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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
μ : 𝕜
v : E
T : E →ₗ[𝕜] E
h : T v = μ • v
⊢ ⟪v, T v⟫ = μ * ↑‖v‖ ^ 2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
μ : ℝ
T : E →ₗ[𝕜] E
hμ : HasEigenvalue T ↑μ
hnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫
v : E
hv₁ : v ∈ (genEigenspace T ↑μ) 1
hv₂ : v ≠ 0
⊢ 0 < ‖v‖ ^ 2

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
μ : ℝ
T : E →ₗ[𝕜] E
hμ : HasEigenvalue T ↑μ
hnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫
⊢ 0 ≤ μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
μ : ℝ
T : E →ₗ[𝕜] E
hμ : HasEigenvalue T ↑μ
hnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫
v : E
hv₁ : v ∈ (genEigenspace T ↑μ) 1
hv₂ : v ≠ 0
⊢ 0 < ‖v‖ ^ 2

Tactic:
by

Goal state after:
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✝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)
⊢ (if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0) =
    (if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0) +
      if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
μ : ℝ
T : E →ₗ[𝕜] E
hμ : HasEigenvalue T ↑μ
hnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫
⊢ 0 < μ

Tactic:
by

Goal state after:
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✝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
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
⊢ (if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0) =
    (RingHom.id 𝕜) c • if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0

Tactic:
by

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:
by

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

Tactic:
by

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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hin : k + ↑i ≤ n
⊢ ⇑((iteratedFDerivLM 𝕜 n k i) f) = iteratedFDeriv ℝ i ⇑f

Tactic:
by

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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
hin : n < k + ↑i
⊢ (iteratedFDerivLM 𝕜 n k i) f = 0

Tactic:
by

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

Tactic:
by

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
K : Compacts E
i : ℕ
f : 𝓓_{K}(E, F)
⊢ ⇑((structureMapLM 𝕜 ⊤ i) f) = iteratedFDeriv ℝ i ⇑f

Tactic:
by

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
f : 𝓓^{n}_{K}(E, F)
x : E
⊢ ((structureMapLM 𝕜 n 0) f) x = ContinuousMultilinearMap.uncurry0 ℝ E (f x)

Tactic:
by

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
S : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯
hS_compact : IsCompactOperator ⇑S
hS_symm : (↑S).IsSymmetric
hS : ∀ (μ : 𝕜), eigenspace (↑S) μ = ⊥
μ : 𝕜
⊢ HasEigenvalue (↑S) μ → μ = 0

Tactic:
by

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
⊢ Injective ⇑(structureMapLM 𝕜 n 0)

Tactic:
by

Goal state after:
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✝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
⊢ IsUniformAddGroup 𝓓^{n}_{K}(E, F)

Tactic:
by

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, eigenspace (↑T) μ)ᗮ = ⊥

Tactic:
by

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
μ : 𝕜
hμ : μ ≠ 0
⊢ FiniteDimensional 𝕜 ↥(eigenspace (↑T) μ)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
⊢ 0 ≤ A ↔ A.PosSemidef

Tactic:
by

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
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ ⇑((structureMapCLM 𝕜 n i) f) = if ↑i ≤ n then iteratedFDeriv ℝ i ⇑f else 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
h₁ : A.PosSemidef
h₂ : (-A).PosSemidef
i✝ j i : n
⊢ A i i ≤ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
h₁ : A.PosSemidef
h₂ : (-A).PosSemidef
i✝ j i : n
⊢ 0 ≤ A i i

Tactic:
by

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
K : Compacts E
i : ℕ
f : 𝓓_{K}(E, F)
⊢ ⇑((structureMapCLM 𝕜 ⊤ i) f) = iteratedFDeriv ℝ i ⇑f

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
h₁ : A.PosSemidef
h₂ : (-A).PosSemidef
⊢ A = 0

Tactic:
by

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:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A B : Matrix n n 𝕜
h₁ : A ≤ B
h₂ : B ≤ A
⊢ (-(B - A)).PosSemidef

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A B : Matrix n n 𝕜
h₁ : A ≤ B
h₂ : B ≤ A
⊢ A = B

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
A B : Matrix n n 𝕜
h₁ : A ≤ B
h₂ : B ≤ A
⊢ A = B

Tactic:
by
    simpa [sub_eq_zero, eq_comm] using le_antisymm_aux h₁
     (by simpa only [← neg_sub B, le_iff] using h₂)

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

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
X : Type u_5
inst✝ : TopologicalSpace X
φ : X → 𝓓^{n}_{K}(E, F)
⊢ Continuous[inst✝, _] φ ↔ ∀ (i : ℕ), Continuous[inst✝, _] (⇑(structureMapCLM ℝ n i) ∘ φ)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝ : RCLike 𝕜
x✝³ x✝² : Matrix n n 𝕜
x✝¹ : x✝³ ≤ x✝²
x✝ : Matrix n n 𝕜
⊢ x✝³ + x✝ ≤ x✝² + x✝

Tactic:
by

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 ∈ quasispectrum ℝ A, 0 ≤ x

Tactic:
by

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
⊢ ∃ s, A = star s * s

Tactic:
by

Goal state after:
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
⊢ (CFC.sqrt A)⁻¹ = CFC.sqrt A⁻¹

Tactic:
by

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 : A.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ A *ᵥ x = 0 ↔ A *ᵥ x = 0

Tactic:
by

Goal state after:
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
x : n → 𝕜
⊢ (((toLinearMap₂' 𝕜) A) (star x)) x = 0 ↔ A *ᵥ x = 0

Tactic:
by

Goal state after:
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
⊢ (CFC.sqrt A).det = RCLike.sqrt A.det

Tactic:
by

Goal state after:
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.IsHermitian
⊢ (CFC.abs A).det = ↑‖A.det‖

Tactic:
by

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

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
X : Type u_5
inst✝ : TopologicalSpace X
φ : X → 𝓓^{n}_{K}(E, F)
⊢ Continuous[inst✝, _] φ ↔ ∀ (i : ℕ), ↑i ≤ n → Continuous[inst✝, _] (⇑(structureMapCLM ℝ n i) ∘ φ)

Tactic:
by

Goal state after:
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
⊢ A.PosSemidef ↔ A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}

Tactic:
by

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
⊢ WithSeminorms (ContDiffMapSupportedIn.seminorm 𝕜 E F n K)

Tactic:
by

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
i : ℕ
hin : n < ↑i
⊢ ¬↑i ≤ n

Tactic:
by

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
i : ℕ
hin : n < ↑i
⊢ ContDiffMapSupportedIn.seminorm 𝕜 E F n K i = ⊥

Tactic:
by

Goal state after:
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
x : Matrix n n 𝕜
hx : x.PosSemidef
⊢ x.PosDef ↔ IsUnit x

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Finite n
m : Type u_3
inst✝ : Finite m
x : Matrix n n 𝕜
y : Matrix m m 𝕜
hx : x.PosSemidef
hy : y.PosSemidef
⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) x y).PosSemidef

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Finite n
m : Type u_3
inst✝ : Finite m
x : Matrix n n 𝕜
y : Matrix m m 𝕜
hx : x.PosDef
hy : y.PosDef
⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) x y).PosDef

Tactic:
by

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
C : ℝ
hC : 0 ≤ C
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ (∀ (x : E), ‖iteratedFDeriv ℝ i (⇑f) x‖ ≤ C) ↔ ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑f) x‖ ≤ C

Tactic:
by

Goal state after:
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.PosSemidef
hB : B.PosSemidef
x : ι →₀ 𝕜
⊢ ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef

Tactic:
by

Goal state after:
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.PosSemidef
hB : B.PosSemidef
⊢ (A ⊙ B).PosSemidef

Tactic:
by

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✝ : DecidableEq n
i : n
x : E
⊢ gram 𝕜 (Pi.single i x) = single i i ⟪x, x⟫_𝕜

Tactic:
by

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
C : ℝ
hC : 0 ≤ C
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K i) f ≤ C ↔ ↑i ≤ n → ∀ x ∈ K, ‖iteratedFDeriv ℝ i (⇑f) x‖ ≤ C

Tactic:
by

Goal state after:
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
hAs : (A.submatrix Subtype.val Subtype.val).PosDef
hBs : (B.submatrix Subtype.val Subtype.val).PosDef
y : ↥x.support → 𝕜
hy : y ≠ 0
⊢ (diagonal y).vec ≠ 0

Tactic:
by

Goal state after:
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
⊢ ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef

Tactic:
by

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
⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ⟪∑ i, x i • v i, ∑ i, y i • v i⟫_𝕜

Tactic:
by

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
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:
by

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
i : ℕ
hin : ↑i ≤ n
f : 𝓓^{n}_{K}(E, F)
x : E
⊢ ‖iteratedFDeriv ℝ i (⇑f) x‖ = ‖((structureMapLM ℝ n i) f) x‖

Tactic:
by

Goal state after:
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
⊢ (A ⊙ B).PosDef

Tactic:
by

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
M : Matrix n n 𝕜
hM : M.PosSemidef
x✝¹ x✝ : Matrix n n 𝕜
⊢ (starRingEnd 𝕜) (x✝¹ * M * x✝ᴴ).trace = (x✝ * M * x✝¹ᴴ).trace

Tactic:
by

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
M : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x y z : Matrix n n 𝕜), (z * M * (x + y)ᴴ).trace = (z * M * xᴴ).trace + (z * M * yᴴ).trace

Tactic:
by

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✝ : Finite n
v : n → E
⊢ (gram 𝕜 v).PosSemidef

Tactic:
by

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
M : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x y : Matrix n n 𝕜) (r : 𝕜), (y * M * (r • x)ᴴ).trace = (starRingEnd 𝕜) r * (y * M * xᴴ).trace

Tactic:
by

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
f : 𝓓^{n}_{K}(E, F)
x : E
⊢ ‖f x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K 0) f

Tactic:
by

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
M : Matrix n n 𝕜
hM : M.PosDef
x : Matrix n n 𝕜
hx : inner 𝕜 x x = 0
⊢ x = 0

Tactic:
by

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
f : 𝓓^{n}_{K}(E, F)
⊢ ‖{ toFun := ⇑f, continuous_toFun := ⋯, map_bounded' := ⋯ }‖ = (ContDiffMapSupportedIn.seminorm 𝕜 E F n K 0) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs

Goal state before:
pp: A : Type u_2
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
⊢ abs (-a) = abs a

Tactic:
by

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✝ : Finite n
v : n → E
h_gram : (gram 𝕜 v).PosDef
⊢ LinearIndependent 𝕜 v

Tactic:
by

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
f : 𝓓^{n}_{K}(E, F)
⊢ ‖{ toFun := ⇑f, continuous_toFun := ⋯, map_bounded' := ⋯ }‖ = (ContDiffMapSupportedIn.seminorm 𝕜 E F n K 0) f

Tactic:
by
  simp [BoundedContinuousFunction.norm_eq_iSup_norm,
    ContDiffMapSupportedIn.seminorm_apply, structureMapCLM_apply]

Goal state after: