mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : ‖↑u - 1‖ < 2
⊢ 2 * (1 - Real.cos ‖argSelfAdjoint u‖) = ‖↑u - 1‖ ^ 2

Tactic:
conv_rhs => rw [← expUnitary_argSelfAdjoint hu]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : ‖↑u - 1‖ < 2
⊢ 2 * (1 - Real.cos ‖argSelfAdjoint u‖) = ‖↑(expUnitary (argSelfAdjoint u)) - 1‖ ^ 2
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: 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
i : Fin (n + 1)
⊢ (↑n + 1) • (s.centroid -ᵥ s.points i) = ∑ x, (s.points x -ᵥ s.points i)

Tactic:
rw [centroid_eq_smul_sum_vsub_vadd s i, vadd_vsub, smul_smul, mul_inv_cancel₀, one_smul]

Goal state after:
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
i : Fin (n + 1)
⊢ ↑n + 1 ≠ 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 : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
ι : Type u_5
b : Basis ι 𝕜 E
A : E →ₗ[𝕜] F
B : F →ₗ[𝕜] E
h : ∀ (i : ι) (y : F), ⟪A (b i), y⟫_𝕜 = ⟪b i, B y⟫_𝕜
i : ι
y : F
⊢ ⟪A (b i), y⟫_𝕜 = ⟪(adjoint B) (b i), y⟫_𝕜

Tactic:
by simp only [h i, adjoint_inner_left]

Goal state after:
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 : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
ι : Type u_5
b : Basis ι 𝕜 F
A : E →ₗ[𝕜] F
B : F →ₗ[𝕜] E
h : ∀ (i : ι) (x : E), ⟪A x, b i⟫_𝕜 = ⟪x, B (b i)⟫_𝕜
x : E
i : ι
⊢ ⟪A x, b i⟫_𝕜 = ⟪(adjoint B) x, b i⟫_𝕜

Tactic:
by simp only [h i, adjoint_inner_left]

Goal state after:
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
E : Type u_5
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Group G
inst✝¹ : MeasurableInv G
f : G → E
μ : Measure G
inst✝ : μ.IsInvInvariant
⊢ ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ

Tactic:
by
  have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
  rw [← h.integral_map, map_inv_eq_self]

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : HasContDiffBump E
c : E
f : ContDiffBump c
inst✝ : FiniteDimensional ℝ E
⊢ HasCompactSupport ↑f

Tactic:
HasCompactSupport,

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : HasContDiffBump E
c : E
f : ContDiffBump c
inst✝ : FiniteDimensional ℝ E
⊢ IsCompact (tsupport ↑f)
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 {p₁, p₂, p₃, p₄}

Tactic:
refine (collinear_insert_insert_insert_of_mem_affineSpan_pair h₁ h₂ h₃).subset ?_
  gcongr; simp

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✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
s : Set P
inst✝ : FiniteDimensional k ↥(vectorSpan k s)
⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2

Tactic:
by
  have h : Coplanar k s ↔ Module.rank k (vectorSpan k s) ≤ 2 := Iff.rfl
  rw [← finrank_eq_rank] at h
  exact mod_cast h

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 : AffineSubspace k P
p : P
hf : ¬FiniteDimensional k ↥s.direction
h : FiniteDimensional k ↥(vectorSpan k (insert p ↑s))
h' : s.direction ≤ vectorSpan k (insert p ↑s)
⊢ False

Tactic:
exact hf (Submodule.finiteDimensional_of_le h')

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

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
c : 𝕜
hf : Memℓp f ∞
⊢ Memℓp (c • f) ∞

pp: case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
⊢ Memℓp (c • f) p

Tactic:
· obtain ⟨A, hA⟩ := hf.bddAbove
    refine memℓp_infty ⟨‖c‖ * A, ?_⟩
    rintro a ⟨i, rfl⟩
    dsimp only [Pi.smul_apply]
    refine (norm_smul_le _ _).trans ?_
    gcongr
    exact hA ⟨i, rfl⟩

Goal state after:
case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
⊢ Memℓp (c • f) p
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E ∞)
h : ‖f‖ = 0
_i : Nonempty α
⊢ IsLUB (Set.range fun i ↦ ‖↑f i‖) 0

Tactic:
simpa [h] using! lp.isLUB_norm f

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
⊢ Continuous fun x ↦ (√(1 + ‖x‖ ^ 2))⁻¹

Tactic:
exact Continuous.inv₀ (by fun_prop) fun x => Real.sqrt_ne_zero'.mpr (by positivity)

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
⊢ (univBall c r).source = univ

Tactic:
unfold univBall; split_ifs <;> rfl

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

Goal state before:
pp: 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)
ι : Type u_5
l : Filter ι
inst✝¹ : l.NeBot
_i : Fact (1 ≤ p)
inst✝ : ∀ (a : α), CompleteSpace (E a)
F : ℕ → ↥(lp E p)
hF : CauchySeq F
f : (i : α) → E i
hf : Tendsto (Subtype.val ∘ F) atTop (𝓝 f)
hf' : Memℓp f p
⊢ ∃ a, Tendsto F atTop (𝓝 a)

Tactic:
exact ⟨⟨f, hf'⟩, tendsto_lp_of_tendsto_pi hF hf⟩

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))
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hfg' : TendstoUniformlyOnFilter (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (𝓝 x)
⊢ ∀ ε > 0,
    ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
      dist (0 n.2) (-((↑‖n.2 - x‖)⁻¹ • (f n.1.1 n.2 - f n.1.1 x)) + (↑‖n.2 - x‖)⁻¹ • (f n.1.2 n.2 - f n.1.2 x)) < ε

Tactic:
rw [Metric.tendstoUniformlyOnFilter_iff] at hfg'

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
hfg' : ∀ ε > 0, ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) (-f' n.1.1 n.2 + f' n.1.2 n.2) < ε
⊢ ∀ ε > 0,
    ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
      dist (0 n.2) (-((↑‖n.2 - x‖)⁻¹ • (f n.1.1 n.2 - f n.1.1 x)) + (↑‖n.2 - x‖)⁻¹ • (f n.1.2 n.2 - f n.1.2 x)) < ε
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✝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
this : ∀ᶠ (x : G) in 𝓝 x₀, ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖
⊢ ∀ᶠ (x : G) in 𝓝 x₀, AEStronglyMeasurable (fun a ↦ (L (f a)) (g (x - a))) μ

pp: case refine_2
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
this : ∀ᶠ (x : G) in 𝓝 x₀, ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖
⊢ ∀ᵐ (a : G) ∂μ, ContinuousAt (fun x ↦ (L (f a)) (g (x - a))) x₀

Tactic:
· exact Eventually.of_forall fun x =>
      hf.aestronglyMeasurable.convolution_integrand_snd' L hg.aestronglyMeasurable

Goal state after:
case refine_2
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
this : ∀ᶠ (x : G) in 𝓝 x₀, ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖
⊢ ∀ᵐ (a : G) ∂μ, ContinuousAt (fun x ↦ (L (f a)) (g (x - a))) x₀
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case neg
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✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this : 𝓝 0 = 𝓝 (0 + 0 + 0)
h1 : Tendsto (fun n ↦ g' n.2 - f' n.1 n.2) (l ×ˢ 𝓝 x) (𝓝 0)
h2 : Tendsto (fun n ↦ g' x - f' n x) l (𝓝 0)
n : ι × E
hx : ¬x = n.2
hnx : 0 < ‖n.2 - x‖
⊢ ‖(f' n.1 x - g' x) (n.2 - x)‖ ≤ ‖f' n.1 x - g' x‖ * ‖n.2 - x‖

Tactic:
exact (f' n.1 x - g' x).le_opNorm (n.2 - x)

Goal state after:
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
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z ↦ ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z ↦ ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.toSpanSingleton 𝕜 (f' n.1 n.2)) n.2
⊢ HasFDerivAt g (ContinuousLinearMap.toSpanSingleton 𝕜 (g' x)) x

Tactic:
have hf' : TendstoUniformlyOnFilter F' G' l (𝓝 x) :=
    (ContinuousLinearMap.smulRightL 𝕜 𝕜 G 1).uniformContinuous.comp_tendstoUniformlyOnFilter hf'

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
hf'✝ : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z ↦ ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z ↦ ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.toSpanSingleton 𝕜 (f' n.1 n.2)) n.2
hf' : TendstoUniformlyOnFilter F' G' l (𝓝 x)
⊢ HasFDerivAt g (ContinuousLinearMap.toSpanSingleton 𝕜 (g' x)) x
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case e_f
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := fun x ↦ ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous[inst✝⁹, _] D
D_pos : ∀ (x : G), 0 < D x
y : G
⊢ ∫ (x : G), f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ = ∫ (x : G), f (y * x) * (D (y * x))⁻¹ * g x ∂μ

Tactic:
rw [← integral_mul_left_eq_self _ y]

Goal state after:
case e_f
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := fun x ↦ ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous[inst✝⁹, _] D
D_pos : ∀ (x : G), 0 < D x
y : G
⊢ ∫ (x : G), f (y * x) * (D (y * x))⁻¹ * g (y⁻¹ * (y * x)) ∂μ = ∫ (x : G), f (y * x) * (D (y * x))⁻¹ * g x ∂μ
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
Dpos : 0 < D
⊢ (D ^ finrank ℝ E * ∫ (x : E), u x ∂μ)⁻¹ • D ^ finrank ℝ E • ∫ (x : E), u x ∂μ = 1

Tactic:
simp [field, (u_int_pos E).ne']

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

Goal state before:
pp: E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
hR : 1 < R
⊢ (support fun x ↦ if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)⁻¹ • x) else 0) = ball 0 R

Tactic:
have A : 0 < (R + 1) / 2 := by linarith

Goal state after:
E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
hR : 1 < R
A : 0 < (R + 1) / 2
⊢ (support fun x ↦ if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)⁻¹ • x) else 0) = ball 0 R
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
h's : IsCompact (closure[inst✝⁸] s)
ν : Measure G := ⋯
⊢ MeasurableSet (toMeasurable ν s ∩ closure[inst✝⁸] s)

Tactic:
exact (measurableSet_toMeasurable ν s).inter isClosed_closure.measurableSet

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

Goal state before:
pp: G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : Group G
inst✝¹⁰ : IsTopologicalGroup G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : BorelSpace G
H : Type u_2
inst✝⁷ : Group H
inst✝⁶ : TopologicalSpace H
inst✝⁵ : IsTopologicalGroup H
inst✝⁴ : CompactSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
ν : Measure H
inst✝ : ν.IsHaarMeasure
f : G →* H
hcont : Continuous[inst✝¹², inst✝⁶] ⇑f
hsurj : Surjective ⇑f
huniv : μ univ = ν univ
this✝¹ : IsFiniteMeasure μ
this✝ : (map (⇑f) μ).IsHaarMeasure
C : ℝ≥0 := (map (⇑f) μ).haarScalarFactor ν
hC : map (⇑f) μ = C • ν
this : ↑C * ν univ = 1 * ν univ
⊢ C = 1

Tactic:
rwa [ENNReal.mul_left_inj (NeZero.ne _) (measure_ne_top _ _), ENNReal.coe_eq_one] at this

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

Goal state before:
pp: case e_a
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
  ∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGroup Fu]
    [inst_3 : NormedSpace 𝕜 Fu] [inst_4 : NormedAddCommGroup Gu] [inst_5 : NormedSpace 𝕜 Gu] (B : Eu →L[𝕜] Fu →L[𝕜] Gu)
    {f : Du → Eu} {g : Du → Fu},
    ContDiffOn 𝕜 (↑n) f s →
      ContDiffOn 𝕜 (↑n) g s →
        ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
          ‖B‖ *
            ∑ i ∈ Finset.range (n + 1),
              ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
Eu Fu Gu : Type u
inst✝⁵ : NormedAddCommGroup Eu
inst✝⁴ : NormedSpace 𝕜 Eu
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
B : Eu →L[𝕜] Fu →L[𝕜] Gu
f : Du → Eu
g : Du → Fu
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hg : ContDiffOn 𝕜 (↑(n + 1)) g s
In : ↑n + 1 ≤ ↑n.succ
I1 :
  ‖iteratedFDerivWithin 𝕜 n (fun y ↦ ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y)) s x‖ ≤
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - i) g s x‖
I2 :
  ‖iteratedFDerivWithin 𝕜 n (fun y ↦ ((precompL Du B) (fderivWithin 𝕜 f s y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 (i + 1) f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
J :
  iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 (fun y ↦ (B (f y)) (g y)) s y) s x =
    iteratedFDerivWithin 𝕜 n
      (fun y ↦ ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y) + ((precompL Du B) (fderivWithin 𝕜 f s y)) (g y)) s x
A : ContDiffOn 𝕜 (↑n) (fun y ↦ ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y)) s
A' : ContDiffOn 𝕜 (↑n) (fun y ↦ ((precompL Du B) (fderivWithin 𝕜 f s y)) (g y)) s
⊢ ∑ x_1 ∈ Finset.range (n + 1),
        ↑(n.choose x_1) * (‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - x_1) g s x‖) +
      ∑ x_1 ∈ Finset.range (n + 1),
        ↑(n.choose x_1) * (‖iteratedFDerivWithin 𝕜 (x_1 + 1) f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g s x‖) =
    ∑ x_1 ∈ Finset.range (n + 1 + 1),
      ↑((n + 1).choose x_1) * (‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - x_1) g s x‖)

Tactic:
exact (Finset.sum_choose_succ_mul
      (fun i j => ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 j g s x‖) n).symm

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

Goal state before:
pp: case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
hf : ∀ i_1 ∈ insert i u, ContDiffOn 𝕜 N (f i_1) s
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ insert i u, f j x) s x‖ ≤
    ∑ p ∈ (insert i u).sym n,
      ↑(↑p).countPerms * ∏ j ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖

Tactic:
conv => lhs; simp only [Finset.prod_insert hi]

Goal state after:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x ↦ ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
hf : ∀ i_1 ∈ insert i u, ContDiffOn 𝕜 N (f i_1) s
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun x ↦ f i x * ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ (insert i u).sym n,
      ↑(↑p).countPerms * ∏ j ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖