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

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
r : ℝ
hr : 0 < r
a : ℝ
⊢ a ∈ T.rayleighQuotient '' sphere 0 r → a ∈ T.rayleighQuotient '' {0}ᶜ

Tactic:
rintro ⟨x, hx, hxT⟩
    exact ⟨x, ne_zero_of_mem_sphere hr.ne' ⟨x, hx⟩, hxT⟩

Goal state after:
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
T : E →L[𝕜] E
inst✝ : CompleteSpace E
hT : IsSelfAdjoint T
a✝ : Nontrivial E
⊢ (algebraMap ℝ 𝕜) ‖T‖ ∈ spectrum 𝕜 T ∨ (algebraMap ℝ 𝕜) (-‖T‖) ∈ spectrum 𝕜 T

Tactic:
[

Goal state after:
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✝ : CompleteSpace E
hT : IsSelfAdjoint T
a✝ : Nontrivial E
⊢ (algebraMap ℝ 𝕜) ‖T‖ ∈ spectrum 𝕜 T ∨ (algebraMap ℝ 𝕜) (-‖T‖) ∈ spectrum 𝕜 T
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
⊢ x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l)

Tactic:
this

Goal state after:
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
⊢ x ^ k * f ≤ ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h₁
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : x ≤ 1
⊢ x ^ k ≤ ((1 + x) / 2) ^ (-↑l)

Tactic:
apply (pow_le_one₀ hx h'x).trans
      apply Real.one_le_rpow_of_pos_of_le_one_of_nonpos
      · positivity
      · linarith
      · simp

Goal state after:
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
hx : x ∈ sphere 0 ‖x₀‖
this : ‖x‖ = ‖x₀‖
⊢ T.reApplyInnerSelf x / ‖x‖ ^ 2 ≤ T.reApplyInnerSelf x₀ / ‖x₀‖ ^ 2

Tactic:
this

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‖
this : ‖x‖ = ‖x₀‖
⊢ T.reApplyInnerSelf x / ‖x₀‖ ^ 2 ≤ T.reApplyInnerSelf x₀ / ‖x₀‖ ^ 2
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
hx : x ∈ sphere 0 ‖x₀‖
this : ‖x‖ = ‖x₀‖
⊢ T.reApplyInnerSelf x₀ / ‖x₀‖ ^ 2 ≤ T.reApplyInnerSelf x / ‖x‖ ^ 2

Tactic:
this

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‖
this : ‖x‖ = ‖x₀‖
⊢ T.reApplyInnerSelf x₀ / ‖x₀‖ ^ 2 ≤ T.reApplyInnerSelf x / ‖x₀‖ ^ 2
Module: Mathlib.LinearAlgebra.Eigenspace.Charpoly

Goal state before:
pp: K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
f : End K V
h : (charpoly f).Splits
b : Basis (Free.ChooseBasisIndex K V) K V := Free.chooseBasis K V
⊢ (trace K V) f = (charpoly f).roots.sum

Tactic:
rw [trace_eq_matrix_trace K (Module.Free.chooseBasis K V),
    Matrix.trace_eq_sum_roots_charpoly_of_splits (by simpa using h), charpoly_toMatrix]

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_3
A : Matrix n n 𝕜
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype n
inst✝² : DecidableEq n
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : OrthonormalBasis n 𝕜 E
this : FiniteDimensional 𝕜 E
⊢ ((toLin b.toBasis b.toBasis) A).IsSymmetric ↔ A.IsHermitian

Tactic:
[

Goal state after:
𝕜 : Type u_1
n : Type u_3
A : Matrix n n 𝕜
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype n
inst✝² : DecidableEq n
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : OrthonormalBasis n 𝕜 E
this : FiniteDimensional 𝕜 E
⊢ ((toLin b.toBasis b.toBasis) A).IsSymmetric ↔ A.IsHermitian
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
j : n
⊢ ↑hA.eigenvectorUnitary *ᵥ Pi.single j 1 = (hA.eigenvectorBasis j).ofLp

Tactic:
[

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
j : n
⊢ ↑hA.eigenvectorUnitary *ᵥ Pi.single j 1 = (hA.eigenvectorBasis j).ofLp
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.re) A.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn hA.eigenvalues₀

Tactic:
[

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 (⇑RCLike.re) A.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn hA.eigenvalues₀
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
⊢ List.Pairwise (fun a b ↦ decide (a ≥ b) = true) (List.ofFn fun x ↦ hA.eigenvalues₀ x)

Tactic:
[

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
⊢ List.Pairwise (fun a b ↦ decide (a ≥ b) = true) (List.ofFn fun x ↦ hA.eigenvalues₀ x)
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 B : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
hB : B.IsHermitian
h : A.charpoly = B.charpoly
this : hA.eigenvalues₀ = hB.eigenvalues₀
⊢ (fun i ↦ hA.eigenvalues₀ ((Fintype.equivOfCardEq ⋯).symm i)) = fun i ↦
    hB.eigenvalues₀ ((Fintype.equivOfCardEq ⋯).symm i)

Tactic:
this

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A B : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
hB : B.IsHermitian
h : A.charpoly = B.charpoly
this : hA.eigenvalues₀ = hB.eigenvalues₀
⊢ (fun i ↦ hB.eigenvalues₀ ((Fintype.equivOfCardEq ⋯).symm i)) = fun i ↦
    hB.eigenvalues₀ ((Fintype.equivOfCardEq ⋯).symm i)
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A B : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
hB : B.IsHermitian
h : A.charpoly = B.charpoly
⊢ hA.eigenvalues₀ = hB.eigenvalues₀

Tactic:
[

Goal state after:
case mpr
𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A B : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
hB : B.IsHermitian
h : A.charpoly = B.charpoly
⊢ hA.eigenvalues₀ = hB.eigenvalues₀
Module: Mathlib.Analysis.Matrix.PosDef

Goal state before:
pp: n : Type u_2
𝕜 : Type u_3
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.PosSemidef
i : n
x✝ : i ∈ Finset.univ
⊢ 0 ≤ ↑(⋯.eigenvalues i)

Tactic:
simpa using hA.eigenvalues_nonneg i

Goal state after:
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
hA' : IsSelfAdjoint A
this : cfcHom hA' = hA.cfcAux
⊢ (cfcHom hA') { toFun := (spectrum ℝ A).domRestrict f, continuous_toFun := ⋯ } = hA.cfc f

Tactic:
this

Goal state after:
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
hA' : IsSelfAdjoint A
this : cfcHom hA' = hA.cfcAux
⊢ hA.cfcAux { toFun := (spectrum ℝ A).domRestrict f, continuous_toFun := ⋯ } = hA.cfc f
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ HasEigenvector T (↑(hT.unsortedEigenvalues hn i)) ((hT.unsortedEigenvectorBasis hn) i)

Tactic:
[

Goal state after:
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ HasEigenvector T (↑(hT.unsortedEigenvalues hn i)) ((hT.unsortedEigenvectorBasis hn) i)
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ v ∈ eigenspace T μ

Tactic:
[

Goal state after:
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ v ∈ eigenspace T μ
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
v : E
i : Fin n
w : EuclideanSpace 𝕜 (Fin n)
⊢ T ((hT.eigenvectorBasis hn).repr.symm w) =
    (hT.eigenvectorBasis hn).repr.symm (toLp 2 fun i ↦ ↑(hT.eigenvalues hn i) * w.ofLp i)

Tactic:
[

Goal state after:
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
v : E
i : Fin n
w : EuclideanSpace 𝕜 (Fin n)
⊢ T ((hT.eigenvectorBasis hn).repr.symm w) =
    (hT.eigenvectorBasis hn).repr.symm (toLp 2 fun i ↦ ↑(hT.eigenvalues hn i) * w.ofLp i)
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
⊢ ((Multiset.map (⇑RCLike.re) T.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn (hT.eigenvalues hn)

Tactic:
[

Goal state after:
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
⊢ ((Multiset.map (⇑RCLike.re) T.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) = List.ofFn (hT.eigenvalues hn)
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: case convert_4
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
this : n = finrank 𝕜 E
⊢ List.Pairwise (fun a b ↦ decide (a ≥ b) = true) (List.ofFn fun x ↦ hT.eigenvalues ⋯ x)

Tactic:
[

Goal state after:
case convert_4
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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
this : n = finrank 𝕜 E
⊢ List.Pairwise (fun a b ↦ decide (a ≥ b) = true) (List.ofFn fun x ↦ hT.eigenvalues ⋯ x)
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : SMulCommClass ℝ 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : NormedSpace 𝕜 F'
inst✝ : SMulCommClass ℝ 𝕜 F'
n n₁ n₂ k : ℕ∞
K K₁ K₂ : Compacts E
i : ℕ
f g : 𝓓^{n}_{K}(E, F)
hi : k + ↑i ≤ n
⊢ ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ =
    ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ + ContDiffMapSupportedIn.of_support_subset ⋯ ⋯

Tactic:
have hi' : (i : ℕ∞ω) ≤ n := mod_cast (le_of_add_le_right hi)
      ext
      simp [iteratedFDeriv_add (f.contDiff.of_le hi') (g.contDiff.of_le hi'), FunLike.coe_add]

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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
⊢ (∀ (μ : 𝕜), HasEigenvalue (↑T) μ → μ = 0) ↔ ↑‖T‖₊ = 0

Tactic:
← T.spectralRadius_eq_nnnorm hT'.isSelfAdjoint,

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 : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
⊢ (∀ (μ : 𝕜), HasEigenvalue (↑T) μ → μ = 0) ↔ spectralRadius 𝕜 T = 0
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 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 : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ ⇑((structureMapLM 𝕜 n i) f) = if ↑i ≤ n then iteratedFDeriv ℝ i ⇑f else 0

Tactic:
simp [structureMapLM]

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

Goal state before:
pp: 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 : ℕ∞
K : Compacts E
⊢ IsUniformInducing ⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))

Tactic:
[

Goal state after:
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 : ℕ∞
K : Compacts E
⊢ IsUniformInducing ⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))
Module: Mathlib.Analysis.Distribution.ContDiffMapSupportedIn

Goal state before:
pp: 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 : ℕ∞
K : Compacts E
⊢ comap (⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))) (⨅ i, comap (eval i) PseudoMetricSpace.toUniformSpace) =
    ⨅ i, comap (⇑(structureMapLM ℝ n i)) inferInstance

Tactic:
comap_iInf,

Goal state after:
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 : ℕ∞
K : Compacts E
⊢ ⨅ i, comap (⇑(ContinuousLinearMap.pi (structureMapCLM 𝕜 n))) (comap (eval i) PseudoMetricSpace.toUniformSpace) =
    ⨅ i, comap (⇑(structureMapLM ℝ n i)) inferInstance
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x : n → 𝕜
B : Matrix n n 𝕜
hA : (star B * B).PosSemidef
h : star x ⬝ᵥ (star B * B) *ᵥ x = 0
⊢ (star B * B) *ᵥ x = 0

Tactic:
[

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x : n → 𝕜
B : Matrix n n 𝕜
hA : (star B * B).PosSemidef
h : star x ⬝ᵥ (star B * B) *ᵥ x = 0
⊢ (star B * B) *ᵥ x = 0
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
v : n → 𝕜
hv : v ≠ 0
y : Matrix n n 𝕜
hx : (star y * y).PosSemidef
h : IsUnit (star y * y)
⊢ 0 < star v ⬝ᵥ (star y * y) *ᵥ v

Tactic:
[

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → 𝕜
hv : v ≠ 0
y : Matrix n n 𝕜
hx : (star y * y).PosSemidef
h : IsUnit (star y * y)
⊢ 0 < star v ⬝ᵥ (star y * y) *ᵥ v
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
⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ∑ i, ∑ j, (starRingEnd 𝕜) (x i) * y j * ⟪v i, v j⟫_𝕜

Tactic:
[

Goal state after:
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
⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ∑ i, ∑ j, (starRingEnd 𝕜) (x i) * y j * ⟪v i, v j⟫_𝕜
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:
[

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

Goal state before:
pp: 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
K : Compacts E
C : ℝ
hC : 0 ≤ C
i : ℕ
f : 𝓓_{K}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F ⊤ K i) f ≤ C ↔ ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑f) x‖ ≤ C

Tactic:
[

Goal state after:
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
K : Compacts E
C : ℝ
hC : 0 ≤ C
i : ℕ
f : 𝓓_{K}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F ⊤ K i) f ≤ C ↔ ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑f) x‖ ≤ C
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
ι : Type u_3
A B : Matrix ι ι 𝕜
hA : A.PosDef
hB : B.PosDef
x : ι →₀ 𝕜
hx : x ≠ 0
hAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef
⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A ⊙ B) i j * xj

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
ι : Type u_3
A B : Matrix ι ι 𝕜
hA : A.PosDef
hB : B.PosDef
x : ι →₀ 𝕜
hx : x ≠ 0
hAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef
⊢ 0 < x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A ⊙ B) i j * xj
Module: Mathlib.Analysis.InnerProductSpace.GramMatrix

Goal state before:
pp: E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
v : n → E
f : E →L[𝕜] F
c : n →₀ 𝕜
⊢ 0 ≤ c.sum fun i xi ↦ c.sum fun j xj ↦ star xi * (‖f‖ ^ 2 • gram 𝕜 v - gram 𝕜 (⇑f ∘ v)) i j * xj

Tactic:
[

Goal state after:
E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
F : Type u_5
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
v : n → E
f : E →L[𝕜] F
c : n →₀ 𝕜
⊢ 0 ≤ c.sum fun i xi ↦ c.sum fun j xj ↦ star xi * (‖f‖ ^ 2 • gram 𝕜 v - gram 𝕜 (⇑f ∘ v)) i j * xj
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✝⁸ : TopologicalSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
inst✝² : Fact (1 ≤ p)
𝕜 : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
⊢ ‖toLp p μ 𝕜‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹

Tactic:
rw [toLp_norm_eq_toLp_norm_coe]

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁸ : TopologicalSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
inst✝² : Fact (1 ≤ p)
𝕜 : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
⊢ ‖BoundedContinuousFunction.toLp p μ 𝕜‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹
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✝⁸ : TopologicalSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SecondCountableTopologyEither α E
inst✝⁴ : CompactSpace α
inst✝³ : IsFiniteMeasure μ
inst✝² : Fact (1 ≤ p)
𝕜 : Type u_4
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
⊢ ‖toLp p μ 𝕜‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹

Tactic:
rw [toLp_norm_eq_toLp_norm_coe]
  exact BoundedContinuousFunction.toLp_norm_le μ

Goal state after:
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫ ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal

Tactic:
[

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫ ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
⊢ AEEqFun.mk (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) ⋯ ∈ Lp 𝕜 1 μ

Tactic:
[

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f g : ↥(Lp E 2 μ)
⊢ AEEqFun.mk (fun x ↦ ⟪↑↑f x, ↑↑g x⟫) ⋯ ∈ Lp 𝕜 1 μ
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f f' g : ↥(Lp E 2 μ)
⊢ ⟪f + f', g⟫ = ⟪f, g⟫ + ⟪f', g⟫

Tactic:
[

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝² : RCLike 𝕜
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f f' g : ↥(Lp E 2 μ)
⊢ ⟪f + f', g⟫ = ⟪f, g⟫ + ⟪f', g⟫