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
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : DecidableEq P
p : ι → P
s : Finset ι
n : ℕ
hc : #s = n + 1
⊢ finrank k ↥(vectorSpan k ↑(Finset.image p s)) ≤ n

Tactic:
have hn : (s.image p).Nonempty := by
    rw [Finset.image_nonempty, ← Finset.card_pos, hc]
    apply Nat.succ_pos

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : DecidableEq P
p : ι → P
s : Finset ι
n : ℕ
hc : #s = n + 1
hn : (Finset.image p s).Nonempty
⊢ finrank k ↥(vectorSpan k ↑(Finset.image p s)) ≤ n
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)
⊢ (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) = (↑n + 1) • ((↑n)⁻¹ - (↑n + 1)⁻¹) • ∑ x, (s.points x -ᵥ s.points i)

Tactic:
smul_smul

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
i : Fin (n + 1)
⊢ (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) = ((↑n + 1) * ((↑n)⁻¹ - (↑n + 1)⁻¹)) • ∑ x, (s.points x -ᵥ s.points i)
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)
this : -(s.points i -ᵥ s.centroid) = -1 • (s.points i -ᵥ s.centroid)
⊢ (↑n)⁻¹ • -1 • (s.points i -ᵥ s.centroid) = (-1 / ↑n) • (s.points i -ᵥ s.centroid)

Tactic:
smul_smul

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
i : Fin (n + 1)
this : -(s.points i -ᵥ s.centroid) = -1 • (s.points i -ᵥ s.centroid)
⊢ ((↑n)⁻¹ * -1) • (s.points i -ᵥ s.centroid) = (-1 / ↑n) • (s.points i -ᵥ s.centroid)
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 : Set P
p₀ : P
h : p₀ ∈ s
v : V
hp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
w : V
hw : w ∈ vectorSpan k s
⊢ Submodule.span k ((fun x ↦ x -ᵥ p₀) '' s) ≤ k ∙ v

Tactic:
Submodule.span_le,

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 : Set P
p₀ : P
h : p₀ ∈ s
v : V
hp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
w : V
hw : w ∈ vectorSpan k s
⊢ (fun x ↦ x -ᵥ p₀) '' s ⊆ ↑(k ∙ 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
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
⊢ affineSpan k (Set.range s.medial.points) = affineSpan k (Set.range s.points)

Tactic:
have hmem2 : s.medial.points 0 ∈ affineSpan k (Set.range s.points) := by
    apply Set.mem_of_mem_of_subset (s.faceOppositeCentroid_mem_affineSpan_face 0)
    exact affineSpan_mono k (by simp)

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)
⊢ affineSpan k (Set.range s.medial.points) = affineSpan k (Set.range s.points)
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
⊢ affineSpan k (Set.range s.medial.points) = affineSpan k (Set.range s.points)

Tactic:
have hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points) :=
    mem_affineSpan _ (by simp)
  have hmem2 : s.medial.points 0 ∈ affineSpan k (Set.range s.points) := by
    apply Set.mem_of_mem_of_subset (s.faceOppositeCentroid_mem_affineSpan_face 0)
    exact affineSpan_mono k (by simp)
  rw [eq_iff_direction_eq_of_mem hmem1 hmem2]
  simp_rw [direction_affineSpan, vectorSpan_def]
  suffices Set.range s.medial.points -ᵥ Set.range s.medial.points
    = (-n : k)⁻¹ • (Set.range s.points -ᵥ Set.range s.points) by
    rw [this, Submodule.span_smul_eq_of_isUnit]
    simpa using NeZero.ne n
  ext v
  suffices (∃ a b, (n : k)⁻¹ • (s.points b -ᵥ s.points a) = v) ↔
    ∃ a b, -((n : k)⁻¹ • (s.points a -ᵥ s.points b)) = v by
    simpa [Set.mem_vsub, Set.mem_smul_set, medial_points,
      faceOppositeCentroid_vsub_faceOppositeCentroid]
  congrm ∃ a b, ?_ = v
  simp [← smul_neg]

Goal state after:
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 : Set P
h : Collinear k s
p₁ p₂ p₃ : P
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₂p₃ : p₂ ≠ p₃
⊢ vectorSpan k (insert p₁ s) = (affineSpan k (insert p₁ ↑line[k, p₂, p₃])).direction

Tactic:
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, h.affineSpan_eq_of_ne hp₂ hp₃ hp₂p₃]

Goal state after:
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
p₁ p₂ p₃ p₄ : P
h₁ : p₁ ∈ line[k, p₃, p₄]
h₂ : p₂ ∈ line[k, p₃, p₄]
⊢ Collinear k {p₁, p₂, p₃, p₄}

Tactic:
rw [collinear_insert_iff_of_mem_affineSpan
      ((AffineSubspace.le_def' _ _).1 (affineSpan_mono k (Set.subset_insert _ _)) _ h₁),
    collinear_insert_iff_of_mem_affineSpan h₂]

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
p₁ p₂ p₃ p₄ : P
h₁ : p₁ ∈ line[k, p₃, p₄]
h₂ : p₂ ∈ line[k, p₃, p₄]
⊢ Collinear k {p₃, p₄}
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
p₁ p₂ p₃ p₄ p₅ : P
h₁ : p₁ ∈ line[k, p₄, p₅]
h₂ : p₂ ∈ line[k, p₄, p₅]
h₃ : p₃ ∈ line[k, p₄, p₅]
⊢ Collinear k {p₂, p₃, p₄, p₅}

Tactic:
collinear_insert_iff_of_mem_affineSpan
      ((AffineSubspace.le_def' _ _).1 (affineSpan_mono k (Set.subset_insert _ _)) _ h₂),

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
p₁ p₂ p₃ p₄ p₅ : P
h₁ : p₁ ∈ line[k, p₄, p₅]
h₂ : p₂ ∈ line[k, p₄, p₅]
h₃ : p₃ ∈ line[k, p₄, p₅]
⊢ Collinear k {p₃, p₄, p₅}
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: case left
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦
    iteratedFDeriv 𝕜 m (fun x ↦ ∑' (i : α), f i x) x

Tactic:
rw [iteratedFDeriv_tsum hf hv h'f hm]

Goal state after:
case left
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
m : ℕ
hm : ↑m ≤ N
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦
    (fun x ↦ ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: I : Type u_5
B : I → Type u_6
inst✝ : (i : I) → NonUnitalNormedRing (B i)
f g : (i : I) → B i
hf : Memℓp f ∞
hg : Memℓp g ∞
⊢ Memℓp (f * g) ∞

Tactic:
by
  rw [memℓp_infty_iff]
  obtain ⟨⟨Cf, hCf⟩, ⟨Cg, hCg⟩⟩ := hf.bddAbove, hg.bddAbove
  refine ⟨Cf * Cg, ?_⟩
  rintro _ ⟨i, rfl⟩
  calc
    ‖(f * g) i‖ ≤ ‖f i‖ * ‖g i‖ := norm_mul_le (f i) (g i)
    _ ≤ Cf * Cg :=
      mul_le_mul (hCf ⟨i, rfl⟩) (hCg ⟨i, rfl⟩) (norm_nonneg _)
        ((norm_nonneg _).trans (hCf ⟨i, rfl⟩))

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

Goal state before:
pp: case inr.left
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 := ⋯
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝ : NormedSpace ℝ E := ⋯
this : l.NeBot
hr : 0 < r
⊢ TendstoUniformlyOn (fun n z ↦ f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (Metric.ball x r)

Tactic:
rw [Metric.tendstoUniformlyOn_iff] at hf' ⊢

Goal state after:
case inr.left
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 > 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 - (f n.1 x - f n.2 x)) < ε
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
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x : E
hx₀ : x₀ ∈ s
hx : x ∈ s
hfg : Cauchy (map (fun n ↦ f n x₀) l)
this : l.NeBot
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n ↦ f n y) l)}
⊢ s ⊆ t

Tactic:
have A : ∀ x ε, x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t := fun x ε xt hx y hy =>
    ⟨hx hy,
      (uniformCauchySeqOn_ball_of_fderiv (hf'.mono hx) (fun n y hy => hf n y (hx hy))
            xt.2).cauchy_map
        hy⟩

Goal state after:
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
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x : E
hx₀ : x₀ ∈ s
hx : x ∈ s
hfg : Cauchy (map (fun n ↦ f n x₀) l)
this : l.NeBot
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n ↦ f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
⊢ s ⊆ t
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✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜 E'
inst✝⁴ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝³ : MeasurableSpace G
μ : Measure G
inst✝² : AddGroup G
inst✝¹ : MeasurableAdd G
inst✝ : MeasurableNeg G
x₀ : G
s : Set G
hs : MeasurableSet s
h2s : (support fun t ↦ (L (f t)) (g (x₀ - t))) ⊆ s
hf : IntegrableOn f s μ
hmg : AEStronglyMeasurable g (Measure.map (fun t ↦ x₀ - t) (μ.restrict s))
s' : Set G := (fun t ↦ -t + x₀) ⁻¹' s
hbg : BddAbove ((fun i ↦ ‖g i‖) '' s')
this : ∀ᵐ (t : G) ∂μ.restrict s, ‖(L (f t)) (g (x₀ - t))‖ ≤ s.indicator (fun t ↦ ‖L‖ * ‖f t‖ * ⨆ i, ‖g ↑i‖) t
⊢ Integrable (s.indicator fun t ↦ ‖L‖ * ‖f t‖ * ⨆ i, ‖g ↑i‖) (μ.restrict s)

Tactic:
rw [integrable_indicator_iff hs]; exact ((hf.norm.const_mul _).mul_const _).integrableOn

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
⊢ ConvolutionExists f g L μ

Tactic:
intro x₀

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
⊢ ConvolutionExistsAt f g x₀ L μ
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
⊢ ConvolutionExists f g L μ

Tactic:
intro x₀

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
⊢ ConvolutionExistsAt f g x₀ L μ
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
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf' : TendstoLocallyUniformlyOn f' g' l s
hf : ∀ᶠ (n : ι) in l, ∀ x ∈ s, HasDerivAt (f n) (f' n x) x
hfg : ∀ x ∈ s, Tendsto (fun n ↦ f n x) l (𝓝 (g x))
hx : x ∈ s
h1 : s ∈ 𝓝 x
⊢ HasDerivAt g (g' x) x

Tactic:
have h2 : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2 :=
    eventually_prod_iff.2 ⟨_, hf, fun x => x ∈ s, h1, fun {n} => id⟩

Goal state after:
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
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf' : TendstoLocallyUniformlyOn f' g' l s
hf : ∀ᶠ (n : ι) in l, ∀ x ∈ s, HasDerivAt (f n) (f' n x) x
hfg : ∀ x ∈ s, Tendsto (fun n ↦ f n x) l (𝓝 (g x))
hx : x ∈ s
h1 : s ∈ 𝓝 x
h2 : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
⊢ HasDerivAt g (g' x) x
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
this : μ ((z * x⁻¹) • ((x * z⁻¹) • k \ k)) = 0
⊢ μ (k \ (z * x⁻¹) • k) = μ ((z * x⁻¹) • (x * z⁻¹) • k \ (z * x⁻¹) • k)

Tactic:
smul_smul

Goal state after:
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
this : μ ((z * x⁻¹) • ((x * z⁻¹) • k \ k)) = 0
⊢ μ (k \ (z * x⁻¹) • k) = μ ((z * x⁻¹ * (x * z⁻¹)) • k \ (z * x⁻¹) • k)
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case refine_4
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
⊢ f x = 1

Tactic:
apply c.one_of_mem_closedBall

Goal state after:
case refine_4
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) := ⋯
f : E → ℝ := ⋯
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ toEuclidean x ∈ closedBall (toEuclidean x) c.rIn
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
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, T = range g0

Tactic:
apply Countable.exists_eq_range T_count

Goal state after:
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) := ⋯
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
⊢ T.Nonempty
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : HasContDiffBump E
inst✝³ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
⊢ ∫ (x : E), ↑f x ∂μ = ∫ (x : E) in closedBall c f.rOut, ↑f x ∂μ

Tactic:
apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)).symm
    apply f.zero_of_le_dist (le_of_lt _)
    simpa using hx

Goal state after:
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.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
⊢ μ.real (closedBall c f.rOut) / K ^ finrank ℝ E ≤ ∫ (x : E), ↑f x ∂μ

Tactic:
have K_pos : 0 < K := by
    simpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)
  apply le_trans _ (f.measure_closedBall_le_integral μ)
  rw [div_le_iff₀ (pow_pos K_pos _), addHaar_real_closedBall' _ _ f.rIn_pos.le,
    addHaar_real_closedBall' _ _ f.rOut_pos.le, mul_assoc, mul_comm _ (K ^ _), ← mul_assoc,
    ← mul_pow, mul_comm _ K]
  gcongr
  exact f.rOut_pos.le

Goal state after: