ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn
theorem eq_smul_self_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E}
    (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
    T x₀ = (↑(T.rayleighQuotient x₀) : 𝕜) • x₀ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) {x₀ : E} (hextr : IsLocalExtrOn T.reApplyInnerSelf (Metric.sphere 0 ‖x₀‖) x₀)
Goal: T x₀ = ↑(T.rayleighQuotient x₀) • x₀
-- IsSelfAdjoint.hasEigenvector_of_isLocalExtrOn
theorem hasEigenvector_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0)
    (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
    HasEigenvector (T : E →ₗ[𝕜] E) (↑(T.rayleighQuotient x₀)) x₀ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0) (hextr : IsLocalExtrOn T.reApplyInnerSelf (Metric.sphere 0 ‖x₀‖) x₀)
Goal: Module.End.HasEigenvector (↑T) (↑(T.rayleighQuotient x₀)) x₀
-- IsSelfAdjoint.hasEigenvector_of_isMaxOn
theorem hasEigenvector_of_isMaxOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0)
    (hextr : IsMaxOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
    HasEigenvector (T : E →ₗ[𝕜] E) (↑(⨆ x : { x : E // x ≠ 0 }, T.rayleighQuotient x)) x₀ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0) (hextr : IsMaxOn T.reApplyInnerSelf (Metric.sphere 0 ‖x₀‖) x₀)
Goal: Module.End.HasEigenvector (↑T) (↑(⨆ x, T.rayleighQuotient ↑x)) x₀
-- IsSelfAdjoint.hasEigenvector_of_isMinOn
theorem hasEigenvector_of_isMinOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0)
    (hextr : IsMinOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
    HasEigenvector (T : E →ₗ[𝕜] E) (↑(⨅ x : { x : E // x ≠ 0 }, T.rayleighQuotient x)) x₀ 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0) (hextr : IsMinOn T.reApplyInnerSelf (Metric.sphere 0 ‖x₀‖) x₀)
Goal: Module.End.HasEigenvector (↑T) (↑(⨅ x, T.rayleighQuotient ↑x)) x₀
-- LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional
theorem hasEigenvalue_iSup_of_finiteDimensional (hT : T.IsSymmetric) :
    HasEigenvalue T ↑(⨆ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [_i : Nontrivial E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: Module.End.HasEigenvalue T ↑(⨆ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)
-- LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional
theorem hasEigenvalue_iInf_of_finiteDimensional (hT : T.IsSymmetric) :
    HasEigenvalue T ↑(⨅ x : { x : E // x ≠ 0 }, RCLike.re ⟪T x, x⟫ / ‖(x : E)‖ ^ 2 : ℝ) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [_i : Nontrivial E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric)
Goal: Module.End.HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)
-- LinearMap.IsSymmetric.subsingleton_of_no_eigenvalue_finiteDimensional
theorem subsingleton_of_no_eigenvalue_finiteDimensional (hT : T.IsSymmetric)
    (hT' : ∀ μ : 𝕜, Module.End.eigenspace (T : E →ₗ[𝕜] E) μ = ⊥) : Subsingleton E 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hT' : ∀ (μ : 𝕜), Module.End.eigenspace T μ = ⊥)
Goal: Subsingleton E
-- FormalMultilinearSeries.leftInv_coeff_zero
@[simp]
theorem leftInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
    p.leftInv i 0 = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
Goal: p.leftInv i 0 = 0
-- FormalMultilinearSeries.leftInv_coeff_one
@[simp]
theorem leftInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
    p.leftInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
Goal: p.leftInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm ↑i.symm
-- FormalMultilinearSeries.leftInv_removeZero
theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
    p.removeZero.leftInv i = p.leftInv i 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
Goal: p.removeZero.leftInv i = p.leftInv i
-- FormalMultilinearSeries.leftInv_comp
theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : (leftInv p i).comp p = id 𝕜 E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i)
Goal: (p.leftInv i).comp p = FormalMultilinearSeries.id 𝕜 E
-- FormalMultilinearSeries.rightInv_coeff_zero
@[simp]
theorem rightInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
    p.rightInv i 0 = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
Goal: p.rightInv i 0 = 0
-- FormalMultilinearSeries.rightInv_coeff_one
@[simp]
theorem rightInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
    p.rightInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
Goal: p.rightInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm ↑i.symm
-- FormalMultilinearSeries.rightInv_removeZero
theorem rightInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
    p.removeZero.rightInv i = p.rightInv i 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
Goal: p.removeZero.rightInv i = p.rightInv i
-- FormalMultilinearSeries.comp_rightInv_aux1
theorem comp_rightInv_aux1 {n : ℕ} (hn : 0 < n) (p : FormalMultilinearSeries 𝕜 E F)
    (q : FormalMultilinearSeries 𝕜 F E) (v : Fin n → F) :
    p.comp q n v =
      ∑ c ∈ {c : Composition n | 1 < c.length}.toFinset,
          p c.length (q.applyComposition c v) +
        p 1 fun _ => q n v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (hn : 0 < n) (p : FormalMultilinearSeries 𝕜 E F) (q : FormalMultilinearSeries 𝕜 F E) (v : Fin n → F)
Goal: (p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v
-- FormalMultilinearSeries.comp_rightInv_aux2
theorem comp_rightInv_aux2 (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ)
    (v : Fin (n + 2) → F) :
    ∑ c ∈ {c : Composition (n + 2) | 1 < c.length}.toFinset,
        p c.length (applyComposition (fun k : ℕ => ite (k < n + 2) (p.rightInv i k) 0) c v) =
      ∑ c ∈ {c : Composition (n + 2) | 1 < c.length}.toFinset,
        p c.length ((p.rightInv i).applyComposition c v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (v : Fin (n + 2) → F)
Goal: ∑ c ∈ {c | 1 < c.length}.toFinset,
    (p c.length) (FormalMultilinearSeries.applyComposition (fun k => if k < n + 2 then p.rightInv i k else 0) c v) =
  ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) ((p.rightInv i).applyComposition c v)
-- FormalMultilinearSeries.comp_rightInv
theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
    p.comp (rightInv p i) = id 𝕜 F 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i) (h0 : p 0 = 0)
Goal: p.comp (p.rightInv i) = FormalMultilinearSeries.id 𝕜 F
-- FormalMultilinearSeries.rightInv_coeff
theorem rightInv_coeff (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (hn : 2 ≤ n) :
    p.rightInv i n =
      -(i.symm : F →L[𝕜] E).compContinuousMultilinearMap
          (∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition n)),
            p.compAlongComposition (p.rightInv i) c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (hn : 2 ≤ n)
Goal: p.rightInv i n =
  -(↑i.symm).compContinuousMultilinearMap (∑ c ∈ {c | 1 < c.length}.toFinset, p.compAlongComposition (p.rightInv i) c)
-- _private.Mathlib.Analysis.Analytic.Inverse.0.FormalMultilinearSeries.leftInv_eq_rightInv_aux
private theorem leftInv_eq_rightInv_aux (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
    leftInv p i = rightInv p i 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i) (h0 : p 0 = 0)
Goal: p.leftInv i = p.rightInv i
-- FormalMultilinearSeries.leftInv_eq_rightInv
theorem leftInv_eq_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : leftInv p i = rightInv p i 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i)
Goal: p.leftInv i = p.rightInv i
-- FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1
theorem radius_right_inv_pos_of_radius_pos_aux1 (n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ}
    (hr : 0 ≤ r) (ha : 0 ≤ a) :
    ∑ k ∈ Ico 2 (n + 1),
        a ^ k *
          ∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
            r ^ c.length * ∏ j, p (c.blocksFun j) ≤
      ∑ j ∈ Ico 2 (n + 1), r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j 
Arguments: (n : ℕ) (p : ℕ → ℝ) (hp : ∀ (k : ℕ), 0 ≤ p k) {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
Goal: ∑ k ∈ Finset.Ico 2 (n + 1),
    a ^ k * ∑ c ∈ {c | 1 < c.length}.toFinset, r ^ c.length * ∏ j : Fin c.length, p (c.blocksFun j) ≤
  ∑ j ∈ Finset.Ico 2 (n + 1), r ^ j * (∑ k ∈ Finset.Ico 1 n, a ^ k * p k) ^ j
-- FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos_aux2
theorem radius_rightInv_pos_of_radius_pos_aux2 {n : ℕ} (hn : 2 ≤ n + 1)
    (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) {r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
    (hC : 0 ≤ C) (hp : ∀ n, ‖p n‖ ≤ C * r ^ n) :
    ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ ≤
      ‖(i.symm : F →L[𝕜] E)‖ * a +
        ‖(i.symm : F →L[𝕜] E)‖ * C *
          ∑ k ∈ Ico 2 (n + 1), (r * ∑ j ∈ Ico 1 n, a ^ j * ‖p.rightInv i j‖) ^ k 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} (hn : 2 ≤ n + 1) (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) {r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a) (hC : 0 ≤ C) (hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n)
Goal: ∑ k ∈ Finset.Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ ≤
  ‖↑i.symm‖ * a + ‖↑i.symm‖ * C * ∑ k ∈ Finset.Ico 2 (n + 1), (r * ∑ j ∈ Finset.Ico 1 n, a ^ j * ‖p.rightInv i j‖) ^ k
-- FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos
theorem radius_rightInv_pos_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
    (hp : 0 < p.radius) : 0 < (p.rightInv i).radius 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (hp : 0 < p.radius)
Goal: 0 < (p.rightInv i).radius
-- SchwartzMap.fourierTransformCLM_apply
@[simp] lemma fourierTransformCLM_apply (f : 𝓢(V, E)) :
    fourierTransformCLM 𝕜 f = 𝓕 f 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] {V : Type u_3} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (f : SchwartzMap V E)
Goal: ⇑((SchwartzMap.fourierTransformCLM 𝕜) f) = 𝓕 ⇑f
-- SchwartzMap.fourierTransformCLE_apply
@[simp] lemma fourierTransformCLE_apply (f : 𝓢(V, E)) :
    fourierTransformCLE 𝕜 f = 𝓕 f 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] {V : Type u_3} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [CompleteSpace E] (f : SchwartzMap V E)
Goal: ⇑((SchwartzMap.fourierTransformCLE 𝕜) f) = 𝓕 ⇑f
-- SchwartzMap.fourierTransformCLE_symm_apply
@[simp] lemma fourierTransformCLE_symm_apply (f : 𝓢(V, E)) :
    (fourierTransformCLE 𝕜).symm f = 𝓕⁻ f 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] {V : Type u_3} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [CompleteSpace E] (f : SchwartzMap V E)
Goal: ⇑((SchwartzMap.fourierTransformCLE 𝕜).symm f) = 𝓕⁻ ⇑f
-- BoxIntegral.Box.measure_Icc_lt_top
theorem measure_Icc_lt_top (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : μ (Box.Icc I) < ∞ 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: μ (BoxIntegral.Box.Icc I) < ⊤
-- BoxIntegral.Box.measure_coe_lt_top
theorem measure_coe_lt_top (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : μ I < ∞ 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: μ ↑I < ⊤
-- BoxIntegral.Box.measurableSet_coe
theorem measurableSet_coe : MeasurableSet (I : Set (ι → ℝ)) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) [Countable ι]
Goal: MeasurableSet ↑I
-- BoxIntegral.Box.measurableSet_Icc
theorem measurableSet_Icc : MeasurableSet (Box.Icc I) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) [Countable ι]
Goal: MeasurableSet (BoxIntegral.Box.Icc I)
-- BoxIntegral.Box.measurableSet_Ioo
theorem measurableSet_Ioo : MeasurableSet (Box.Ioo I) 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) [Countable ι]
Goal: MeasurableSet (BoxIntegral.Box.Ioo I)
-- BoxIntegral.Box.coe_ae_eq_Icc
theorem coe_ae_eq_Icc : (I : Set (ι → ℝ)) =ᵐ[volume] Box.Icc I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) [Fintype ι]
Goal: ↑I =ᵐ[MeasureTheory.volume] BoxIntegral.Box.Icc I
-- BoxIntegral.Box.Ioo_ae_eq_Icc
theorem Ioo_ae_eq_Icc : Box.Ioo I =ᵐ[volume] Box.Icc I 
Arguments: {ι : Type u_1} (I : BoxIntegral.Box ι) [Fintype ι]
Goal: BoxIntegral.Box.Ioo I =ᵐ[MeasureTheory.volume] BoxIntegral.Box.Icc I
-- BoxIntegral.Prepartition.measure_iUnion_toReal
theorem Prepartition.measure_iUnion_toReal [Finite ι] {I : Box ι} (π : Prepartition I)
    (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] :
    (μ π.iUnion).toReal = ∑ J ∈ π.boxes, (μ J).toReal 
Arguments: {ι : Type u_1} [Finite ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: (μ π.iUnion).toReal = ∑ J ∈ π.boxes, (μ ↑J).toReal
-- MeasureTheory.Measure.toBoxAdditive_apply
@[simps]
def toBoxAdditive [Finite ι] (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : ι →ᵇᵃ[⊤] ℝ 
Arguments: {ι : Type u_1} [Finite ι] (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] (J : BoxIntegral.Box ι)
Goal: μ.toBoxAdditive J = (μ ↑J).toReal
-- BoxIntegral.Box.volume_apply
theorem volume_apply (I : Box ι) :
    (volume : Measure (ι → ℝ)).toBoxAdditive I = ∏ i, (I.upper i - I.lower i) 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι)
Goal: MeasureTheory.volume.toBoxAdditive I = ∏ i : ι, (I.upper i - I.lower i)
-- BoxIntegral.Box.volume_apply'
@[simp]
theorem volume_apply' (I : Box ι) :
    ((volume : Measure (ι → ℝ)) I).toReal = ∏ i, (I.upper i - I.lower i) 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι)
Goal: (MeasureTheory.volume ↑I).toReal = ∏ i : ι, (I.upper i - I.lower i)
-- BoxIntegral.Box.volume_face_mul
theorem volume_face_mul {n} (i : Fin (n + 1)) (I : Box (Fin (n + 1))) :
    (∏ j, ((I.face i).upper j - (I.face i).lower j)) * (I.upper i - I.lower i) =
      ∏ j, (I.upper j - I.lower j) 
Arguments: {n : ℕ} (i : Fin (n + 1)) (I : BoxIntegral.Box (Fin (n + 1)))
Goal: (∏ j : Fin n, ((I.face i).upper j - (I.face i).lower j)) * (I.upper i - I.lower i) =
  ∏ j : Fin (n + 1), (I.upper j - I.lower j)
-- BoxIntegral.BoxAdditiveMap.volume_apply
theorem volume_apply {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (I : Box ι) (x : E) :
    BoxAdditiveMap.volume I x = (∏ j, (I.upper j - I.lower j)) • x 
Arguments: {ι : Type u_1} [Fintype ι] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (I : BoxIntegral.Box ι) (x : E)
Goal: (BoxIntegral.BoxAdditiveMap.volume I) x = (∏ j : ι, (I.upper j - I.lower j)) • x
-- gramSchmidt_def
theorem gramSchmidt_def (f : ι → E) (n : ι) :
    gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) (n : ι)
Goal: gramSchmidt 𝕜 f n = f n - ∑ i ∈ Finset.Iio n, ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 f i})) (f n))
-- gramSchmidt_def'
theorem gramSchmidt_def' (f : ι → E) (n : ι) :
    f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) (n : ι)
Goal: f n = gramSchmidt 𝕜 f n + ∑ i ∈ Finset.Iio n, ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 f i})) (f n))
-- gramSchmidt_def''
theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
    f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n,
      (⟪gramSchmidt 𝕜 f i, f n⟫ / (‖gramSchmidt 𝕜 f i‖ : 𝕜) ^ 2) • gramSchmidt 𝕜 f i 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) (n : ι)
Goal: f n =
  gramSchmidt 𝕜 f n + ∑ i ∈ Finset.Iio n, (⟪gramSchmidt 𝕜 f i, f n⟫_𝕜 / ↑‖gramSchmidt 𝕜 f i‖ ^ 2) • gramSchmidt 𝕜 f i
-- gramSchmidt_zero
@[simp]
theorem gramSchmidt_zero {ι : Type*} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι]
    [IsWellOrder ι (· < ·)] (f : ι → E) : gramSchmidt 𝕜 f ⊥ = f ⊥ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [LinearOrder ι] [LocallyFiniteOrder ι] [OrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E)
Goal: gramSchmidt 𝕜 f ⊥ = f ⊥
-- gramSchmidt_orthogonal
theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
    ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) {a b : ι} (h₀ : a ≠ b)
Goal: ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫_𝕜 = 0
-- gramSchmidt_pairwise_orthogonal
theorem gramSchmidt_pairwise_orthogonal (f : ι → E) :
    Pairwise fun a b => ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E)
Goal: Pairwise fun a b => ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫_𝕜 = 0
-- gramSchmidt_inv_triangular
theorem gramSchmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
    ⟪gramSchmidt 𝕜 v j, v i⟫ = 0 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (v : ι → E) {i j : ι} (hij : i < j)
Goal: ⟪gramSchmidt 𝕜 v j, v i⟫_𝕜 = 0
-- mem_span_gramSchmidt
theorem mem_span_gramSchmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
    f i ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) {i j : ι} (hij : i ≤ j)
Goal: f i ∈ Submodule.span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j)
-- gramSchmidt_mem_span
theorem gramSchmidt_mem_span (f : ι → E) :
    ∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) {j i : ι}
Goal: i ≤ j → gramSchmidt 𝕜 f i ∈ Submodule.span 𝕜 (f '' Set.Iic j)
-- span_gramSchmidt_Iic
theorem span_gramSchmidt_Iic (f : ι → E) (c : ι) :
    span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic c) = span 𝕜 (f '' Set.Iic c) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) (c : ι)
Goal: Submodule.span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic c) = Submodule.span 𝕜 (f '' Set.Iic c)
-- span_gramSchmidt_Iio
theorem span_gramSchmidt_Iio (f : ι → E) (c : ι) :
    span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio c) = span 𝕜 (f '' Set.Iio c) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E) (c : ι)
Goal: Submodule.span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio c) = Submodule.span 𝕜 (f '' Set.Iio c)
-- span_gramSchmidt
theorem span_gramSchmidt (f : ι → E) : span 𝕜 (range (gramSchmidt 𝕜 f)) = span 𝕜 (range f) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] (f : ι → E)
Goal: Submodule.span 𝕜 (Set.range (gramSchmidt 𝕜 f)) = Submodule.span 𝕜 (Set.range f)
-- gramSchmidt_of_orthogonal
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι fun x x_1 => x < x_1] {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫_𝕜 = 0)
Goal: gramSchmidt 𝕜 f = f