ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Orientation.volumeForm_zero_pos
@[simp]
theorem volumeForm_zero_pos [_i : Fact (finrank ℝ E = 0)] :
    Orientation.volumeForm (positiveOrientation : Orientation ℝ E (Fin 0)) =
      AlternatingMap.constLinearEquivOfIsEmpty 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [_i : Fact (FiniteDimensional.finrank ℝ E = 0)]
Goal: positiveOrientation.volumeForm = AlternatingMap.constLinearEquivOfIsEmpty 1
-- Orientation.volumeForm_zero_neg
theorem volumeForm_zero_neg [_i : Fact (finrank ℝ E = 0)] :
    Orientation.volumeForm (-positiveOrientation : Orientation ℝ E (Fin 0)) =
      -AlternatingMap.constLinearEquivOfIsEmpty 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [_i : Fact (FiniteDimensional.finrank ℝ E = 0)]
Goal: (-positiveOrientation).volumeForm = -AlternatingMap.constLinearEquivOfIsEmpty 1
-- Orientation.volumeForm_robust
theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.orientation = o) :
    o.volumeForm = b.toBasis.det 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.orientation = o)
Goal: o.volumeForm = b.toBasis.det
-- Orientation.volumeForm_robust_neg
theorem volumeForm_robust_neg (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.orientation ≠ o) :
    o.volumeForm = -b.toBasis.det 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.orientation ≠ o)
Goal: o.volumeForm = -b.toBasis.det
-- Orientation.volumeForm_neg_orientation
@[simp]
theorem volumeForm_neg_orientation : (-o).volumeForm = -o.volumeForm 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
Goal: (-o).volumeForm = -o.volumeForm
-- Orientation.volumeForm_robust'
theorem volumeForm_robust' (b : OrthonormalBasis (Fin n) ℝ E) (v : Fin n → E) :
    |o.volumeForm v| = |b.toBasis.det v| 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (b : OrthonormalBasis (Fin n) ℝ E) (v : Fin n → E)
Goal: |o.volumeForm v| = |b.toBasis.det v|
-- Orientation.abs_volumeForm_apply_le
theorem abs_volumeForm_apply_le (v : Fin n → E) : |o.volumeForm v| ≤ ∏ i : Fin n, ‖v i‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (v : Fin n → E)
Goal: |o.volumeForm v| ≤ ∏ i : Fin n, ‖v i‖
-- Orientation.volumeForm_apply_le
theorem volumeForm_apply_le (v : Fin n → E) : o.volumeForm v ≤ ∏ i : Fin n, ‖v i‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (v : Fin n → E)
Goal: o.volumeForm v ≤ ∏ i : Fin n, ‖v i‖
-- Orientation.abs_volumeForm_apply_of_pairwise_orthogonal
theorem abs_volumeForm_apply_of_pairwise_orthogonal {v : Fin n → E}
    (hv : Pairwise fun i j => ⟪v i, v j⟫ = 0) : |o.volumeForm v| = ∏ i : Fin n, ‖v i‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) {v : Fin n → E} (hv : Pairwise fun i j => ⟪v i, v j⟫_ℝ = 0)
Goal: |o.volumeForm v| = ∏ i : Fin n, ‖v i‖
-- Orientation.abs_volumeForm_apply_of_orthonormal
theorem abs_volumeForm_apply_of_orthonormal (v : OrthonormalBasis (Fin n) ℝ E) :
    |o.volumeForm v| = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (v : OrthonormalBasis (Fin n) ℝ E)
Goal: |o.volumeForm ⇑v| = 1
-- Orientation.volumeForm_map
theorem volumeForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
    [Fact (finrank ℝ F = n)] (φ : E ≃ₗᵢ[ℝ] F) (x : Fin n → F) :
    (Orientation.map (Fin n) φ.toLinearEquiv o).volumeForm x = o.volumeForm (φ.symm ∘ x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [Fact (FiniteDimensional.finrank ℝ F = n)] (φ : E ≃ₗᵢ[ℝ] F) (x : Fin n → F)
Goal: ((Orientation.map (Fin n) φ.toLinearEquiv) o).volumeForm x = o.volumeForm (⇑φ.symm ∘ x)
-- Orientation.volumeForm_comp_linearIsometryEquiv
theorem volumeForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
    (hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x : Fin n → E) :
    o.volumeForm (φ ∘ x) = o.volumeForm x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ E = n)] (o : Orientation ℝ E (Fin n)) (φ : E ≃ₗᵢ[ℝ] E) (hφ : 0 < LinearMap.det ↑φ.toLinearEquiv) (x : Fin n → E)
Goal: o.volumeForm (⇑φ ∘ x) = o.volumeForm x
-- InnerProductSpace.toDualMap_apply
@[simp]
theorem toDualMap_apply {x y : E} : toDualMap 𝕜 E x y = ⟪x, y⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E}
Goal: ((InnerProductSpace.toDualMap 𝕜 E) x) y = ⟪x, y⟫_𝕜
-- InnerProductSpace.innerSL_norm
theorem innerSL_norm [Nontrivial E] : ‖(innerSL 𝕜 : E →L⋆[𝕜] E →L[𝕜] 𝕜)‖ = 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Nontrivial E]
Goal: ‖innerSL 𝕜‖ = 1
-- InnerProductSpace.ext_inner_left_basis
theorem ext_inner_left_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
    (h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} {x y : E} (b : Basis ι 𝕜 E) (h : ∀ (i : ι), ⟪b i, x⟫_𝕜 = ⟪b i, y⟫_𝕜)
Goal: x = y
-- InnerProductSpace.ext_inner_right_basis
theorem ext_inner_right_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
    (h : ∀ i : ι, ⟪x, b i⟫ = ⟪y, b i⟫) : x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_3} {x y : E} (b : Basis ι 𝕜 E) (h : ∀ (i : ι), ⟪x, b i⟫_𝕜 = ⟪y, b i⟫_𝕜)
Goal: x = y
-- InnerProductSpace.toDual_apply
@[simp]
theorem toDual_apply {x y : E} : toDual 𝕜 E x y = ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {x y : E}
Goal: ((InnerProductSpace.toDual 𝕜 E) x) y = ⟪x, y⟫_𝕜
-- InnerProductSpace.toDual_symm_apply
@[simp]
theorem toDual_symm_apply {x : E} {y : NormedSpace.Dual 𝕜 E} : ⟪(toDual 𝕜 E).symm y, x⟫ = y x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {x : E} {y : NormedSpace.Dual 𝕜 E}
Goal: ⟪(InnerProductSpace.toDual 𝕜 E).symm y, x⟫_𝕜 = y x
-- InnerProductSpace.continuousLinearMapOfBilin_apply
@[simp]
theorem continuousLinearMapOfBilin_apply (v w : E) : ⟪B♯ v, w⟫ = B v w 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (B : E →L⋆[𝕜] E →L[𝕜] 𝕜) (v w : E)
Goal: ⟪(InnerProductSpace.continuousLinearMapOfBilin B) v, w⟫_𝕜 = (B v) w
-- InnerProductSpace.unique_continuousLinearMapOfBilin
theorem unique_continuousLinearMapOfBilin {v f : E} (is_lax_milgram : ∀ w, ⟪f, w⟫ = B v w) :
    f = B♯ v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (B : E →L⋆[𝕜] E →L[𝕜] 𝕜) {v f : E} (is_lax_milgram : ∀ (w : E), ⟪f, w⟫_𝕜 = (B v) w)
Goal: f = (InnerProductSpace.continuousLinearMapOfBilin B) v
-- ContinuousLinearMap.IsPositive.isSelfAdjoint
theorem IsPositive.isSelfAdjoint {T : E →L[𝕜] E} (hT : IsPositive T) : IsSelfAdjoint T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive)
Goal: IsSelfAdjoint T
-- ContinuousLinearMap.IsPositive.inner_nonneg_left
theorem IsPositive.inner_nonneg_left {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
    0 ≤ re ⟪T x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) (x : E)
Goal: 0 ≤ RCLike.re ⟪T x, x⟫_𝕜
-- ContinuousLinearMap.IsPositive.inner_nonneg_right
theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
    0 ≤ re ⟪x, T x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) (x : E)
Goal: 0 ≤ RCLike.re ⟪x, T x⟫_𝕜
-- ContinuousLinearMap.isPositive_zero
theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]
Goal: ContinuousLinearMap.IsPositive 0
-- ContinuousLinearMap.isPositive_one
theorem isPositive_one : IsPositive (1 : E →L[𝕜] E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]
Goal: ContinuousLinearMap.IsPositive 1
-- ContinuousLinearMap.IsPositive.add
theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
    (T + S).IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive)
Goal: (T + S).IsPositive
-- ContinuousLinearMap.IsPositive.conj_adjoint
theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
    (S ∘L T ∘L S†).IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F)
Goal: (S.comp (T.comp (ContinuousLinearMap.adjoint S))).IsPositive
-- ContinuousLinearMap.IsPositive.adjoint_conj
theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
    (S† ∘L T ∘L S).IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E)
Goal: ((ContinuousLinearMap.adjoint S).comp (T.comp S)).IsPositive
-- ContinuousLinearMap.IsPositive.conj_orthogonalProjection
theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
    [CompleteSpace U] :
    (U.subtypeL ∘L
        orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive) [CompleteSpace ↥U]
Goal: (U.subtypeL.comp ((orthogonalProjection U).comp (T.comp (U.subtypeL.comp (orthogonalProjection U))))).IsPositive
-- ContinuousLinearMap.IsPositive.orthogonalProjection_comp
theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
    [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E) [CompleteSpace ↥U]
Goal: ((orthogonalProjection U).comp (T.comp U.subtypeL)).IsPositive
-- ContinuousLinearMap.isPositive_iff_complex
theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
    IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ 
Arguments: {E' : Type u_4} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E'] (T : E' →L[ℂ] E')
Goal: T.IsPositive ↔ ∀ (x : E'), ↑(RCLike.re ⟪T x, x⟫_ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ RCLike.re ⟪T x, x⟫_ℂ
-- ContinuousLinearMap.le_def
lemma le_def (f g : E →L[𝕜] E) : f ≤ g ↔ (g - f).IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (f g : E →L[𝕜] E)
Goal: f ≤ g ↔ (g - f).IsPositive
-- ContinuousLinearMap.nonneg_iff_isPositive
lemma nonneg_iff_isPositive (f : E →L[𝕜] E) : 0 ≤ f ↔ f.IsPositive 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] (f : E →L[𝕜] E)
Goal: 0 ≤ f ↔ f.IsPositive
-- RCLike.tendsto_inverse_atTop_nhds_zero_nat
theorem RCLike.tendsto_inverse_atTop_nhds_zero_nat :
    Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (𝓝 0) 
Arguments: (𝕜 : Type u_1) [RCLike 𝕜]
Goal: Filter.Tendsto (fun n => (↑n)⁻¹) Filter.atTop (𝓝 0)
-- hasFDerivAt_norm_rpow
theorem hasFDerivAt_norm_rpow (x : E) {p : ℝ} (hp : 1 < p) :
    HasFDerivAt (fun x : E ↦ ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • innerSL ℝ x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (x : E) {p : ℝ} (hp : 1 < p)
Goal: HasFDerivAt (fun x => ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) x
-- differentiable_norm_rpow
theorem differentiable_norm_rpow {p : ℝ} (hp : 1 < p) :
    Differentiable ℝ (fun x : E ↦ ‖x‖ ^ p) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {p : ℝ} (hp : 1 < p)
Goal: Differentiable ℝ fun x => ‖x‖ ^ p
-- hasDerivAt_norm_rpow
theorem hasDerivAt_norm_rpow (x : ℝ) {p : ℝ} (hp : 1 < p) :
    HasDerivAt (fun x : ℝ ↦ ‖x‖ ^ p) (p * ‖x‖ ^ (p - 2) * x) x 
Arguments: (x : ℝ) {p : ℝ} (hp : 1 < p)
Goal: HasDerivAt (fun x => ‖x‖ ^ p) (p * ‖x‖ ^ (p - 2) * x) x
-- hasDerivAt_abs_rpow
theorem hasDerivAt_abs_rpow (x : ℝ) {p : ℝ} (hp : 1 < p) :
    HasDerivAt (fun x : ℝ ↦ |x| ^ p) (p * |x| ^ (p - 2) * x) x 
Arguments: (x : ℝ) {p : ℝ} (hp : 1 < p)
Goal: HasDerivAt (fun x => |x| ^ p) (p * |x| ^ (p - 2) * x) x
-- fderiv_norm_rpow
theorem fderiv_norm_rpow (x : E) {p : ℝ} (hp : 1 < p) :
    fderiv ℝ (fun x ↦ ‖x‖ ^ p) x = (p * ‖x‖ ^ (p - 2)) • innerSL ℝ x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (x : E) {p : ℝ} (hp : 1 < p)
Goal: fderiv ℝ (fun x => ‖x‖ ^ p) x = (p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x
-- Differentiable.fderiv_norm_rpow
theorem Differentiable.fderiv_norm_rpow {f : F → E} (hf : Differentiable ℝ f)
    {x : F} {p : ℝ} (hp : 1 < p) :
    fderiv ℝ (fun x ↦ ‖f x‖ ^ p) x =
    (p * ‖f x‖ ^ (p - 2)) • (innerSL ℝ (f x)).comp (fderiv ℝ f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : F → E} (hf : Differentiable ℝ f) {x : F} {p : ℝ} (hp : 1 < p)
Goal: fderiv ℝ (fun x => ‖f x‖ ^ p) x = (p * ‖f x‖ ^ (p - 2)) • ((innerSL ℝ) (f x)).comp (fderiv ℝ f x)
-- norm_fderiv_norm_rpow_le
theorem norm_fderiv_norm_rpow_le {f : F → E} (hf : Differentiable ℝ f) {x : F}
    {p : ℝ} (hp : 1 < p) :
    ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ p) x‖ ≤ p * ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : F → E} (hf : Differentiable ℝ f) {x : F} {p : ℝ} (hp : 1 < p)
Goal: ‖fderiv ℝ (fun x => ‖f x‖ ^ p) x‖ ≤ p * ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖
-- norm_fderiv_norm_id_rpow
theorem norm_fderiv_norm_id_rpow (x : E) {p : ℝ} (hp : 1 < p) :
    ‖fderiv ℝ (fun x ↦ ‖x‖ ^ p) x‖ = p * ‖x‖ ^ (p - 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] (x : E) {p : ℝ} (hp : 1 < p)
Goal: ‖fderiv ℝ (fun x => ‖x‖ ^ p) x‖ = p * ‖x‖ ^ (p - 1)
-- nnnorm_fderiv_norm_rpow_le
theorem nnnorm_fderiv_norm_rpow_le {f : F → E} (hf : Differentiable ℝ f)
    {x : F} {p : ℝ≥0} (hp : 1 < p) :
    ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ (p : ℝ)) x‖₊ ≤ p * ‖f x‖₊ ^ ((p : ℝ) - 1) * ‖fderiv ℝ f x‖₊ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : F → E} (hf : Differentiable ℝ f) {x : F} {p : NNReal} (hp : 1 < p)
Goal: ‖fderiv ℝ (fun x => ‖f x‖ ^ ↑p) x‖₊ ≤ p * ‖f x‖₊ ^ (↑p - 1) * ‖fderiv ℝ f x‖₊
-- contDiff_norm_rpow
theorem contDiff_norm_rpow {p : ℝ} (hp : 1 < p) : ContDiff ℝ 1 (fun x : E ↦ ‖x‖ ^ p) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {p : ℝ} (hp : 1 < p)
Goal: ContDiff ℝ 1 fun x => ‖x‖ ^ p
-- ContDiff.norm_rpow
theorem ContDiff.norm_rpow {f : F → E} (hf : ContDiff ℝ 1 f) {p : ℝ} (hp : 1 < p) :
    ContDiff ℝ 1 (fun x ↦ ‖f x‖ ^ p) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : F → E} (hf : ContDiff ℝ 1 f) {p : ℝ} (hp : 1 < p)
Goal: ContDiff ℝ 1 fun x => ‖f x‖ ^ p
-- Differentiable.norm_rpow
theorem Differentiable.norm_rpow {f : F → E} (hf : Differentiable ℝ f) {p : ℝ} (hp : 1 < p) :
    Differentiable ℝ (fun x ↦ ‖f x‖ ^ p) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : F → E} (hf : Differentiable ℝ f) {p : ℝ} (hp : 1 < p)
Goal: Differentiable ℝ fun x => ‖f x‖ ^ p
-- Finset.le_sum_schlomilch'
theorem le_sum_schlomilch' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ Ico (u 0) (u n), f k) ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} {u : ℕ → ℕ} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu : Monotone u) (n : ℕ)
Goal: ∑ k ∈ Finset.Ico (u 0) (u n), f k ≤ ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u k)
-- Finset.le_sum_condensed'
theorem le_sum_condensed' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ Ico 1 (2 ^ n), f k) ≤ ∑ k ∈ range n, 2 ^ k • f (2 ^ k) 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ)
Goal: ∑ k ∈ Finset.Ico 1 (2 ^ n), f k ≤ ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ k)
-- Finset.le_sum_schlomilch
theorem le_sum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ range (u n), f k) ≤
      ∑ k ∈ range (u 0), f k + ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} {u : ℕ → ℕ} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu : Monotone u) (n : ℕ)
Goal: ∑ k ∈ Finset.range (u n), f k ≤ ∑ k ∈ Finset.range (u 0), f k + ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u k)
-- Finset.le_sum_condensed
theorem le_sum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ range (2 ^ n), f k) ≤ f 0 + ∑ k ∈ range n, 2 ^ k • f (2 ^ k) 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ)
Goal: ∑ k ∈ Finset.range (2 ^ n), f k ≤ f 0 + ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ k)
-- Finset.sum_schlomilch_le'
theorem sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : Monotone u) (n : ℕ) :
    (∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} {u : ℕ → ℕ} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu : Monotone u) (n : ℕ)
Goal: ∑ k ∈ Finset.range n, (u (k + 1) - u k) • f (u (k + 1)) ≤ ∑ k ∈ Finset.Ico (u 0 + 1) (u n + 1), f k
-- Finset.sum_condensed_le'
theorem sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))) ≤ ∑ k ∈ Ico 2 (2 ^ n + 1), f k 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ)
Goal: ∑ k ∈ Finset.range n, 2 ^ k • f (2 ^ (k + 1)) ≤ ∑ k ∈ Finset.Ico 2 (2 ^ n + 1), f k
-- Finset.sum_schlomilch_le
theorem sum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) (n : ℕ) :
    ∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u k) ≤
    (u 1 - u 0) • f (u 0) + C • ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} {u : ℕ → ℕ} {C : ℕ} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (h_nonneg : ∀ (n : ℕ), 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) (n : ℕ)
Goal: ∑ k ∈ Finset.range (n + 1), (u (k + 1) - u k) • f (u k) ≤
  (u 1 - u 0) • f (u 0) + C • ∑ k ∈ Finset.Ico (u 0 + 1) (u n + 1), f k
-- Finset.sum_condensed_le
theorem sum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
    (∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k)) ≤ f 1 + 2 • ∑ k ∈ Ico 2 (2 ^ n + 1), f k 
Arguments: {M : Type u_1} [OrderedAddCommMonoid M] {f : ℕ → M} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ)
Goal: ∑ k ∈ Finset.range (n + 1), 2 ^ k • f (2 ^ k) ≤ f 1 + 2 • ∑ k ∈ Finset.Ico 2 (2 ^ n + 1), f k
-- ENNReal.le_tsum_schlomilch
theorem le_tsum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu : StrictMono u) :
    ∑' k , f k ≤ ∑ k ∈ range (u 0), f k + ∑' k : ℕ, (u (k + 1) - u k) * f (u k) 
Arguments: {u : ℕ → ℕ} {f : ℕ → ENNReal} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu : StrictMono u)
Goal: ∑' (k : ℕ), f k ≤ ∑ k ∈ Finset.range (u 0), f k + ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k)
-- ENNReal.le_tsum_condensed
theorem le_tsum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
    ∑' k, f k ≤ f 0 + ∑' k : ℕ, 2 ^ k * f (2 ^ k) 
Arguments: {f : ℕ → ENNReal} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m)
Goal: ∑' (k : ℕ), f k ≤ f 0 + ∑' (k : ℕ), 2 ^ k * f (2 ^ k)
-- ENNReal.tsum_schlomilch_le
theorem tsum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (h_nonneg : ∀ n, 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u) :
    ∑' k : ℕ, (u (k + 1) - u k) * f (u k) ≤ (u 1 - u 0) * f (u 0) + C * ∑' k, f k 
Arguments: {u : ℕ → ℕ} {f : ℕ → ENNReal} {C : ℕ} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (h_nonneg : ∀ (n : ℕ), 0 ≤ f n) (hu : Monotone u) (h_succ_diff : SuccDiffBounded C u)
Goal: ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k
-- ENNReal.tsum_condensed_le
theorem tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) :
    (∑' k : ℕ, 2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' k, f k 
Arguments: {f : ℕ → ENNReal} (hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m)
Goal: ∑' (k : ℕ), 2 ^ k * f (2 ^ k) ≤ f 1 + 2 * ∑' (k : ℕ), f k
-- NNReal.summable_schlomilch_iff
theorem summable_schlomilch_iff {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ≥0}
    (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m)
    (h_pos : ∀ n, 0 < u n) (hu_strict : StrictMono u)
    (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
    (Summable fun k : ℕ => (u (k + 1) - (u k : ℝ≥0)) * f (u k)) ↔ Summable f 
Arguments: {C : ℕ} {u : ℕ → ℕ} {f : ℕ → NNReal} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u)
Goal: (Summable fun k => (↑(u (k + 1)) - ↑(u k)) * f (u k)) ↔ Summable f
-- NNReal.summable_condensed_iff
theorem summable_condensed_iff {f : ℕ → ℝ≥0} (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
    (Summable fun k : ℕ => (2 : ℝ≥0) ^ k * f (2 ^ k)) ↔ Summable f 
Arguments: {f : ℕ → NNReal} (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m)
Goal: (Summable fun k => 2 ^ k * f (2 ^ k)) ↔ Summable f
-- summable_schlomilch_iff_of_nonneg
theorem summable_schlomilch_iff_of_nonneg {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
    (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
    (hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u) :
    (Summable fun k : ℕ => (u (k + 1) - (u k : ℝ)) * f (u k)) ↔ Summable f 
Arguments: {C : ℕ} {u : ℕ → ℕ} {f : ℕ → ℝ} (h_nonneg : ∀ (n : ℕ), 0 ≤ f n) (hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ (n : ℕ), 0 < u n) (hu_strict : StrictMono u) (hC_nonzero : C ≠ 0) (h_succ_diff : SuccDiffBounded C u)
Goal: (Summable fun k => (↑(u (k + 1)) - ↑(u k)) * f (u k)) ↔ Summable f