mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ Antitone ((hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn))) ∘ ⇑Fin.revPerm)

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
⊢ Antitone ((hT.unsortedEigenvalues hn ∘ ⇑(Tuple.sort (hT.unsortedEigenvalues hn))) ∘ ⇑Fin.revPerm)
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)
a : Fin n
⊢ (↑(hT.eigenvalues hn a) * w.ofLp a) • (hT.eigenvectorBasis hn) a =
    (↑(hT.eigenvalues hn a) * w.ofLp a) • (hT.eigenvectorBasis hn) a

Tactic:
]

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
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
⊢ (Finset.univ.val.bind fun i ↦ (Polynomial.X - Polynomial.C ((RCLike.ofReal ∘ hT.eigenvalues hn) i)).roots) =
    Multiset.map (RCLike.ofReal ∘ hT.eigenvalues hn) Finset.univ.val

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
⊢ (Finset.univ.val.bind fun i ↦ (Polynomial.X - Polynomial.C ((RCLike.ofReal ∘ hT.eigenvalues hn) i)).roots) =
    Multiset.map (RCLike.ofReal ∘ hT.eigenvalues hn) Finset.univ.val
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
c : 𝕜
f : 𝓓^{n}_{K}(E, F)
⊢ (if hk : k + 1 ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0) =
    (RingHom.id 𝕜) c • if hk : k + 1 ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0

Tactic:
by
    split_ifs with hk
    · have hk' : 0 < (n : ℕ∞ω) := mod_cast (add_pos_of_right zero_lt_one k).trans_le hk
      ext
      simp [fderiv_const_smul (f.contDiff.differentiable hk'.ne').differentiableAt,
        FunLike.coe_smul]
    · simp

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
f : 𝓓^{n}_{K}(E, F)
⊢ ⇑({ toFun := fun f ↦ if hk : k + 1 ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0, map_add' := ⋯,
          map_smul' := ⋯ }
        f) =
    if k + 1 ≤ n then fderiv ℝ ⇑f else 0

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 k : ℕ∞
K : Compacts E
f : 𝓓^{n}_{K}(E, F)
⊢ ⇑({ toFun := fun f ↦ if hk : k + 1 ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0, map_add' := ⋯,
          map_smul' := ⋯ }
        f) =
    if k + 1 ≤ n then fderiv ℝ ⇑f else 0
Module: Mathlib.Analysis.InnerProductSpace.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝³ : FiniteDimensional 𝕜 E
n : ℕ
E' : Type u_3
inst✝² : NormedAddCommGroup E'
inst✝¹ : InnerProductSpace 𝕜 E'
inst✝ : FiniteDimensional 𝕜 E'
T' : E' →ₗ[𝕜] E'
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
hT' : T'.IsSymmetric
hn' : finrank 𝕜 E' = n
h : hT.eigenvalues hn = hT'.eigenvalues hn'
⊢ ∏ i, (Polynomial.X - Polynomial.C ↑(hT'.eigenvalues hn' i)) =
    ∏ i, (Polynomial.X - Polynomial.C ↑(hT'.eigenvalues hn' i))

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝³ : FiniteDimensional 𝕜 E
n : ℕ
E' : Type u_3
inst✝² : NormedAddCommGroup E'
inst✝¹ : InnerProductSpace 𝕜 E'
inst✝ : FiniteDimensional 𝕜 E'
T' : E' →ₗ[𝕜] E'
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
hT' : T'.IsSymmetric
hn' : finrank 𝕜 E' = n
h : T.charpoly = T'.charpoly
⊢ ((Multiset.map (⇑RCLike.re) T'.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2) =
    (Multiset.map (⇑RCLike.re) T'.charpoly.roots).sort fun x1 x2 ↦ x1 ≥ x2

Tactic:
]

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) ↔ ¬∀ (i : 𝕜), ⨆ (_ : i ∈ spectrum 𝕜 T), ↑‖i‖₊ = 0

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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
⊢ (¬∀ (μ : 𝕜), HasEigenvalue (↑T) μ → μ = 0) ↔ ¬∀ (i : 𝕜), ⨆ (_ : i ∈ spectrum 𝕜 T), ↑‖i‖₊ = 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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ ⇑({ toFun := fun f ↦ if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0, map_add' := ⋯,
          map_smul' := ⋯ }
        f) =
    if k + ↑i ≤ n then iteratedFDeriv ℝ i ⇑f else 0

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 k : ℕ∞
K : Compacts E
i : ℕ
f : 𝓓^{n}_{K}(E, F)
⊢ ⇑({ toFun := fun f ↦ if hi : k + ↑i ≤ n then ContDiffMapSupportedIn.of_support_subset ⋯ ⋯ else 0, map_add' := ⋯,
          map_smul' := ⋯ }
        f) =
    if k + ↑i ≤ n then iteratedFDeriv ℝ i ⇑f else 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 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
  simp [hin]

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 = ⊥
h : S = 0
⊢ (⨆ μ, eigenspace (↑T) μ)ᗮ = ⊥

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
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 = ⊥
h : S = 0
⊢ (⨆ μ, eigenspace (↑T) μ)ᗮ = ⊥
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 = ⊥
h : S = 0
⊢ Subsingleton ↥(⨆ μ, eigenspace (↑T) μ)ᗮ

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
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 = ⊥
h : S = 0
⊢ Subsingleton ↥(⨆ μ, eigenspace (↑T) μ)ᗮ
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:
]

Goal state after:
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)
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
μ : 𝕜
hμ : μ ≠ 0
hT : IsCompactOperator ⇑LinearMap.id
⊢ FiniteDimensional 𝕜 ↥(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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
μ : 𝕜
hμ : μ ≠ 0
hT : IsCompactOperator ⇑LinearMap.id
⊢ FiniteDimensional 𝕜 ↥(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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
μ : 𝕜
hμ : μ ≠ 0
hT : IsCompactOperator ⇑LinearMap.id
⊢ IsCompactOperator id

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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
μ : 𝕜
hμ : μ ≠ 0
hT : IsCompactOperator ⇑LinearMap.id
⊢ IsCompactOperator id
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
⊢ A.PosSemidef ↔ A.PosSemidef

Tactic:
]

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✝² - x✝³).PosSemidef

Tactic:
]

Goal state after:
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✝² - x✝³).PosSemidef
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)
⊢ (∀ (i : ℕ), Continuous[inst✝, _] (⇑(structureMapCLM ℝ n i) ∘ φ)) ↔
    ∀ (i : ℕ), ↑i ≤ n → Continuous[inst✝, _] (⇑(structureMapCLM ℝ n i) ∘ φ)

Tactic:
]

Goal state after:
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)
⊢ (∀ (i : ℕ), Continuous[inst✝, _] (⇑(structureMapCLM ℝ n i) ∘ φ)) ↔
    ∀ (i : ℕ), ↑i ≤ n → Continuous[inst✝, _] (⇑(structureMapCLM ℝ n i) ∘ φ)
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⁻¹ = A⁻¹

Tactic:
]

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
x : n → 𝕜
B : Matrix n n 𝕜
hA : (star B * B).PosSemidef
h : B *ᵥ x = 0
⊢ 0 = 0

Tactic:
]

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 fun x ↦ ↑(NNReal.sqrt x.toNNReal)).det = ↑√(RCLike.re A.det)

Tactic:
]

Goal state after:
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 fun x ↦ ↑(NNReal.sqrt x.toNNReal)).det = ↑√(RCLike.re A.det)
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
⊢ (hA.cfc fun x ↦ ‖x‖).det = ↑‖A.det‖

Tactic:
]

Goal state after:
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
⊢ (hA.cfc fun x ↦ ‖x‖).det = ↑‖A.det‖
Module: Mathlib.Analysis.Matrix.Order

Goal state before:
pp: case refine_2
𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
x✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}
h1 : A.IsHermitian
h2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}
⊢ 0 ≤ h1.eigenvalues

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
n : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
x✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}
h1 : A.IsHermitian
h2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}
⊢ 0 ≤ h1.eigenvalues
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
y : Matrix n n 𝕜
hx : (star y * y).PosSemidef
h : IsUnit (star y * y)
hv : y *ᵥ v = 0
⊢ 0 = 0

Tactic:
]

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:
have hAs := hA.submatrix ((↑) : x.support → ι)

Goal state after:
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
hAs : (A.submatrix Subtype.val Subtype.val).PosSemidef
⊢ ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef
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
hAs : (A.submatrix Subtype.val Subtype.val).PosSemidef
hBs : (B.submatrix Subtype.val Subtype.val).PosSemidef
⊢ (A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val).IsHermitian ∧
    ∀ (x_1 : ↥x.support → 𝕜),
      0 ≤ star x_1 ⬝ᵥ A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val *ᵥ x_1

Tactic:
]

Goal state after:
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
hAs : (A.submatrix Subtype.val Subtype.val).PosSemidef
hBs : (B.submatrix Subtype.val Subtype.val).PosSemidef
⊢ (A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val).IsHermitian ∧
    ∀ (x_1 : ↥x.support → 𝕜),
      0 ≤ star x_1 ⬝ᵥ A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val *ᵥ x_1
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
hAs : (A.submatrix Subtype.val Subtype.val).PosSemidef
hBs : (B.submatrix Subtype.val Subtype.val).PosSemidef
y : ↥x.support → 𝕜
⊢ 0 ≤
    star (diagonal y).vec ⬝ᵥ
      kroneckerMap (fun x1 x2 ↦ x1 * x2) (A.submatrix Subtype.val Subtype.val) (B.submatrix Subtype.val Subtype.val) *ᵥ
        (diagonal y).vec

Tactic:
]

Goal state after:
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
hAs : (A.submatrix Subtype.val Subtype.val).PosSemidef
hBs : (B.submatrix Subtype.val Subtype.val).PosSemidef
y : ↥x.support → 𝕜
⊢ 0 ≤
    star (diagonal y).vec ⬝ᵥ
      kroneckerMap (fun x1 x2 ↦ x1 * x2) (A.submatrix Subtype.val Subtype.val) (B.submatrix Subtype.val Subtype.val) *ᵥ
        (diagonal y).vec
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
⊢ (A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val).IsHermitian ∧
    ∀ ⦃x_1 : ↥x.support → 𝕜⦄,
      x_1 ≠ 0 → 0 < star x_1 ⬝ᵥ A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val *ᵥ x_1

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
hAs : (A.submatrix Subtype.val Subtype.val).PosDef
hBs : (B.submatrix Subtype.val Subtype.val).PosDef
⊢ (A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val).IsHermitian ∧
    ∀ ⦃x_1 : ↥x.support → 𝕜⦄,
      x_1 ≠ 0 → 0 < star x_1 ⬝ᵥ A.submatrix Subtype.val Subtype.val ⊙ B.submatrix Subtype.val Subtype.val *ᵥ x_1
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
⊢ 0 <
    star (diagonal y).vec ⬝ᵥ
      kroneckerMap (fun x1 x2 ↦ x1 * x2) (A.submatrix Subtype.val Subtype.val) (B.submatrix Subtype.val Subtype.val) *ᵥ
        (diagonal y).vec

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
hAs : (A.submatrix Subtype.val Subtype.val).PosDef
hBs : (B.submatrix Subtype.val Subtype.val).PosDef
y : ↥x.support → 𝕜
hy : y ≠ 0
⊢ 0 <
    star (diagonal y).vec ⬝ᵥ
      kroneckerMap (fun x1 x2 ↦ x1 * x2) (A.submatrix Subtype.val Subtype.val) (B.submatrix Subtype.val Subtype.val) *ᵥ
        (diagonal y).vec
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
this : Fintype n
x : n → 𝕜
⊢ re 0 ≤ re ⟪∑ i, x i • v i, ∑ i, x i • v i⟫_𝕜 ∧ im 0 = im ⟪∑ i, x i • v i, ∑ i, x 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✝ : Finite n
v : n → E
this : Fintype n
x : n → 𝕜
⊢ re 0 ≤ re ⟪∑ i, x i • v i, ∑ i, x i • v i⟫_𝕜 ∧ im 0 = im ⟪∑ i, x i • v i, ∑ i, x 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
n : ℕ∞
K : Compacts E
f : 𝓓^{n}_{K}(E, F)
x : E
⊢ ‖iteratedFDeriv ℝ 0 (⇑f) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K 0) f

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
f : 𝓓^{n}_{K}(E, F)
x : E
⊢ ‖iteratedFDeriv ℝ 0 (⇑f) x‖ ≤ (ContDiffMapSupportedIn.seminorm 𝕜 E F n K 0) f
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
this : Fintype n
⊢ ∀ (g : n → 𝕜), ∑ i, g i • v i = 0 → ∀ (i : n), g i = 0

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✝ : Finite n
v : n → E
h_gram : (gram 𝕜 v).PosDef
this : Fintype n
⊢ ∀ (g : n → 𝕜), ∑ i, g i • v i = 0 → ∀ (i : n), g i = 0
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 y : Matrix n n 𝕜
hy : IsUnit y
hM : (star y * y).PosDef
__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace
hx : y * xᴴ = 0
⊢ x = 0

Tactic:
]

Goal state after:
𝕜 : Type u_1
n : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : Fintype n
x y : Matrix n n 𝕜
hy : IsUnit y
hM : (star y * y).PosDef
__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace
hx : y * xᴴ = 0
⊢ 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
A : Matrix n n 𝕜
hA✝ : IsSelfAdjoint A
hA : A.IsHermitian
⊢ Isometry ⇑(cfcHom hA✝)

Tactic:
]

Goal state after:
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✝ : IsSelfAdjoint A
hA : A.IsHermitian
⊢ Isometry ⇑(cfcHom hA✝)
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✝ : IsSelfAdjoint A
hA : A.IsHermitian
⊢ ∀ (x : C(↑(spectrum ℝ A), ℝ)), ‖hA.cfcAux x‖ = ‖x‖

Tactic:
]

Goal state after:
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✝ : IsSelfAdjoint A
hA : A.IsHermitian
⊢ ∀ (x : C(↑(spectrum ℝ A), ℝ)), ‖hA.cfcAux x‖ = ‖x‖
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 : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Finite n
v : n → E
this : Fintype n
h_li : ∀ (g : n → 𝕜), ∑ i, g i • v i = 0 → ∀ (i : n), g i = 0
⊢ (gram 𝕜 v).PosDef

Tactic:
]

Goal state after:
E : Type u_1
n : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Finite n
v : n → E
this : Fintype n
h_li : ∀ (g : n → 𝕜), ∑ i, g i • v i = 0 → ∀ (i : n), g i = 0
⊢ (gram 𝕜 v).PosDef
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✝ : IsSelfAdjoint A
hA : A.IsHermitian
f : C(↑(spectrum ℝ A), ℝ)
⊢ ‖⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩‖ = ‖f‖

Tactic:
]

Goal state after:
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✝ : IsSelfAdjoint A
hA : A.IsHermitian
f : C(↑(spectrum ℝ A), ℝ)
⊢ ‖⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩‖ = ‖f‖