ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.someExistsOneDivLT_lt
private theorem someExistsOneDivLT_lt (hi : ¬s ≤[i] 0) :
    (1 / (findExistsOneDivLT s i + 1) : ℝ) < s (someExistsOneDivLT s i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi : ¬MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: 1 / (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s i) + 1) <
  ↑s (MeasureTheory.SignedMeasure.someExistsOneDivLT s i)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_succ
private theorem restrictNonposSeq_succ (n : ℕ) :
    restrictNonposSeq s i n.succ = someExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (n : ℕ)
Goal: MeasureTheory.SignedMeasure.restrictNonposSeq s i n.succ =
  MeasureTheory.SignedMeasure.someExistsOneDivLT s
    (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_subset
private theorem restrictNonposSeq_subset (n : ℕ) : restrictNonposSeq s i n ⊆ i 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (n : ℕ)
Goal: MeasureTheory.SignedMeasure.restrictNonposSeq s i n ⊆ i
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_lt
private theorem restrictNonposSeq_lt (n : ℕ) (hn : ¬s ≤[i \ ⋃ k ≤ n, restrictNonposSeq s i k] 0) :
    (1 / (findExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq s i k) + 1) : ℝ) <
      s (restrictNonposSeq s i n.succ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (n : ℕ) (hn :
  ¬MeasureTheory.VectorMeasure.restrict s
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      MeasureTheory.VectorMeasure.restrict 0
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
Goal: 1 /
    (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
          (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
      1) <
  ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n.succ)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq
private theorem measure_of_restrictNonposSeq (hi₂ : ¬s ≤[i] 0) (n : ℕ)
    (hn : ¬s ≤[i \ ⋃ k < n, restrictNonposSeq s i k] 0) : 0 < s (restrictNonposSeq s i n) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi₂ : ¬MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i) (n : ℕ) (hn :
  ¬MeasureTheory.VectorMeasure.restrict s
        (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      MeasureTheory.VectorMeasure.restrict 0
        (i \ ⋃ k, ⋃ (_ : k < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k))
Goal: 0 < ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet
private theorem restrictNonposSeq_measurableSet (n : ℕ) :
    MeasurableSet (restrictNonposSeq s i n) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (n : ℕ)
Goal: MeasurableSet (MeasureTheory.SignedMeasure.restrictNonposSeq s i n)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint'
private theorem restrictNonposSeq_disjoint' {n m : ℕ} (h : n < m) :
    restrictNonposSeq s i n ∩ restrictNonposSeq s i m = ∅ 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} {n m : ℕ} (h : n < m)
Goal: MeasureTheory.SignedMeasure.restrictNonposSeq s i n ∩ MeasureTheory.SignedMeasure.restrictNonposSeq s i m = ∅
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint
private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonposSeq s i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α}
Goal: Pairwise (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i)
-- _private.Mathlib.MeasureTheory.Decomposition.SignedHahn.0.MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos'
private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
    (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi₁ : MeasurableSet i) (hi₂ : ↑s i < 0) (hn :
  ¬∀ (n : ℕ),
      ¬MeasureTheory.VectorMeasure.restrict s
            (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤
          MeasureTheory.VectorMeasure.restrict 0
            (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l))
Goal: ∃ j,
  MeasurableSet j ∧
    j ⊆ i ∧ MeasureTheory.VectorMeasure.restrict s j ≤ MeasureTheory.VectorMeasure.restrict 0 j ∧ ↑s j < 0
-- MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos
theorem exists_subset_restrict_nonpos (hi : s i < 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i : Set α} (hi : ↑s i < 0)
Goal: ∃ j,
  MeasurableSet j ∧
    j ⊆ i ∧ MeasureTheory.VectorMeasure.restrict s j ≤ MeasureTheory.VectorMeasure.restrict 0 j ∧ ↑s j < 0
-- MeasureTheory.SignedMeasure.zero_mem_measureOfNegatives
theorem zero_mem_measureOfNegatives : (0 : ℝ) ∈ s.measureOfNegatives 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α}
Goal: 0 ∈ s.measureOfNegatives
-- MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives
theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α}
Goal: BddBelow s.measureOfNegatives
-- MeasureTheory.SignedMeasure.exists_compl_positive_negative
theorem exists_compl_positive_negative (s : SignedMeasure α) :
    ∃ i : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ s ≤[iᶜ] 0 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: ∃ i,
  MeasurableSet i ∧
    MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i ∧
      MeasureTheory.VectorMeasure.restrict s iᶜ ≤ MeasureTheory.VectorMeasure.restrict 0 iᶜ
-- MeasureTheory.SignedMeasure.exists_isCompl_positive_negative
theorem exists_isCompl_positive_negative (s : SignedMeasure α) :
    ∃ i j : Set α, MeasurableSet i ∧ 0 ≤[i] s ∧ MeasurableSet j ∧ s ≤[j] 0 ∧ IsCompl i j 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : MeasureTheory.SignedMeasure α)
Goal: ∃ i j,
  MeasurableSet i ∧
    MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i ∧
      MeasurableSet j ∧
        MeasureTheory.VectorMeasure.restrict s j ≤ MeasureTheory.VectorMeasure.restrict 0 j ∧ IsCompl i j
-- MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative
theorem of_symmDiff_compl_positive_negative {s : SignedMeasure α} {i j : Set α}
    (hi : MeasurableSet i) (hj : MeasurableSet j) (hi' : 0 ≤[i] s ∧ s ≤[iᶜ] 0)
    (hj' : 0 ≤[j] s ∧ s ≤[jᶜ] 0) : s (i ∆ j) = 0 ∧ s (iᶜ ∆ jᶜ) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {i j : Set α} (hi : MeasurableSet i) (hj : MeasurableSet j) (hi' :
  MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i ∧
    MeasureTheory.VectorMeasure.restrict s iᶜ ≤ MeasureTheory.VectorMeasure.restrict 0 iᶜ) (hj' :
  MeasureTheory.VectorMeasure.restrict 0 j ≤ MeasureTheory.VectorMeasure.restrict s j ∧
    MeasureTheory.VectorMeasure.restrict s jᶜ ≤ MeasureTheory.VectorMeasure.restrict 0 jᶜ)
Goal: ↑s (symmDiff i j) = 0 ∧ ↑s (symmDiff iᶜ jᶜ) = 0
-- LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure
theorem LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure [NormedSpace 𝕜 E]
    (f : E →ₗ[𝕜] E) (hf : LipschitzWith 1 f) (g : E →L[𝕜] LinearMap.eqLocus f 1)
    (hg_proj : ∀ x : LinearMap.eqLocus f 1, g x = x)
    (hg_ker : (LinearMap.ker g : Set E) ⊆ closure (LinearMap.range (f - 1))) (x : E) :
    Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop (𝓝 (g x)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : E →ₗ[𝕜] E) (hf : LipschitzWith 1 ⇑f) (g : E →L[𝕜] ↥(LinearMap.eqLocus f 1)) (hg_proj : ∀ (x : ↥(LinearMap.eqLocus f 1)), g ↑x = x) (hg_ker : ↑(LinearMap.ker g) ⊆ closure ↑(LinearMap.range (f - 1))) (x : E)
Goal: Filter.Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) id x_1 x) Filter.atTop (𝓝 ↑(g x))
-- ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection
theorem ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection (f : E →L[𝕜] E)
    (hf : ‖f‖ ≤ 1) (x : E) :
    Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop
      (𝓝 <| orthogonalProjection (LinearMap.eqLocus f 1) x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (f : E →L[𝕜] E) (hf : ‖f‖ ≤ 1) (x : E)
Goal: Filter.Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) id x_1 x) Filter.atTop
  (𝓝 ↑((orthogonalProjection (LinearMap.eqLocus f 1)) x))
-- LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace
theorem invariant_orthogonalComplement_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
    T v ∈ (eigenspace T μ)ᗮ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (μ : 𝕜) (v : E) (hv : v ∈ (Module.End.eigenspace T μ)ᗮ)
Goal: T v ∈ (Module.End.eigenspace T μ)ᗮ
-- LinearMap.IsSymmetric.conj_eigenvalue_eq_self
theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) {μ : 𝕜} (hμ : Module.End.HasEigenvalue T μ)
Goal: (starRingEnd 𝕜) μ = μ
-- LinearMap.IsSymmetric.orthogonalFamily_eigenspaces
theorem orthogonalFamily_eigenspaces :
    OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: OrthogonalFamily 𝕜 (fun μ => ↥(Module.End.eigenspace T μ)) fun μ => (Module.End.eigenspace T μ).subtypeₗᵢ
-- LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'
theorem orthogonalFamily_eigenspaces' :
    OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
      (eigenspace T μ).subtypeₗᵢ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: OrthogonalFamily 𝕜 (fun μ => ↥(Module.End.eigenspace T (↑T μ))) fun μ => (Module.End.eigenspace T (↑T μ)).subtypeₗᵢ
-- LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_invariant
theorem orthogonalComplement_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
    T v ∈ (⨆ μ, eigenspace T μ)ᗮ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) ⦃v : E⦄ (hv : v ∈ (⨆ μ, Module.End.eigenspace T μ)ᗮ)
Goal: T v ∈ (⨆ μ, Module.End.eigenspace T μ)ᗮ
-- LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces
theorem orthogonalComplement_iSup_eigenspaces (μ : 𝕜) :
    eigenspace (T.restrict hT.orthogonalComplement_iSup_eigenspaces_invariant) μ = ⊥ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (μ : 𝕜)
Goal: Module.End.eigenspace (T.restrict ⋯) μ = ⊥
-- LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot
theorem orthogonalComplement_iSup_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E]
Goal: (⨆ μ, Module.End.eigenspace T μ)ᗮ = ⊥
-- LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot'
theorem orthogonalComplement_iSup_eigenspaces_eq_bot' :
    (⨆ μ : Eigenvalues T, eigenspace T μ)ᗮ = ⊥ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E]
Goal: (⨆ μ, Module.End.eigenspace T (↑T μ))ᗮ = ⊥
-- LinearMap.IsSymmetric.directSum_decompose_apply
theorem directSum_decompose_apply [_hT : Fact T.IsSymmetric] (x : E) (μ : Eigenvalues T) :
    DirectSum.decompose (fun μ : Eigenvalues T => eigenspace T μ) x μ =
      orthogonalProjection (eigenspace T μ) x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} [FiniteDimensional 𝕜 E] [_hT : Fact T.IsSymmetric] (x : E) (μ : Module.End.Eigenvalues T)
Goal: ((DirectSum.decompose fun μ => Module.End.eigenspace T (↑T μ)) x) μ =
  (orthogonalProjection (Module.End.eigenspace T (↑T μ))) x
-- LinearMap.IsSymmetric.direct_sum_isInternal
theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E]
Goal: DirectSum.IsInternal fun μ => Module.End.eigenspace T (↑T μ)
-- LinearMap.IsSymmetric.diagonalization_symm_apply
@[simp]
theorem diagonalization_symm_apply (w : PiLp 2 fun μ : Eigenvalues T => eigenspace T μ) :
    hT.diagonalization.symm w = ∑ μ, w μ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] (w : PiLp 2 fun μ => ↥(Module.End.eigenspace T (↑T μ)))
Goal: hT.diagonalization.symm w = ∑ μ : Module.End.Eigenvalues T, ↑(w μ)
-- LinearMap.IsSymmetric.diagonalization_apply_self_apply
theorem diagonalization_apply_self_apply (v : E) (μ : Eigenvalues T) :
    hT.diagonalization (T v) μ = (μ : 𝕜) • hT.diagonalization v μ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] (v : E) (μ : Module.End.Eigenvalues T)
Goal: hT.diagonalization (T v) μ = ↑T μ • hT.diagonalization v μ
-- LinearMap.IsSymmetric.eigenvectorBasis_def
noncomputable irreducible_def eigenvectorBasis : OrthonormalBasis (Fin n) 𝕜 E 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n)
Goal: hT.eigenvectorBasis hn = DirectSum.IsInternal.subordinateOrthonormalBasis hn ⋯ ⋯
-- LinearMap.IsSymmetric.eigenvalues_def
noncomputable irreducible_def eigenvalues (i : Fin n) : ℝ 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) (i : Fin n)
Goal: hT.eigenvalues hn i = RCLike.re (↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯))
-- LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis
theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
    HasEigenvector T (hT.eigenvalues hn i) (hT.eigenvectorBasis hn i) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) (i : Fin n)
Goal: Module.End.HasEigenvector T (↑(hT.eigenvalues hn i)) ((hT.eigenvectorBasis hn) i)
-- LinearMap.IsSymmetric.hasEigenvalue_eigenvalues
theorem hasEigenvalue_eigenvalues (i : Fin n) : HasEigenvalue T (hT.eigenvalues hn i) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) (i : Fin n)
Goal: Module.End.HasEigenvalue T ↑(hT.eigenvalues hn i)
-- LinearMap.IsSymmetric.apply_eigenvectorBasis
@[simp]
theorem apply_eigenvectorBasis (i : Fin n) :
    T (hT.eigenvectorBasis hn i) = (hT.eigenvalues hn i : 𝕜) • hT.eigenvectorBasis hn i 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) (i : Fin n)
Goal: T ((hT.eigenvectorBasis hn) i) = ↑(hT.eigenvalues hn i) • (hT.eigenvectorBasis hn) i
-- LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply
theorem eigenvectorBasis_apply_self_apply (v : E) (i : Fin n) :
    (hT.eigenvectorBasis hn).repr (T v) i =
      hT.eigenvalues hn i * (hT.eigenvectorBasis hn).repr v i 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) (v : E) (i : Fin n)
Goal: (hT.eigenvectorBasis hn).repr (T v) i = ↑(hT.eigenvalues hn i) * (hT.eigenvectorBasis hn).repr v i
-- inner_product_apply_eigenvector
@[simp]
theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜] E}
    (h : v ∈ Module.End.eigenspace T μ) : ⟪v, T v⟫ = μ * (‖v‖ : 𝕜) ^ 2 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {μ : 𝕜} {v : E} {T : E →ₗ[𝕜] E} (h : v ∈ Module.End.eigenspace T μ)
Goal: ⟪v, T v⟫_𝕜 = μ * ↑‖v‖ ^ 2
-- eigenvalue_nonneg_of_nonneg
theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
    (hnn : ∀ x : E, 0 ≤ RCLike.re ⟪x, T x⟫) : 0 ≤ μ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Module.End.HasEigenvalue T ↑μ) (hnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫_𝕜)
Goal: 0 ≤ μ
-- eigenvalue_pos_of_pos
theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
    (hnn : ∀ x : E, 0 < RCLike.re ⟪x, T x⟫) : 0 < μ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : Module.End.HasEigenvalue T ↑μ) (hnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫_𝕜)
Goal: 0 < μ
-- OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_same_orientation
    (h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] (e f : OrthonormalBasis ι ℝ E) (h : e.toBasis.orientation = f.toBasis.orientation)
Goal: e.toBasis.det ⇑f = 1
-- OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
    (h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] (e f : OrthonormalBasis ι ℝ E) (h : e.toBasis.orientation ≠ f.toBasis.orientation)
Goal: e.toBasis.det ⇑f = -1
-- OrthonormalBasis.same_orientation_iff_det_eq_det
theorem same_orientation_iff_det_eq_det :
    e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] {e f : OrthonormalBasis ι ℝ E}
Goal: e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation
-- OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
    e.toBasis.det = -f.toBasis.det 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] (e f : OrthonormalBasis ι ℝ E) (h : e.toBasis.orientation ≠ f.toBasis.orientation)
Goal: e.toBasis.det = -f.toBasis.det
-- OrthonormalBasis.orthonormal_adjustToOrientation
theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι)
Goal: Orthonormal ℝ ⇑(e.toBasis.adjustToOrientation x)
-- OrthonormalBasis.toBasis_adjustToOrientation
theorem toBasis_adjustToOrientation :
    (e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι)
Goal: (e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x
-- OrthonormalBasis.orientation_adjustToOrientation
@[simp]
theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.orientation = x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι)
Goal: (e.adjustToOrientation x).toBasis.orientation = x
-- OrthonormalBasis.adjustToOrientation_apply_eq_or_eq_neg
theorem adjustToOrientation_apply_eq_or_eq_neg (i : ι) :
    e.adjustToOrientation x i = e i ∨ e.adjustToOrientation x i = -e i 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι) (i : ι)
Goal: (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i
-- OrthonormalBasis.det_adjustToOrientation
theorem det_adjustToOrientation :
    (e.adjustToOrientation x).toBasis.det = e.toBasis.det ∨
      (e.adjustToOrientation x).toBasis.det = -e.toBasis.det 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι)
Goal: (e.adjustToOrientation x).toBasis.det = e.toBasis.det ∨ (e.adjustToOrientation x).toBasis.det = -e.toBasis.det
-- OrthonormalBasis.abs_det_adjustToOrientation
theorem abs_det_adjustToOrientation (v : ι → E) :
    |(e.adjustToOrientation x).toBasis.det v| = |e.toBasis.det v| 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {ι : Type u_2} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e : OrthonormalBasis ι ℝ E) (x : Orientation ℝ E ι) (v : ι → E)
Goal: |(e.adjustToOrientation x).toBasis.det v| = |e.toBasis.det v|
-- Orientation.finOrthonormalBasis_orientation
@[simp]
theorem finOrthonormalBasis_orientation (hn : 0 < n) (h : finrank ℝ E = n)
    (x : Orientation ℝ E (Fin n)) : (x.finOrthonormalBasis hn h).toBasis.orientation = x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} (hn : 0 < n) (h : FiniteDimensional.finrank ℝ E = n) (x : Orientation ℝ E (Fin n))
Goal: (Orientation.finOrthonormalBasis hn h x).toBasis.orientation = x
-- Orientation.volumeForm_def
irreducible_def volumeForm : E [⋀^Fin n]→ₗ[ℝ] ℝ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
Goal: o.volumeForm =
  Nat.casesAuxOn (motive := fun a => n = a → E [⋀^Fin n]→ₗ[ℝ] ℝ) n
    (fun h =>
      Eq.ndrec (motive := fun {n} =>
        [_i : Fact (FiniteDimensional.finrank ℝ E = n)] → Orientation ℝ E (Fin n) → E [⋀^Fin n]→ₗ[ℝ] ℝ)
        (fun [Fact (FiniteDimensional.finrank ℝ E = 0)] o =>
          let opos := AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1;
          ⋯.by_cases (fun x => opos) fun x => -opos)
        ⋯ o)
    (fun n_1 h =>
      Eq.ndrec (motive := fun {n} =>
        [_i : Fact (FiniteDimensional.finrank ℝ E = n)] → Orientation ℝ E (Fin n) → E [⋀^Fin n]→ₗ[ℝ] ℝ)
        (fun [Fact (FiniteDimensional.finrank ℝ E = n_1 + 1)] o => (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det)
        ⋯ o)
    ⋯