a.r i ≤ ‖a.c i‖

Tactic:
convert! (a.hlast i this).1 using 1

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
i j : Fin N.succ
inej : i ≠ j
hi : 2 < ‖a.c i‖
hij : ‖a.c i‖ ≤ ‖a.c 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
hcrj : ‖a.c j‖ ≤ a.r j + 1
this : i < last N
⊢ ‖a.c i‖ = dist (a.c i) (a.c (last N))
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case h0
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
S : ∑ _i, μ s / ↑N ≤ ∑ i, μ (s ∩ v i)
i : Fin N
hi : μ s / (↑N + 1) < μ (s ∩ v i)
B : μ (o ∩ v i) = ∑' (x : ↑(u i)), μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : μ s / (↑N + 1) < ∑ x ∈ w, μ (o ∩ closedBall (↑↑x) (r ↑↑x))
⊢ ↑N + 1 ≠ 0

Tactic:
simp

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

Goal state before:
pp: case hI
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
S : ∑ _i, μ s / ↑N ≤ ∑ i, μ (s ∩ v i)
i : Fin N
hi : μ s / (↑N + 1) < μ (s ∩ v i)
B : μ (o ∩ v i) = ∑' (x : ↑(u i)), μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : μ s / (↑N + 1) < ∑ x ∈ w, μ (o ∩ closedBall (↑↑x) (r ↑↑x))
⊢ ↑N + 1 ≠ ∞

Tactic:
simp

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.Regular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
this : map Inv.inv (map Inv.inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
⊢ c ^ 2 * μ ↑K = 1 ^ 2 * (c ^ 2 • μ) ↑K

Tactic:
simp

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

Goal state before:
pp: G : Type u_1
inst✝⁷ : CommGroup G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
c : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)
hc : μ.inv = c • μ
this : map Inv.inv (map Inv.inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
⊢ c ^ 2 * μ ↑K = 1 ^ 2 * (c ^ 2 • μ) ↑K

Tactic:
simp

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

Goal state before:
pp: case zero
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
h : ContDiffWithinAt 𝕜' (↑0) f s x
a : E
m : Fin 0 → E
⊢ ((restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' 0 f s) a) m = (iteratedFDerivWithin 𝕜 0 f s a) m

Tactic:
simp

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

Goal state before:
pp: 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
S : ∑ _i, μ s / ↑N ≤ ∑ i, μ (s ∩ v i)
i : Fin N
hi : μ s / (↑N + 1) < μ (s ∩ v i)
B : μ (o ∩ v i) = ∑' (x : ↑(u i)), μ (o ∩ closedBall (↑↑x) (r ↑↑x))
w : Finset ↑(u i)
hw : μ s / (↑N + 1) < ∑ x ∈ w, μ (o ∩ closedBall (↑↑x) (r ↑↑x))
⊢ 1 / (↑N + 1) * μ s + ↑N / (↑N + 1) * μ s ≤ μ ((⋃ x ∈ w, closedBall (↑↑x) (r ↑↑x)) ∩ o) + ↑N / (↑N + 1) * μ s

Tactic:
by
        gcongr
        rw [one_div, mul_comm, ← div_eq_mul_inv]
        apply hw.le.trans (le_of_eq _)
        rw [← Finset.set_biUnion_coe, inter_comm _ o, inter_iUnion₂, Finset.set_biUnion_coe,
          measure_biUnion_finset]
        · have : (w : Set (u i)).PairwiseDisjoint
              fun b : u i => closedBall (b : α) (r (b : α)) := by
            intro k _ l _ hkl; exact hu i k.2 l.2 (Subtype.val_injective.ne hkl)
          exact this.mono fun k => inter_subset_right
        · intro b _
          apply omeas.inter measurableSet_closedBall

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
⊢ ↑(n + 1) ≠ ↑⊤

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y
a : E
h₁a : restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] a] iteratedFDerivWithin 𝕜 n f s
h₃a : ContDiffWithinAt 𝕜' (↑(n + 1)) f s a
h₄a : a ∈ s
m : Fin (n + 1) → E
⊢ ((fderivWithin 𝕜' (iteratedFDerivWithin 𝕜' n f s) s a) (m 0)) (Fin.tail m) =
    ((restrictScalars 𝕜 ∘
          ⇑(ContinuousLinearMap.restrictScalars 𝕜 (fderivWithin 𝕜' (iteratedFDerivWithin 𝕜' n f s) s a)))
        (m 0))
      (Fin.tail m)

Tactic:
simp

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

Goal state before:
pp: case h.hmn
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y
a : E
h₁a : restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] a] iteratedFDerivWithin 𝕜 n f s
h₃a : ContDiffWithinAt 𝕜' (↑(n + 1)) f s a
h₄a : a ∈ s
m : Fin (n + 1) → E
⊢ n < n + 1

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ 0 ≤ ↑r

Tactic:
simp

Goal state after: