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

Goal state before:
pp: case mpr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
p : P
h : ∑ i, (s.points i -ᵥ p) = 0
⊢ p -ᵥ s.centroid = 0

Tactic:
have : ∑ i, (s.points i -ᵥ p) = ∑ i, ((s.points i -ᵥ s.centroid) - (p -ᵥ s.centroid)) := by
      apply sum_congr rfl
      intro x hx
      rw [vsub_sub_vsub_cancel_right _ _ s.centroid]

Goal state after:
case mpr
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : CharZero k
s : Simplex k P n
p : P
h : ∑ i, (s.points i -ᵥ p) = 0
this : ∑ i, (s.points i -ᵥ p) = ∑ i, (s.points i -ᵥ s.centroid - (p -ᵥ s.centroid))
⊢ p -ᵥ s.centroid = 0
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
inst✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
⊢ (adjoint T * T).IsSymmetric

Tactic:
by
  intro x y
  simp [adjoint_inner_left, adjoint_inner_right]

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

Goal state before:
pp: case inl
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
s₂ : AffineSubspace k P
inst✝¹ : FiniteDimensional k ↥s₂.direction
inst✝ : FiniteDimensional k ↥⊥.direction
⊢ FiniteDimensional k ↥(⊥ ⊔ s₂).direction

Tactic:
bot_sup_eq

Goal state after:
case inl
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
s₂ : AffineSubspace k P
inst✝¹ : FiniteDimensional k ↥s₂.direction
inst✝ : FiniteDimensional k ↥⊥.direction
⊢ FiniteDimensional k ↥s₂.direction
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
x✝ : ∃ α, ∀ (x : H), f x = (α • g) x
y : ↥(unitary 𝕜)
h : ∀ (x : H), f x = (y • g) x
⊢ ↑y • 1 = ↑(↑g).symm ∘SL ↑↑f

Tactic:
by ext; simp [h]

Goal state after:
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
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
i j : ↥s'
hij : i ≠ j
⊢ Disjoint {i} {j}

Tactic:
simp [hij]

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

Goal state before:
pp: case inr.mpr
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 : Set P
p₁ : P
hp₁ : p₁ ∈ s
⊢ (∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀) → ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁

Tactic:
rintro ⟨p, v, hv⟩

Goal state after:
case inr.mpr
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 : Set P
p₁ : P
hp₁ : p₁ ∈ s
p : P
v : V
hv : ∀ p_1 ∈ s, ∃ r, p_1 = r • v +ᵥ p
⊢ ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
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
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
hmem2 : s.medial.points 0 ∈ affineSpan k (Set.range s.points)
v : V
⊢ (∃ a b, (↑n)⁻¹ • (s.points b -ᵥ s.points a) = v) ↔ ∃ a b, -((↑n)⁻¹ • (s.points a -ᵥ s.points b)) = v

Tactic:
congrm ∃ a b, ?_ = v

Goal state after:
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
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
hmem2 : s.medial.points 0 ∈ affineSpan k (Set.range s.points)
v : V
a b : Fin (n + 1)
⊢ (↑n)⁻¹ • (s.points b -ᵥ s.points a) = -((↑n)⁻¹ • (s.points a -ᵥ s.points b))
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)
hp : 0 < p.toReal
⊢ ‖0‖ = 0

Tactic:
rw [lp.norm_eq_tsum_rpow hp]

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

Tactic:
cases isEmpty_or_nonempty α
    · simp only [lp.eq_zero' f, neg_zero, norm_zero]
    apply (lp.isLUB_norm (-f)).unique
    simpa only [coeFn_neg, Pi.neg_apply, norm_neg] using lp.isLUB_norm f

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

Goal state before:
pp: case inr.right
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
this : 2 * √x ^ (2 - 1) ≠ 0
⊢ ∀ (n : WithTop ℕ∞), ContDiffAt ℝ n (fun x ↦ √x) x

Tactic:
exact fun n => sqPartialHomeomorph.contDiffAt_symm_deriv this hx (hasDerivAt_pow 2 (√x))
        (contDiffAt_id.pow 2)

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

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : ι × ι) in l ×ˢ l, ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
⊢ ∃ q, 0 < q ∧ q * r < ε

Tactic:
by
      simp_rw [mul_comm]
      exact exists_pos_mul_lt hε.lt r

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
| Tendsto (fun y ↦ ‖y.2 - x‖⁻¹ * ‖g y.2 - g x - (g' x) (y.2 - x)‖) (l.curry (𝓝 x)) (𝓝 0)

Tactic:
congr
    ext
    rw [← abs_norm, ← abs_inv, ← @RCLike.norm_ofReal 𝕜 _ _, RCLike.ofReal_inv, ← norm_smul]

Goal state after:
case f
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
x✝ : ι × E
| ‖(↑‖x✝.2 - x‖)⁻¹ • (g x✝.2 - g x - (g' x) (x✝.2 - x))‖

case l₁
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
| l.curry (𝓝 x)

case l₂
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
| 𝓝 0
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 := fun a ↦ |bound a|
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‖
a : α
ha : HasFDerivAt (fun x ↦ F x a) (F' a) x₀
⊢ Tendsto (fun x ↦ ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀))) (𝓝 x₀) (𝓝 0)

Tactic:
tendsto_zero_iff_norm_tendsto_zero

Goal state after:
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 := fun a ↦ |bound a|
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‖
a : α
ha : HasFDerivAt (fun x ↦ F x a) (F' a) x₀
⊢ Tendsto (fun x ↦ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀))‖) (𝓝 x₀) (𝓝 0)
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
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf' : TendstoLocallyUniformlyOn (fderiv 𝕜 ∘ f) g' l s
hf : ∀ (n : ι), DifferentiableOn 𝕜 (f n) s
hfg : ∀ x ∈ s, Tendsto (fun n ↦ f n x) l (𝓝 (g x))
hx : x ∈ s
n : ι
z : E
hz : z ∈ s
⊢ HasFDerivAt (f n) ((fderiv 𝕜 ∘ f) n z) z

Tactic:
exact ((hf n z hz).differentiableAt (hs.mem_nhds hz)).hasFDerivAt

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

Goal state before:
pp: case pos.inl.inr
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
hF : CompleteSpace F
hE : finrank ℝ E > 0
⊢ μ.real univ • f 0 = |(0 ^ finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ

Tactic:
have : Nontrivial E := finrank_pos_iff.1 hE
      simp [zero_pow hE.ne', measure_univ_of_isAddLeftInvariant, measureReal_def]

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

Goal state before:
pp: F : Type u_1
inst✝⁶ : NormedAddCommGroup F
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E → F
R : ℝ
hR : R ≠ 0
⊢ Integrable (fun x ↦ f (R • x)) μ ↔ Integrable f μ

Tactic:
suffices
    ∀ {g : E → F} (_ : Integrable g μ) {S : ℝ} (_ : S ≠ 0), Integrable (fun x => g (S • x)) μ by
    refine ⟨fun hf => ?_, fun hf => this hf hR⟩
    convert! this hf (inv_ne_zero hR)
    rw [← mul_smul, mul_inv_cancel₀ hR, one_smul]
  -- now prove
  intro g hg S hS
  let t := ((Homeomorph.smul (isUnit_iff_ne_zero.2 hS).unit).toMeasurableEquiv : E ≃ᵐ E)
  refine (integrable_map_equiv t g).mp (?_ : Integrable g (map (S • ·) μ))
  rwa [map_addHaar_smul μ hS, integrable_smul_measure _ ENNReal.ofReal_ne_top]
  simpa only [Ne, ENNReal.ofReal_eq_zero, not_le, abs_pos] using inv_ne_zero (pow_ne_zero _ hS)

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

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

Tactic:
rintro t ⟨y, rfl⟩
    exact ⟨c.nonneg, c.le_one⟩

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

Goal state before:
pp: case inr.refine_1.h₁
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
x : E
hx : x ∉ s
⊢ ∑' (n : ℕ), r n * g n x = 0

Tactic:
have : ∀ n, g n x = 0 := by
        intro n
        contrapose! hx
        exact g_s n hx

Goal state after:
case inr.refine_1.h₁
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
x : E
hx : x ∉ s
this : ∀ (n : ℕ), g n x = 0
⊢ ∑' (n : ℕ), r n * g n x = 0
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
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
⊢ G ↑a < G ↑b →
    p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
      p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))

Tactic:
G_lt

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
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
a b : Fin N.succ
G_lt : G ↑a < G ↑b
⊢ p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧
    p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case right
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))
ID : ‖D / (1 + D) - 1‖ = 1 / (1 + D)
⊢ ‖x - y‖ ≤ 1

Tactic:
rw [← mem_closedBall_iff_norm']

Goal state after:
case right
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))
ID : ‖D / (1 + D) - 1‖ = 1 / (1 + D)
⊢ y ∈ closedBall x 1
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
t_comp : IsCompact t
st : s ⊆ t
A : ν t - ν s ≤ μ' (t \ s)
B : μ' t = ν t
⊢ μ' s ≠ ∞

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
t_comp : IsCompact t
st : s ⊆ t
A : ν (t \ s) ≤ μ' (t \ s)
B : μ' t = ν t
⊢ ν s ≠ ∞

Tactic:
· exact ((measure_mono st).trans_lt t_comp.measure_lt_top).ne

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
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
t_comp : IsCompact t
st : s ⊆ t
A : ν (t \ s) ≤ μ' (t \ s)
B : μ' t = ν t
⊢ ν s ≠ ∞
Module: Mathlib.MeasureTheory.Covering.Besicovitch

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

Tactic:
{ exact (x_ne_y rfl).elim }

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

Goal state before:
pp: G : Type u_1
inst✝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
t_comp : IsCompact t
st : s ⊆ t
A : ν (t \ s) ≤ μ' (t \ s)
B : μ' t = ν t
⊢ ν s ≠ ∞

Tactic:
· exact ((measure_mono st).trans_lt t_comp.measure_lt_top).ne

Goal state after: