mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B1 :
  ∫⁻ (t : ℝ) in Ioc 0 M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
B2 :
  ∫⁻ (t : ℝ) in Ioi M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
have I : Ioi (0 : ℝ) = Ioc (0 : ℝ) M ∪ Ioi M := (Ioc_union_Ioi_eq_Ioi M_nonneg).symm

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B1 :
  ∫⁻ (t : ℝ) in Ioc 0 M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
B2 :
  ∫⁻ (t : ℝ) in Ioi M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
I : Ioi 0 = Ioc 0 M ∪ Ioi M
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
cst : ℝ → ℝ := fun x ↦ 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (x : ℝ) in 0..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal 1
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}

Tactic:
ENNReal.ofReal_one,

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
cst : ℝ → ℝ := fun x ↦ 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (x : ℝ) in 0..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * 1
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case insert
ι : Type u_1
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ι → E → F
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, HasTemperateGrowth (f i)) → HasTemperateGrowth fun x ↦ ∑ i ∈ s, f i x
hf : ∀ i ∈ insert a s, HasTemperateGrowth (f i)
⊢ HasTemperateGrowth fun x ↦ ∑ i ∈ insert a s, f i x

Tactic:
| insert a s has ih =>

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

Goal state before:
pp: case e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
aux :
  ∫ (a : ℝ) in Ioi 0, μ.real {a_1 | a < f a_1} = (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ.real {a_1 | a < f a_1})).toReal
⊢ ∫ (t : ℝ) in Ioi 0, μ.real {a | t < f a} = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

Tactic:
aux

Goal state after:
case e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
aux :
  ∫ (a : ℝ) in Ioi 0, μ.real {a_1 | a < f a_1} = (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ.real {a_1 | a < f a_1})).toReal
⊢ (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ.real {a_1 | a < f a_1})).toReal =
    (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
⊢ x ∈ t →
    ‖iteratedFDerivWithin ℝ n (fun x ↦ x ^ r) t x‖ ≤
      (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k

Tactic:
hx

Goal state after:
case h
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
⊢ ‖iteratedFDerivWithin ℝ n (fun x ↦ x ^ r) t x‖ ≤
    (∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖) * (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: R₁ : Type u_1
R₂ : Type u_2
inst✝⁸ : Semiring R₁
inst✝⁷ : Semiring R₂
σ₁₂ : R₁ →+* R₂
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁶ : TopologicalSpace M₁
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : TopologicalSpace M₃
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
f : M₂ → M₃
hf : IsCompactOperator f
g : M₁ →SL[σ₁₂] M₂
this : Tendsto (⇑g) (𝓝 0) (𝓝 (g 0))
⊢ IsCompactOperator (f ∘ ⇑g)

Tactic:
map_zero

Goal state after:
R₁ : Type u_1
R₂ : Type u_2
inst✝⁸ : Semiring R₁
inst✝⁷ : Semiring R₂
σ₁₂ : R₁ →+* R₂
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁶ : TopologicalSpace M₁
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : TopologicalSpace M₃
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
f : M₂ → M₃
hf : IsCompactOperator f
g : M₁ →SL[σ₁₂] M₂
this : Tendsto (⇑g) (𝓝 0) (𝓝 0)
⊢ IsCompactOperator (f ∘ ⇑g)
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module 𝕜₁ M₁
inst✝⁴ : Module 𝕜₂ M₂
inst✝³ : IsTopologicalAddGroup M₁
inst✝² : ContinuousConstSMul 𝕜₁ M₁
inst✝¹ : IsTopologicalAddGroup M₂
inst✝ : ContinuousSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
U : Set M₂
hU : U ∈ 𝓝 (f 0)
⊢ U ∈ map (⇑f) (𝓝 0)

Tactic:
map_zero

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module 𝕜₁ M₁
inst✝⁴ : Module 𝕜₂ M₂
inst✝³ : IsTopologicalAddGroup M₁
inst✝² : ContinuousConstSMul 𝕜₁ M₁
inst✝¹ : IsTopologicalAddGroup M₂
inst✝ : ContinuousSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
U : Set M₂
hU : U ∈ 𝓝 0
⊢ U ∈ map (⇑f) (𝓝 0)
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
U : Set M₂
hU : U ∈ 𝓝 0
V : Set M₂
hV : V ∈ 𝓝 0
hVU : ∀ v ∈ V, ∀ w ∈ V, v + w ∈ U
SV : Set M₁ × Set M₂ := (closedBall 0 1, -V)
v : M₁ →SL[σ₁₂] M₂
hv : v ∈ {f | IsCompactOperator ⇑f}
huv : v - u ∈ {f | ∀ x ∈ SV.1, f x ∈ SV.2}
T : Set M₂
hT : T.Finite
hTv✝ : closure[inst✝⁴.toTopologicalSpace] (⇑↑v '' closedBall 0 1) ⊆ ⋃ y ∈ T, y +ᵥ V
hTv : closedBall 0 1 ⊆ ⋃ i ∈ T, ⇑v ⁻¹' (i +ᵥ V)
x : M₁
⊢ x ∈ closedBall 0 1 → x ∈ ⋃ i ∈ T, ⇑u ⁻¹' (i +ᵥ U)

Tactic:
hx

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁸ : SeminormedAddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : NormedSpace 𝕜₁ M₁
inst✝⁵ : Module 𝕜₂ M₂
inst✝⁴ : UniformSpace M₂
inst✝³ : IsUniformAddGroup M₂
inst✝² : ContinuousConstSMul 𝕜₂ M₂
inst✝¹ : T2Space M₂
inst✝ : CompleteSpace M₂
u : M₁ →SL[σ₁₂] M₂
hu : ∀ U ∈ 𝓝 0, ∃ y ∈ {f | IsCompactOperator ⇑f}, y - u ∈ U
U : Set M₂
hU : U ∈ 𝓝 0
V : Set M₂
hV : V ∈ 𝓝 0
hVU : ∀ v ∈ V, ∀ w ∈ V, v + w ∈ U
SV : Set M₁ × Set M₂ := (closedBall 0 1, -V)
v : M₁ →SL[σ₁₂] M₂
hv : v ∈ {f | IsCompactOperator ⇑f}
huv : v - u ∈ {f | ∀ x ∈ SV.1, f x ∈ SV.2}
T : Set M₂
hT : T.Finite
hTv✝ : closure[inst✝⁴.toTopologicalSpace] (⇑↑v '' closedBall 0 1) ⊆ ⋃ y ∈ T, y +ᵥ V
hTv : closedBall 0 1 ⊆ ⋃ i ∈ T, ⇑v ⁻¹' (i +ᵥ V)
x : M₁
hx : x ∈ closedBall 0 1
⊢ x ∈ ⋃ i ∈ T, ⇑u ⁻¹' (i +ᵥ U)
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hx₀ : x₀ ≠ 0
hextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀' : 0 < ‖x₀‖
hx₀'' : x₀ ∈ sphere 0 ‖x₀‖
x : E
⊢ x ∈ sphere 0 ‖x₀‖ → x ∈ {x | (fun x ↦ T.rayleighQuotient x ≤ T.rayleighQuotient x₀) x}

Tactic:
hx

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hx₀ : x₀ ≠ 0
hextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀' : 0 < ‖x₀‖
hx₀'' : x₀ ∈ sphere 0 ‖x₀‖
x : E
hx : x ∈ sphere 0 ‖x₀‖
⊢ x ∈ {x | (fun x ↦ T.rayleighQuotient x ≤ T.rayleighQuotient x₀) x}
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hx₀ : x₀ ≠ 0
hextr : IsMinOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀' : 0 < ‖x₀‖
hx₀'' : x₀ ∈ sphere 0 ‖x₀‖
x : E
⊢ x ∈ sphere 0 ‖x₀‖ → x ∈ {x | (fun x ↦ T.rayleighQuotient x₀ ≤ T.rayleighQuotient x) x}

Tactic:
hx

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hx₀ : x₀ ≠ 0
hextr : IsMinOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀' : 0 < ‖x₀‖
hx₀'' : x₀ ∈ sphere 0 ‖x₀‖
x : E
hx : x ∈ sphere 0 ‖x₀‖
⊢ x ∈ {x | (fun x ↦ T.rayleighQuotient x₀ ≤ T.rayleighQuotient x) x}
Module: Mathlib.Data.Fin.Tuple.Sort

Goal state before:
pp: case H
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
⊢ Function.Injective fun i ↦ (f i, i)

Tactic:
intro _ _
    -- Porting note: proof was `simp`
    rw [Prod.ext_iff]
    simp

Goal state after:
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
⊢ Multiset.map (RCLike.ofReal ∘ hA.eigenvalues) Finset.univ.val =
    Multiset.map (RCLike.ofReal ∘ hA.eigenvalues₀) Finset.univ.val

Tactic:
simp only [← Multiset.map_map, eigenvalues, ← Function.comp_apply (f := hA.eigenvalues₀)]

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
⊢ Multiset.map (fun x ↦ ↑x)
      (Multiset.map hA.eigenvalues₀ (Multiset.map (⇑(Fintype.equivOfCardEq ⋯).symm) Finset.univ.val)) =
    Multiset.map RCLike.ofReal (Multiset.map hA.eigenvalues₀ Finset.univ.val)
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : hA.eigenvalues = 0
⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) 0 = 0

Tactic:
map_zero

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
h : hA.eigenvalues = 0
⊢ 0 = 0
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_2
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.IsHermitian
h_ne : hA.eigenvalues = 0
this : A = ((Unitary.conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) 0
⊢ A = 0

Tactic:
map_zero

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_2
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : A.IsHermitian
h_ne : hA.eigenvalues = 0
this : A = 0
⊢ A = 0
Module: Mathlib.Analysis.Matrix.HermitianFunctionalCalculus

Goal state before:
pp: case a
n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
a : Matrix n n 𝕜
ha : a.IsHermitian
f : C(↑(spectrum ℝ a), ℝ)
x : ℝ
hx : x ∈ spectrum ℝ a
⊢ ∃ y, ↑(f ⟨ha.eigenvalues y, ⋯⟩) = (algebraMap ℝ 𝕜) (f ⟨x, hx⟩)

Tactic:
obtain ⟨i, rfl⟩ := ha.spectrum_real_eq_range_eigenvalues ▸ hx

Goal state after:
case a
n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
a : Matrix n n 𝕜
ha : a.IsHermitian
f : C(↑(spectrum ℝ a), ℝ)
i : n
hx : ha.eigenvalues i ∈ spectrum ℝ a
⊢ ∃ y, ↑(f ⟨ha.eigenvalues y, ⋯⟩) = (algebraMap ℝ 𝕜) (f ⟨ha.eigenvalues i, hx⟩)
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
f : 𝓓^{n}_{K}(E, F)
i : ℕ
⊢ EqOn (iteratedFDeriv ℝ i ⇑f) 0 (↑K)ᶜ

Tactic:
intro x (hx : x ∉ K)

Goal state after:
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
f : 𝓓^{n}_{K}(E, F)
i : ℕ
x : E
hx : x ∉ K
⊢ iteratedFDeriv ℝ i (⇑f) x = 0 x
Module: Mathlib.Analysis.Matrix.HermitianFunctionalCalculus

Goal state before:
pp: n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
f : ℝ → ℝ
⊢ cfc f A = hA.cfc f

Tactic:
have hA' : IsSelfAdjoint A := hA
  have := cfcHom_eq_of_continuous_of_map_id hA' hA.cfcAux hA.isClosedEmbedding_cfcAux.continuous
    hA.cfcAux_id
  rw [cfc_apply f A hA' (by rw [continuousOn_iff_continuous_domRestrict]; fun_prop), this]
  simp only [cfcAux_apply, ContinuousMap.coe_mk, Function.comp_def, Set.domRestrict_apply,
    IsHermitian.cfc]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hT : T.IsSymmetric
⊢ T.charpoly.Splits

Tactic:
by
  refine Polynomial.splits_iff_card_roots.mpr ?_
  simp [hT.roots_charpoly_eq_eigenvalues rfl, LinearMap.charpoly_natDegree]

Goal state after:
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : 0 ≤ A
x : ℝ
⊢ x ∈ spectrum ℝ A → 0 ≤ x

Tactic:
hx

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
A : Matrix n n 𝕜
hA : 0 ≤ A
x : ℝ
hx : x ∈ spectrum ℝ A
⊢ 0 ≤ x
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.PosSemidef
⊢ A.PosDef ↔ A.det ≠ 0

Tactic:
simp [hA.posDef_iff_isUnit, isUnit_iff_isUnit_det]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.GramMatrix

Goal state before:
pp: E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype n
v : n → E
x y : n → 𝕜
⊢ ∑ i, ∑ j, (starRingEnd 𝕜) (x i) * y j * ⟪v i, v j⟫_𝕜 = ⟪∑ i, x i • v i, ∑ i, y i • v i⟫_𝕜

Tactic:
· simp_rw [sum_inner, inner_sum, inner_smul_left, inner_smul_right, mul_assoc]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : SecondCountableTopologyEither α E
inst✝ : IsFiniteMeasure μ
f : α →ᵇ E
x : α
⊢ ↑(ContinuousMap.toAEEqFun μ f.toContinuousMap) x = f.toContinuousMap x →
    ‖↑↑⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩ x‖₊ ≤ ‖f‖₊

Tactic:
hx

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : SecondCountableTopologyEither α E
inst✝ : IsFiniteMeasure μ
f : α →ᵇ E
x : α
hx : ↑(ContinuousMap.toAEEqFun μ f.toContinuousMap) x = f.toContinuousMap x
⊢ ‖↑↑⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩ x‖₊ ≤ ‖f‖₊
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
hi : ↑i ≤ n
x : E
⊢ x ∈ K → ‖iteratedFDeriv ℝ i (⇑((postcompLM T') f)) x‖ ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f

Tactic:
hx

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace ℝ E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : SMulCommClass ℝ 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : SMulCommClass ℝ 𝕜 F'
n : ℕ∞
K : Compacts E
inst✝ : LinearMap.CompatibleSMul F F' ℝ 𝕜
i : ℕ
T : F →L[𝕜] F'
f : 𝓓^{n}_{K}(E, F)
T' : F →L[ℝ] F' := ContinuousLinearMap.restrictScalars ℝ T
hi : ↑i ≤ n
x : E
hx : x ∈ K
⊢ ‖iteratedFDeriv ℝ i (⇑((postcompLM T') f)) x‖ ≤ ‖T'‖ * (ContDiffMapSupportedIn.seminorm ℝ E F n K i) f
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
hi : ↑i ≤ k
x : E
⊢ x ∈ K → ‖iteratedFDeriv ℝ i (⇑((fderivLM 𝕜 n k) f)) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f

Tactic:
hx

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
n k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hk : k + 1 ≤ n
hi : ↑i ≤ k
x : E
hx : x ∈ K
⊢ ‖iteratedFDeriv ℝ i (⇑((fderivLM 𝕜 n k) f)) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K (i + 1)) f
Module: Mathlib.Geometry.Manifold.SmoothApprox

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
ε : E → ℝ
n : ℕ∞
f_cont :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
ε_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ε
ε_pos : ∀ (x : E), 0 < ε x
g : E → F
g_contDiff : ContDiff ℝ (↑n) g
g_approx : ∀ (x : E), dist (g x) (f x) < ε x
g_supp : support g ⊆ support f
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ (∀ (x : E), dist (g x) (f x) < ε x) ∧ support g ⊆ support f

Tactic:
exact ⟨g, g_contDiff, g_approx, g_supp⟩

Goal state after:
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : IntegrableOn φ (↑K) μ
f : 𝓓^{n}_{K}(E, F₁)
x : E
⊢ x ∉ ↑K → (B (f x)) (φ x) = 0

Tactic:
hx

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
n : ℕ∞
K : Compacts E
m : MeasurableSpace E
inst✝¹⁰ : OpensMeasurableSpace E
F₁ : Type u_5
F₂ : Type u_6
F₃ : Type u_7
inst✝⁹ : NormedAddCommGroup F₁
inst✝⁸ : NormedSpace 𝕜 F₁
inst✝⁷ : NormedSpace ℝ F₁
inst✝⁶ : NormedAddCommGroup F₂
inst✝⁵ : NormedSpace 𝕜 F₂
inst✝⁴ : NormedAddCommGroup F₃
inst✝³ : NormedSpace 𝕜 F₃
inst✝² : SMulCommClass ℝ 𝕜 F₁
inst✝¹ : NormedSpace ℝ F₃
inst✝ : SMulCommClass ℝ 𝕜 F₃
B : F₁ →L[𝕜] F₂ →L[𝕜] F₃
μ : Measure E
φ : E → F₂
hφ : IntegrableOn φ (↑K) μ
f : 𝓓^{n}_{K}(E, F₁)
x : E
hx : x ∉ ↑K
⊢ (B (f x)) (φ x) = 0
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: case pos
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
p : ℝ≥0∞
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
hp : p ≠ ∞
hf : f =ᵐ[μ] 0
⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) p μ ≤ ENNReal.ofReal ε

Tactic:
use 0
    simpa [HasCompactSupport.zero, eLpNorm_congr_ae hf] using! contDiff_const

Goal state after: