mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : FiniteDimensional k ↥s.direction
hs : s = ⊥
⊢ finrank k ↥(vectorSpan k (insert p ∅)) = 0

Tactic:
rw [← finrank_bot k V]

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : FiniteDimensional k ↥s.direction
hs : s = ⊥
⊢ finrank k ↥(vectorSpan k (insert p ∅)) = finrank k ↥⊥
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ ∞.toReal
H : Summable fun x ↦ 1
⊢ BddAbove (Set.range fun i ↦ ‖f i‖)

Tactic:
simpa using ((Set.Finite.of_summable_const (by simp) H).image fun i => ‖f i‖).bddAbove

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)
h : ‖f‖ = 0
hp : 0 < p.toReal
hf : (fun i ↦ ‖↑f i‖ ^ p.toReal) = 0
this✝ : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal
i : α
this : ↑f i = 0 ∧ p.toReal ≠ 0
⊢ ↑f i = ↑0 i

Tactic:
exact this.1

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

Goal state before:
pp: I : Type u_5
B : I → Type u_6
inst✝¹ : (i : I) → NormedRing (B i)
inst✝ : ∀ (i : I), NormOneClass (B i)
i : I
⊢ (fun i ↦ ‖1 i‖) i ≤ 1

Tactic:
exact norm_one.le

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

Goal state before:
pp: case top.refine_1.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
i : α
x : E i
this : Nonempty α
hp : 0 < ∞
j : α
hij : i ≠ j
⊢ ‖0‖ ≤ ‖x‖

Tactic:
_root_.norm_zero

Goal state after:
case top.refine_1.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
i : α
x : E i
this : Nonempty α
hp : 0 < ∞
j : α
hij : i ≠ j
⊢ 0 ≤ ‖x‖
Module: Mathlib.Analysis.InnerProductSpace.Calculus

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
inst✝ : NormedSpace ℝ E
f g : ℝ → E
f' g' : E
x : ℝ
⊢ HasDerivAt f f' x → HasDerivAt g g' x → HasDerivAt (fun t ↦ ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x

Tactic:
by
  simpa only [← hasDerivWithinAt_univ] using HasDerivWithinAt.inner 𝕜

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ : G
⊢ ConvolutionExistsAt f g x₀ L μ

Tactic:
refine HasCompactSupport.convolutionExistsAt L ?_ hf hg

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ : G
⊢ HasCompactSupport fun t ↦ (L (f t)) (g (x₀ - t))
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
hcf : HasCompactSupport f
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ : G
⊢ ConvolutionExistsAt f g x₀ L μ

Tactic:
refine HasCompactSupport.convolutionExistsAt L ?_ hf hg

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
hcf : HasCompactSupport f
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ : G
⊢ HasCompactSupport fun t ↦ (L (f t)) (g (x₀ - t))
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
⊢ x ∈ support g + support f

Tactic:
by_contra hx

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : x ∉ support g + support f
⊢ False
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
x₀ : H
s : Set H
inst✝ : NormedSpace ℝ H
μ : Measure ℝ
F : H → ℝ → E
F' : ℝ → H →L[ℝ] E
a b : ℝ
bound : ℝ → ℝ
hs : s ∈ 𝓝 x₀
hF_int : IntervalIntegrable (F x₀) μ a b
h_lip :
  (∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s) ∧
    ∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff :
  (∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), HasFDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀) ∧
    ∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), HasFDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Set.Ioc a b)) ∧ AEStronglyMeasurable F' (μ.restrict (Set.Ioc b a))
hF_meas :
  (∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Set.Ioc a b))) ∧
    ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Set.Ioc b a))
⊢ IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀

Tactic:
have H₁ :=
    hasFDerivAt_integral_of_dominated_loc_of_lip hs hF_meas.1 hF_int.1 hF'_meas.1 h_lip.1
      bound_integrable.1 h_diff.1

Goal state after:
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
x₀ : H
s : Set H
inst✝ : NormedSpace ℝ H
μ : Measure ℝ
F : H → ℝ → E
F' : ℝ → H →L[ℝ] E
a b : ℝ
bound : ℝ → ℝ
hs : s ∈ 𝓝 x₀
hF_int : IntervalIntegrable (F x₀) μ a b
h_lip :
  (∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s) ∧
    ∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 ↦ F x_1 x) s
bound_integrable : IntervalIntegrable bound μ a b
h_diff :
  (∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), HasFDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀) ∧
    ∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), HasFDerivAt (fun x_1 ↦ F x_1 x) (F' x) x₀
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Set.Ioc a b)) ∧ AEStronglyMeasurable F' (μ.restrict (Set.Ioc b a))
hF_meas :
  (∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Set.Ioc a b))) ∧
    ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Set.Ioc b a))
H₁ :
  Integrable F' (μ.restrict (Set.Ioc a b)) ∧
    HasFDerivAt (fun x ↦ ∫ (a : ℝ) in Set.Ioc a b, F x a ∂μ) (∫ (a : ℝ) in Set.Ioc a b, F' a ∂μ) x₀
⊢ IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x ↦ ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
Module: Mathlib.Topology.MetricSpace.ProperSpace.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : ProperSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a z : α
r : ℝ
hf : ContinuousOn f (closedBall a r)
hz : z ∈ closedBall a r
hf1 : ∀ z' ∈ sphere a r, f z < f z'
⊢ ∃ z ∈ ball a r, IsLocalMin f z

Tactic:
simp_rw [← closedBall_sdiff_ball] at hf1

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : ProperSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a z : α
r : ℝ
hf : ContinuousOn f (closedBall a r)
hz : z ∈ closedBall a r
hf1 : ∀ z' ∈ closedBall a r \ ball a r, f z < f z'
⊢ ∃ z ∈ ball a r, IsLocalMin f z
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case pos
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : SeminormedAddCommGroup G
x₀ : G
R : ℝ
hf : support f ⊆ ball 0 R
t : G
hg : ‖t‖ < R → g (x₀ - t) = g x₀
ht : t ∈ support f
h2t : ‖t‖ < R
⊢ (L (f t)) (g (x₀ - t)) = (L (f t)) (g x₀)

Tactic:
rw [hg h2t]

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case refine_1
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 : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : SFinite μ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hif : Integrable f μ
hf : support f ⊆ ball 0 R
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
⊢ ∀ y ∈ (fun i ↦ ‖g i‖) '' (fun t ↦ x₀ - t) ⁻¹' ball 0 R, y ≤ ‖z₀‖ + ε

Tactic:
rintro _ ⟨x, hx, rfl⟩

Goal state after:
case refine_1
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 : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : SFinite μ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hif : Integrable f μ
hf : support f ⊆ ball 0 R
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
x : G
hx : x ∈ (fun t ↦ x₀ - t) ⁻¹' ball 0 R
⊢ (fun i ↦ ‖g i‖) x ≤ ‖z₀‖ + ε
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
A : ∀ (n : ℕ), μ ((x * z⁻¹) • k \ k) ≤ u n
B : μ ((x * z⁻¹) • k \ k) = 0
C : μ (k \ (z * x⁻¹) • k) = 0
H : z ∈ (z * x⁻¹) • k
⊢ x ∈ k

Tactic:
simpa [mem_smul_set_iff_inv_smul_mem] using H

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

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

Tactic:
have : k ∈ A := by
      simpa only [true_and, mem_univ, Classical.not_not, Set.mem_sdiff] using
        Nat.notMem_of_lt_sInf hk

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)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
k : ℕ
hk : k < sInf (univ \ A)
this : k ∈ A
⊢ ∃ j < i,
    (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
      k = p.color j
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

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

Tactic:
obtain ⟨⟨f, f_cont⟩, hf⟩ := exists_continuous_nonneg_pos (1 : G)

Goal state after:
case pos
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
⊢ (map (⇑φ) μ').haarScalarFactor (map (⇑φ) μ) = μ'.haarScalarFactor μ
Module: Mathlib.Analysis.Calculus.BumpFunction.Convolution

Goal state before:
pp: G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : CompleteSpace E'
inst✝² : BorelSpace G
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddHaarMeasure
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
x₀ : G
h₀ : Tendsto (fun a ↦ ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
⊢ Tendsto (fun i ↦ ∫ (t : G), (φ i).normed μ (x₀ - t) • g t ∂μ) l (𝓝 (g x₀))

Tactic:
have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, Eventually.of_forall (fun i ↦ (φ i).rOut_pos)⟩

Goal state after:
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ E'
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace ℝ G
inst✝³ : CompleteSpace E'
inst✝² : BorelSpace G
inst✝¹ : FiniteDimensional ℝ G
inst✝ : μ.IsAddHaarMeasure
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g μ
x₀ : G
h₀ : Tendsto (fun a ↦ ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)
hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0)
⊢ Tendsto (fun i ↦ ∫ (t : G), (φ i).normed μ (x₀ - t) • g t ∂μ) l (𝓝 (g x₀))
Module: Mathlib.Analysis.Calculus.ContDiff.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝¹ : CompleteSpace 𝕜
f : D → E →L[𝕜] F
s : Set D
inst✝ : FiniteDimensional 𝕜 E
h : ∀ (y : E), ContDiffOn 𝕜 n (fun x ↦ (f x) y) s
d : ℕ := finrank 𝕜 E
⊢ ContDiffOn 𝕜 n f s

Tactic:
have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm

Goal state after:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
n : ℕ∞ω
inst✝¹ : CompleteSpace 𝕜
f : D → E →L[𝕜] F
s : Set D
inst✝ : FiniteDimensional 𝕜 E
h : ∀ (y : E), ContDiffOn 𝕜 n (fun x ↦ (f x) y) s
d : ℕ := finrank 𝕜 E
hd : d = finrank 𝕜 (Fin d → 𝕜)
⊢ ContDiffOn 𝕜 n f s
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
k : ℕ
f : E → F
a : E
⊢ ContDiffPointwiseHolderAt k 0 f a ↔ ContDiffAt ℝ (↑k) f a

Tactic:
by
  refine ⟨contDiffAt, fun h ↦ ⟨h, ?_⟩⟩
  simpa using ((h.continuousAt_iteratedFDeriv le_rfl).sub_const _).norm.isBoundedUnder_le

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

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : MetricSpace Y
r : ℝ≥0
f : X → Y
x₁ x₂ : X
h : edist (f x₁) (f x₂) ≤ ↑0 * edist x₁ x₂ ^ ↑r
⊢ f x₁ = f x₂

Tactic:
simp [ENNReal.coe_zero, zero_mul, nonpos_iff_eq_zero, edist_eq_zero] at h

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : MetricSpace Y
r : ℝ≥0
f : X → Y
x₁ x₂ : X
h : f x₁ = f x₂
⊢ f x₁ = f x₂
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : SeminormedAddCommGroup Y
C C' r : ℝ≥0
f g : X → Y
hf : HolderWith C r f
hg : HolderWith C' r g
x₁ x₂ : X
⊢ ↑C * edist x₁ x₂ ^ ↑r + edist (g x₁) (g x₂) ≤ (↑C + ↑C') * edist x₁ x₂ ^ ↑r

Tactic:
hg x₁ x₂

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : SeminormedAddCommGroup Y
C C' r : ℝ≥0
f g : X → Y
hf : HolderWith C r f
hg : HolderWith C' r g
x₁ x₂ : X
⊢ ↑C * edist x₁ x₂ ^ ↑r + ↑C' * edist x₁ x₂ ^ ↑r ≤ (↑C + ↑C') * edist x₁ x₂ ^ ↑r
Module: Mathlib.Analysis.Calculus.Deriv.Abs

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : StrongDual ℝ E
x : E
hf : HasStrictFDerivAt f f' x
h₀ : f x ≠ 0
⊢ HasStrictFDerivAt (fun x ↦ |f x|) (↑(SignType.sign (f x)) • f') x

Tactic:
convert! (hasStrictDerivAt_abs h₀).hasStrictFDerivAt.comp x hf using 1
  ext y
  simp [mul_comm]

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

Goal state before:
pp: f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ IsMaxOn f (Iic c) b

Tactic:
by
  refine isMaxOn_Iic_of_mono_anti ?_ ?_
  · apply monotoneOn_of_deriv_nonneg (convex_Iic b) (continuousOn_Iic hb hd₀) <;> simp_all
  · apply antitoneOn_of_deriv_nonpos (convex_Icc b c) (continuousOn_Icc hb hc hd₁) <;> simp_all

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
L : F →L[𝕜] G
f : E → F
x : E
N : ℕ∞ω
n : ℕ
hf : ContDiffAt 𝕜 N f x
hn : ↑n ≤ N
⊢ ‖iteratedFDeriv 𝕜 n (⇑L ∘ f) x‖ ≤ ‖L‖ * ‖iteratedFDeriv 𝕜 n f x‖

Tactic:
simp only [← iteratedFDerivWithin_univ]
  exact L.norm_iteratedFDerivWithin_comp_left hf.contDiffWithinAt uniqueDiffOn_univ (Set.mem_univ x)
    hn

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

Goal state before:
pp: α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : μ v ≤ μ s' + ε / 2 / ↑N
r1 : α → ℝ
hr1 : ∀ x ∈ s', r1 x ∈ f x ∩ Ioo 0 1 ∧ closedBall x (r1 x) ⊆ v
q : BallPackage (↑s') α := { c := fun x ↦ ↑x, r := fun x ↦ r1 ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }
S : Fin N → Set ↑s'
S_disj : ∀ (i : Fin N), (S i).PairwiseDisjoint fun j ↦ closedBall (q.c j) (q.r j)
hS : range q.c ⊆ ⋃ i, ⋃ j ∈ S i, ball (q.c j) (q.r j)
S_count : ∀ (i : Fin N), (S i).Countable
r : α → ℝ := fun x ↦ if x ∈ s' then r1 x else r0 x
r_t0 : ∀ x ∈ t0, r x = r0 x
⊢ μ (⋃ x, closedBall (↑x) (r0 ↑x)) ≤ μ u

Tactic:
apply measure_mono
          simp only [SetCoe.forall, iUnion_subset_iff]
          intro x hx
          apply Subset.trans (closedBall_subset_ball (hr0 x hx).2.2) (hR x (t0s hx)).2

Goal state after: