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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
y : F
x : E
⊢ ↑(adjoint A) (y - A x) = 0 ↔ (adjoint A) (y - A x) = 0

Tactic:
coe_coe

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
y : F
x : E
⊢ (adjoint A) (y - A x) = 0 ↔ (adjoint A) (y - A x) = 0
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
T : E →L[𝕜] E
hT : IsSelfAdjoint T
S : E →L[𝕜] F
⊢ IsSelfAdjoint (S ∘SL T ∘SL adjoint S)

Tactic:
isSelfAdjoint_iff'

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
T : E →L[𝕜] E
hT : adjoint T = T
S : E →L[𝕜] F
⊢ adjoint (S ∘SL T ∘SL adjoint S) = S ∘SL T ∘SL adjoint S
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
T : E →L[𝕜] E
hT : IsSelfAdjoint T
S : F →L[𝕜] E
⊢ IsSelfAdjoint (adjoint S ∘SL T ∘SL S)

Tactic:
isSelfAdjoint_iff'

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
T : E →L[𝕜] E
hT : adjoint T = T
S : F →L[𝕜] E
⊢ adjoint (adjoint S ∘SL T ∘SL S) = adjoint S ∘SL T ∘SL S
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 • (s.faceOppositeCentroid i -ᵥ s.points i) = ∑ x, (s.points x -ᵥ s.points i)

Tactic:
simp [faceOppositeCentroid_eq_sum_vsub_vadd, smul_smul, mul_inv_cancel₀ (NeZero.ne (n : k)),
    one_smul]

Goal state after:
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
⊢ (∀ (a : H →L[𝕜] H), g.conjStarAlgEquiv a = f.conjStarAlgEquiv a) ↔ ∃ α, ∀ (x : H), f x = (α • g) x

Tactic:
conjStarAlgEquiv_apply,

Goal state after:
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
⊢ (∀ (a : H →L[𝕜] H), ↑↑g ∘SL a ∘SL ↑↑g.symm = ↑↑f ∘SL a ∘SL ↑↑f.symm) ↔ ∃ α, ∀ (x : H), f x = (α • g) x
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:
classical
  have hn : (s.image p).Nonempty := by
    rw [Finset.image_nonempty, ← Finset.card_pos, hc]
    apply Nat.succ_pos
  rcases hn with ⟨p₁, hp₁⟩
  rw [vectorSpan_eq_span_vsub_finset_right_ne k hp₁]
  refine le_trans (finrank_span_finset_le_card (((s.image p).erase p₁).image fun p => p -ᵥ p₁)) ?_
  rw [Finset.card_image_of_injective _ (vsub_left_injective p₁), Finset.card_erase_of_mem hp₁,
    tsub_le_iff_right, ← hc]
  apply Finset.card_image_le

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
h : LinearIndependent k fun i ↦ s.points ↑i -ᵥ s.points 0
i : { x // x ≠ 0 }
⊢ (↑n)⁻¹ • (s.points 0 -ᵥ s.points ↑i) = ((fun x ↦ Units.mk0 (-↑n)⁻¹ ⋯) • fun i ↦ s.points ↑i -ᵥ s.points 0) i

Tactic:
simp [← smul_neg]

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
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))

Tactic:
simp [← smul_neg]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
h : affineSpan ℝ s = ⊤
t : Set V
hts : t ⊆ s
b : AffineBasis ↑t ℝ V
hb : ⇑b = Subtype.val
this : (interior ((convexHull ℝ) (range ⇑b))).Nonempty
⊢ (interior ((convexHull ℝ) s)).Nonempty

Tactic:
rw [hb, Subtype.range_coe_subtype, ofPred_mem_eq] at this
    refine this.mono (by gcongr)

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
p₁ : P
hp₁ : p₁ ∈ s
v : V
h : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
r₂ : k
hp₂ : r₂ • v +ᵥ p₁ ∈ s
hp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁
r₃ : k
hp₃ : r₃ • v +ᵥ p₁ ∈ s
h₂ : r₂ ≠ 0
⊢ (r₃ / r₂) • ((r₂ • v +ᵥ p₁) -ᵥ p₁) = r₃ • v

Tactic:
simp [smul_smul, h₂]

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

Goal state before:
pp: case 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‖ ^ ENNReal.toReal 0
⊢ {i | f i ≠ 0}.Finite

Tactic:
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf

Goal state after:
case 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‖ ^ ENNReal.toReal 0
H : Summable fun x ↦ 1
⊢ {i | f i ≠ 0}.Finite
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
⊢ BddAbove (Set.range fun i ↦ ‖f i‖)

Tactic:
have H : Summable fun _ : α => (1 : ℝ) := by simpa using hf

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

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

Tactic:
refine ⟨fun hf ↦ ?_, fun ⟨C, _, hC⟩ ↦ memℓp_gen' hC⟩
  exact ⟨_, tsum_nonneg fun i ↦ (by positivity), memℓp_gen_iff' hp |>.mp hf⟩

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

Goal state before:
pp: 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
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
x₀ : E
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ HasFDerivAt (fun y ↦ ∑' (n : α), f n y) (∑' (n : α), f' n x) x

Tactic:
let A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _

Goal state after:
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
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
x₀ : E
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n ↦ f n x₀
x : E
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ HasFDerivAt (fun y ↦ ∑' (n : α), f n y) (∑' (n : α), f' n x) x
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : (i : α) → StarAddMonoid (E i)
inst✝ : ∀ (i : α), NormedStarGroup (E i)
hp : Fact (1 ≤ p)
f : ↥(lp E p)
h : 0 < p.toReal
⊢ ‖star f‖ = ‖f‖

Tactic:
· simp only [lp.norm_eq_tsum_rpow h, lp.star_apply, norm_star]

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

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_6
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : 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
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))
⊢ TendstoUniformlyOnFilter (fun n y ↦ (↑‖y - x‖)⁻¹ • (f n y - f n x)) (fun y ↦ (↑‖y - x‖)⁻¹ • (g y - g x)) l (𝓝 x)

Tactic:
let A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _

Goal state after:
ι : Type u_1
l : Filter ι
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_6
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : 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
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))
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n y ↦ (↑‖y - x‖)⁻¹ • (f n y - f n x)) (fun y ↦ (↑‖y - x‖)⁻¹ • (g y - g x)) l (𝓝 x)
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
x₀ : G
h : HasCompactSupport fun t ↦ (L (f t)) (g (x₀ - t))
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
u : G ≃ₜ G := (Homeomorph.neg G).trans (Homeomorph.addRight x₀)
v : G ≃ₜ G := (Homeomorph.neg G).trans (Homeomorph.addLeft x₀)
A : AEStronglyMeasurable (g ∘ ⇑v) (μ.restrict (tsupport fun t ↦ (L (f t)) (g (x₀ - t))))
x✝ x : G
⊢ ↑(toAddUnits x₀) - x = v x

Tactic:
simp only [v, Homeomorph.neg, sub_eq_add_neg, val_toAddUnits_apply, Homeomorph.trans_apply,
    Equiv.neg_apply, Homeomorph.homeomorph_mk_coe, Homeomorph.coe_addLeft]

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
x✝ : ι × E
⊢ (↑‖x✝.2 - x‖)⁻¹ • (g x✝.2 - g x - (g' x) (x✝.2 - x)) =
    (↑‖x✝.2 - x‖)⁻¹ • (g x✝.2 - g x - (f x✝.1 x✝.2 - f x✝.1 x)) +
        (↑‖x✝.2 - x‖)⁻¹ • (f x✝.1 x✝.2 - f x✝.1 x - ((f' x✝.1 x) x✝.2 - (f' x✝.1 x) x)) +
      (↑‖x✝.2 - x‖)⁻¹ • (f' x✝.1 x - g' x) (x✝.2 - x)

Tactic:
← smul_add,

Goal state after:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : 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)) =
    (↑‖x✝.2 - x‖)⁻¹ •
        (g x✝.2 - g x - (f x✝.1 x✝.2 - f x✝.1 x) + (f x✝.1 x✝.2 - f x✝.1 x - ((f' x✝.1 x) x✝.2 - (f' x✝.1 x) x))) +
      (↑‖x✝.2 - x‖)⁻¹ • (f' x✝.1 x - g' x) (x✝.2 - x)
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x ↦ ‖g x‖)
hf : Integrable f μ
hg : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ : G
⊢ ∀ᶠ (x : G) in 𝓝 x₀, ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖

Tactic:
filter_upwards with x

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x ↦ ‖g x‖)
hf : Integrable f μ
hg : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ x : G
⊢ ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖
Module: Mathlib.Topology.Metrizable.Urysohn

Goal state before:
pp: case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
⊢ F ⁻¹' closedBall (F x) δ ∈ 𝓝 x

Tactic:
have h_fin : { UV : s | δ ≤ ε UV }.Finite := by simpa only [← not_lt] using! hε (gt_mem_nhds δ0)

Goal state after:
case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : {UV | δ ≤ ε UV}.Finite
⊢ F ⁻¹' closedBall (F x) δ ∈ 𝓝 x
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : HasContDiffBump E
inst✝ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
x : E
⊢ f.normed μ (c - x) = f.normed μ (c + x)

Tactic:
by
  simp_rw [f.normed_def, f.sub]

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
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
ι : Type u_2
φ : ι → ContDiffBump c
l : Filter ι
hφ : ∀ ε > 0, ∀ᶠ (x : ι) in l, (φ x).rOut < ε
ε : ℝ
hε : 0 < ε
i : ι
hi : (φ i).rOut < ε
⊢ (support fun x ↦ (φ i).normed μ x) ∈ 𝒫 ball c ε

Tactic:
rw [(φ i).support_normed_eq]

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
ι : Type u_2
φ : ι → ContDiffBump c
l : Filter ι
hφ : ∀ ε > 0, ∀ᶠ (x : ι) in l, (φ x).rOut < ε
ε : ℝ
hε : 0 < ε
i : ι
hi : (φ i).rOut < ε
⊢ ball c (φ i).rOut ∈ 𝒫 ball c ε
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case inr
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
n : ℕ
hn : n ≤ N
H : n < N
⊢ G n < p.lastStep

Tactic:
simp only [G]; simp only [H.ne, (hg n H).left.trans hi, if_false]

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

Goal state before:
pp: case h1
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
⊢ ∀ᵐ (x : E) ∂μ, w D (-x) = w D x

Tactic:
filter_upwards with x

Goal state after:
case h1
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✝ x : E
⊢ w D (-x) = w D x
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case h2
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
⊢ ∀ᵐ (x : E) ∂μ, φ (-x) = φ x

Tactic:
filter_upwards with x

Goal state after:
case h2
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✝ x : E
⊢ φ (-x) = φ x
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case h_lim
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 := ⋯
vf_cont : ∀ (n : ℕ), Continuous[inst✝⁷, _] (v n ∘ f)
ν : Measure G
hν : IsFiniteMeasureOnCompacts ν
⊢ ∀ᵐ (a : G) ∂ν, Tendsto (fun n ↦ v n (f a)) atTop (𝓝 ({1}.indicator (fun x ↦ 1) (f a)))

Tactic:
filter_upwards with x

Goal state after:
case h_lim
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
⊢ Tendsto (fun n ↦ v n (f x)) atTop (𝓝 ({1}.indicator (fun x ↦ 1) (f x)))
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: case e_a.e_a
X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C D s : ℝ≥0
A : Set X
hA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D
hf : HolderOnWith C r f A
hsr : ↑s ≤ ↑r
ht : 0 < s
hr : 0 < ↑r
θ₁ : ℝ≥0 := NNReal.mk (↑s / ↑r) ⋯
θ₂ : ℝ≥0 := NNReal.mk (1 - ↑s / ↑r) ⋯
hθ : θ₁ + θ₂ = 1
hθt : r * θ₁ + 0 * θ₂ = s
⊢ ↑r - ↑s = ↑r * ↑θ₂

Tactic:
simp [mul_sub, θ₂, mul_div_cancel₀ _ hr.ne']

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
C₁ C₂ s t : ℝ≥0
A : Set X
hf₁ : HolderOnWith C₁ r f A
hf₂ : HolderOnWith C₂ s f A
hrt : r ≤ t
hts : t ≤ s
⊢ HolderOnWith (max C₁ C₂) t f A

Tactic:
by
  replace hf₁ := hf₁.mono_const (le_max_left C₁ C₂)
  replace hf₂ := hf₂.mono_const (le_max_right C₁ C₂)
  exact convex_setOfPred_holderOnWith f (max C₁ C₂) A |>.segment_subset hf₁ hf₂
    (NNReal.Icc_subset_segment ⟨hrt, hts⟩)

Goal state after: