ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- gramSchmidt_ne_zero_coe
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 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} (n : ι) (h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val))
Goal: gramSchmidt 𝕜 f n ≠ 0
-- gramSchmidt_ne_zero
theorem gramSchmidt_ne_zero {f : ι → E} (n : ι) (h₀ : LinearIndependent 𝕜 f) :
    gramSchmidt 𝕜 f n ≠ 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} (n : ι) (h₀ : LinearIndependent 𝕜 f)
Goal: gramSchmidt 𝕜 f n ≠ 0
-- gramSchmidt_triangular
theorem gramSchmidt_triangular {i j : ι} (hij : i < j) (b : Basis ι 𝕜 E) :
    b.repr (gramSchmidt 𝕜 b i) j = 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] {i j : ι} (hij : i < j) (b : Basis ι 𝕜 E)
Goal: (b.repr (gramSchmidt 𝕜 (⇑b) i)) j = 0
-- gramSchmidt_linearIndependent
theorem gramSchmidt_linearIndependent {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
    LinearIndependent 𝕜 (gramSchmidt 𝕜 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} (h₀ : LinearIndependent 𝕜 f)
Goal: LinearIndependent 𝕜 (gramSchmidt 𝕜 f)
-- coe_gramSchmidtBasis
theorem coe_gramSchmidtBasis (b : Basis ι 𝕜 E) : (gramSchmidtBasis b : ι → E) = gramSchmidt 𝕜 b 
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] (b : Basis ι 𝕜 E)
Goal: ⇑(gramSchmidtBasis b) = gramSchmidt 𝕜 ⇑b
-- gramSchmidtNormed_unit_length_coe
theorem gramSchmidtNormed_unit_length_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : ‖gramSchmidtNormed 𝕜 f n‖ = 1 
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) (h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val))
Goal: ‖gramSchmidtNormed 𝕜 f n‖ = 1
-- gramSchmidtNormed_unit_length
theorem gramSchmidtNormed_unit_length {f : ι → E} (n : ι) (h₀ : LinearIndependent 𝕜 f) :
    ‖gramSchmidtNormed 𝕜 f n‖ = 1 
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) (h₀ : LinearIndependent 𝕜 f)
Goal: ‖gramSchmidtNormed 𝕜 f n‖ = 1
-- gramSchmidtNormed_unit_length'
theorem gramSchmidtNormed_unit_length' {f : ι → E} {n : ι} (hn : gramSchmidtNormed 𝕜 f n ≠ 0) :
    ‖gramSchmidtNormed 𝕜 f n‖ = 1 
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} (hn : gramSchmidtNormed 𝕜 f n ≠ 0)
Goal: ‖gramSchmidtNormed 𝕜 f n‖ = 1
-- gramSchmidt_orthonormal
theorem gramSchmidt_orthonormal {f : ι → E} (h₀ : LinearIndependent 𝕜 f) :
    Orthonormal 𝕜 (gramSchmidtNormed 𝕜 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} (h₀ : LinearIndependent 𝕜 f)
Goal: Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f)
-- gramSchmidt_orthonormal'
theorem gramSchmidt_orthonormal' (f : ι → E) :
    Orthonormal 𝕜 fun i : { i | gramSchmidtNormed 𝕜 f i ≠ 0 } => gramSchmidtNormed 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)
Goal: Orthonormal 𝕜 fun i => gramSchmidtNormed 𝕜 f ↑i
-- span_gramSchmidtNormed
theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
    span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s) 
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) (s : Set ι)
Goal: Submodule.span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = Submodule.span 𝕜 (gramSchmidt 𝕜 f '' s)
-- span_gramSchmidtNormed_range
theorem span_gramSchmidtNormed_range (f : ι → E) :
    span 𝕜 (range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (range (gramSchmidt 𝕜 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 (gramSchmidtNormed 𝕜 f)) = Submodule.span 𝕜 (Set.range (gramSchmidt 𝕜 f))
-- gramSchmidtOrthonormalBasis_apply
theorem gramSchmidtOrthonormalBasis_apply {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0) :
    gramSchmidtOrthonormalBasis h f i = gramSchmidtNormed 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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i ≠ 0)
Goal: (gramSchmidtOrthonormalBasis h f) i = gramSchmidtNormed 𝕜 f i
-- gramSchmidtOrthonormalBasis_apply_of_orthogonal
theorem gramSchmidtOrthonormalBasis_apply_of_orthogonal {f : ι → E}
    (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) {i : ι} (hi : f i ≠ 0) :
    gramSchmidtOrthonormalBasis h f i = (‖f i‖⁻¹ : 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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫_𝕜 = 0) {i : ι} (hi : f i ≠ 0)
Goal: (gramSchmidtOrthonormalBasis h f) i = (↑‖f i‖)⁻¹ • f i
-- inner_gramSchmidtOrthonormalBasis_eq_zero
theorem inner_gramSchmidtOrthonormalBasis_eq_zero {f : ι → E} {i : ι}
    (hi : gramSchmidtNormed 𝕜 f i = 0) (j : ι) : ⟪gramSchmidtOrthonormalBasis h f i, f j⟫ = 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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) {f : ι → E} {i : ι} (hi : gramSchmidtNormed 𝕜 f i = 0) (j : ι)
Goal: ⟪(gramSchmidtOrthonormalBasis h f) i, f j⟫_𝕜 = 0
-- gramSchmidtOrthonormalBasis_inv_triangular
theorem gramSchmidtOrthonormalBasis_inv_triangular {i j : ι} (hij : i < j) :
    ⟪gramSchmidtOrthonormalBasis h f j, f 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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) (f : ι → E) {i j : ι} (hij : i < j)
Goal: ⟪(gramSchmidtOrthonormalBasis h f) j, f i⟫_𝕜 = 0
-- gramSchmidtOrthonormalBasis_inv_triangular'
theorem gramSchmidtOrthonormalBasis_inv_triangular' {i j : ι} (hij : i < j) :
    (gramSchmidtOrthonormalBasis h f).repr (f i) j = 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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) (f : ι → E) {i j : ι} (hij : i < j)
Goal: (gramSchmidtOrthonormalBasis h f).repr (f i) j = 0
-- gramSchmidtOrthonormalBasis_inv_blockTriangular
theorem gramSchmidtOrthonormalBasis_inv_blockTriangular :
    ((gramSchmidtOrthonormalBasis h f).toBasis.toMatrix f).BlockTriangular id 
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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) (f : ι → E)
Goal: ((gramSchmidtOrthonormalBasis h f).toBasis.toMatrix f).BlockTriangular id
-- gramSchmidtOrthonormalBasis_det
theorem gramSchmidtOrthonormalBasis_det [DecidableEq ι] :
    (gramSchmidtOrthonormalBasis h f).toBasis.det f =
      ∏ i, ⟪gramSchmidtOrthonormalBasis h f i, 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] [Fintype ι] [FiniteDimensional 𝕜 E] (h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) (f : ι → E) [DecidableEq ι]
Goal: (gramSchmidtOrthonormalBasis h f).toBasis.det f = ∏ i : ι, ⟪(gramSchmidtOrthonormalBasis h f) i, f i⟫_𝕜
-- HasFiniteFPowerSeriesOnBall.finite
structure HasFiniteFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E)
    (n : ℕ) (r : ℝ≥0∞) extends HasFPowerSeriesOnBall f p x r : Prop where
  finite : ∀ (m : ℕ), n ≤ m → p m = 0
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} {r : ℝ≥0∞} (self : HasFiniteFPowerSeriesOnBall f p x n r) (m : ℕ)
Goal: n ≤ m → p m = 0
-- HasFiniteFPowerSeriesOnBall.mk'
theorem HasFiniteFPowerSeriesOnBall.mk' {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E}
    {n : ℕ} {r : ℝ≥0∞} (finite : ∀ (m : ℕ), n ≤ m → p m = 0) (pos : 0 < r)
    (sum_eq : ∀ y ∈ EMetric.ball 0 r, (∑ i ∈ Finset.range n, p i fun _ ↦ y) = f (x + y)) :
    HasFiniteFPowerSeriesOnBall f p x n r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} {r : ℝ≥0∞} (finite : ∀ (m : ℕ), n ≤ m → p m = 0) (pos : 0 < r) (sum_eq : ∀ y ∈ EMetric.ball 0 r, (∑ i ∈ Finset.range n, (p i) fun x => y) = f (x + y))
Goal: HasFiniteFPowerSeriesOnBall f p x n r
-- HasFiniteFPowerSeriesAt.toHasFPowerSeriesAt
theorem HasFiniteFPowerSeriesAt.toHasFPowerSeriesAt
    (hf : HasFiniteFPowerSeriesAt f p x n) : HasFPowerSeriesAt f p x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f p x n)
Goal: HasFPowerSeriesAt f p x
-- HasFiniteFPowerSeriesAt.finite
theorem HasFiniteFPowerSeriesAt.finite (hf : HasFiniteFPowerSeriesAt f p x n) :
    ∀ m : ℕ, n ≤ m → p m = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f p x n) (m : ℕ)
Goal: n ≤ m → p m = 0
-- HasFiniteFPowerSeriesOnBall.hasFiniteFPowerSeriesAt
theorem HasFiniteFPowerSeriesOnBall.hasFiniteFPowerSeriesAt
    (hf : HasFiniteFPowerSeriesOnBall f p x n r) :
    HasFiniteFPowerSeriesAt f p x n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: HasFiniteFPowerSeriesAt f p x n
-- HasFiniteFPowerSeriesAt.cPolynomialAt
theorem HasFiniteFPowerSeriesAt.cPolynomialAt (hf : HasFiniteFPowerSeriesAt f p x n) :
    CPolynomialAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f p x n)
Goal: CPolynomialAt 𝕜 f x
-- HasFiniteFPowerSeriesOnBall.cPolynomialAt
theorem HasFiniteFPowerSeriesOnBall.cPolynomialAt (hf : HasFiniteFPowerSeriesOnBall f p x n r) :
    CPolynomialAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: CPolynomialAt 𝕜 f x
-- CPolynomialAt.analyticAt
theorem CPolynomialAt.analyticAt (hf : CPolynomialAt 𝕜 f x) : AnalyticAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (hf : CPolynomialAt 𝕜 f x)
Goal: AnalyticAt 𝕜 f x
-- CPolynomialOn.analyticOn
theorem CPolynomialOn.analyticOn {s : Set E} (hf : CPolynomialOn 𝕜 f s) : AnalyticOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : CPolynomialOn 𝕜 f s)
Goal: AnalyticOn 𝕜 f s
-- HasFiniteFPowerSeriesOnBall.congr
theorem HasFiniteFPowerSeriesOnBall.congr (hf : HasFiniteFPowerSeriesOnBall f p x n r)
    (hg : EqOn f g (EMetric.ball x r)) : HasFiniteFPowerSeriesOnBall g p x n r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (hg : Set.EqOn f g (EMetric.ball x r))
Goal: HasFiniteFPowerSeriesOnBall g p x n r
-- HasFiniteFPowerSeriesOnBall.comp_sub
theorem HasFiniteFPowerSeriesOnBall.comp_sub (hf : HasFiniteFPowerSeriesOnBall f p x n r) (y : E) :
    HasFiniteFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) n r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (y : E)
Goal: HasFiniteFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) n r
-- HasFiniteFPowerSeriesOnBall.mono
theorem HasFiniteFPowerSeriesOnBall.mono (hf : HasFiniteFPowerSeriesOnBall f p x n r)
    (r'_pos : 0 < r') (hr : r' ≤ r) : HasFiniteFPowerSeriesOnBall f p x n r' 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (r'_pos : 0 < r') (hr : r' ≤ r)
Goal: HasFiniteFPowerSeriesOnBall f p x n r'
-- HasFiniteFPowerSeriesAt.congr
theorem HasFiniteFPowerSeriesAt.congr (hf : HasFiniteFPowerSeriesAt f p x n) (hg : f =ᶠ[𝓝 x] g) :
    HasFiniteFPowerSeriesAt g p x n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f p x n) (hg : f =ᶠ[𝓝 x] g)
Goal: HasFiniteFPowerSeriesAt g p x n
-- HasFiniteFPowerSeriesAt.eventually
protected theorem HasFiniteFPowerSeriesAt.eventually (hf : HasFiniteFPowerSeriesAt f p x n) :
    ∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFiniteFPowerSeriesOnBall f p x n r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f p x n)
Goal: ∀ᶠ (r : ℝ≥0∞) in 𝓝[>] 0, HasFiniteFPowerSeriesOnBall f p x n r
-- hasFiniteFPowerSeriesOnBall_const
theorem hasFiniteFPowerSeriesOnBall_const {c : F} {e : E} :
    HasFiniteFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e 1 ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : F} {e : E}
Goal: HasFiniteFPowerSeriesOnBall (fun x => c) (constFormalMultilinearSeries 𝕜 E c) e 1 ⊤
-- hasFiniteFPowerSeriesAt_const
theorem hasFiniteFPowerSeriesAt_const {c : F} {e : E} :
    HasFiniteFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : F} {e : E}
Goal: HasFiniteFPowerSeriesAt (fun x => c) (constFormalMultilinearSeries 𝕜 E c) e 1
-- CPolynomialAt_const
theorem CPolynomialAt_const {v : F} : CPolynomialAt 𝕜 (fun _ => v) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {v : F}
Goal: CPolynomialAt 𝕜 (fun x => v) x
-- CPolynomialOn_const
theorem CPolynomialOn_const {v : F} {s : Set E} : CPolynomialOn 𝕜 (fun _ => v) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {v : F} {s : Set E}
Goal: CPolynomialOn 𝕜 (fun x => v) s
-- HasFiniteFPowerSeriesOnBall.add
theorem HasFiniteFPowerSeriesOnBall.add (hf : HasFiniteFPowerSeriesOnBall f pf x n r)
    (hg : HasFiniteFPowerSeriesOnBall g pg x m r) :
    HasFiniteFPowerSeriesOnBall (f + g) (pf + pg) x (max n m) r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n m : ℕ} (hf : HasFiniteFPowerSeriesOnBall f pf x n r) (hg : HasFiniteFPowerSeriesOnBall g pg x m r)
Goal: HasFiniteFPowerSeriesOnBall (f + g) (pf + pg) x (max n m) r
-- HasFiniteFPowerSeriesAt.add
theorem HasFiniteFPowerSeriesAt.add (hf : HasFiniteFPowerSeriesAt f pf x n)
    (hg : HasFiniteFPowerSeriesAt g pg x m) :
    HasFiniteFPowerSeriesAt (f + g) (pf + pg) x (max n m) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {n m : ℕ} (hf : HasFiniteFPowerSeriesAt f pf x n) (hg : HasFiniteFPowerSeriesAt g pg x m)
Goal: HasFiniteFPowerSeriesAt (f + g) (pf + pg) x (max n m)
-- CPolynomialAt.congr
theorem CPolynomialAt.congr (hf : CPolynomialAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : CPolynomialAt 𝕜 g x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {x : E} (hf : CPolynomialAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g)
Goal: CPolynomialAt 𝕜 g x
-- CPolynomialAt_congr
theorem CPolynomialAt_congr (h : f =ᶠ[𝓝 x] g) : CPolynomialAt 𝕜 f x ↔ CPolynomialAt 𝕜 g x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {x : E} (h : f =ᶠ[𝓝 x] g)
Goal: CPolynomialAt 𝕜 f x ↔ CPolynomialAt 𝕜 g x
-- CPolynomialAt.add
theorem CPolynomialAt.add (hf : CPolynomialAt 𝕜 f x) (hg : CPolynomialAt 𝕜 g x) :
    CPolynomialAt 𝕜 (f + g) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {x : E} (hf : CPolynomialAt 𝕜 f x) (hg : CPolynomialAt 𝕜 g x)
Goal: CPolynomialAt 𝕜 (f + g) x
-- HasFiniteFPowerSeriesOnBall.neg
theorem HasFiniteFPowerSeriesOnBall.neg (hf : HasFiniteFPowerSeriesOnBall f pf x n r) :
    HasFiniteFPowerSeriesOnBall (-f) (-pf) x n r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f pf x n r)
Goal: HasFiniteFPowerSeriesOnBall (-f) (-pf) x n r
-- HasFiniteFPowerSeriesAt.neg
theorem HasFiniteFPowerSeriesAt.neg (hf : HasFiniteFPowerSeriesAt f pf x n) :
    HasFiniteFPowerSeriesAt (-f) (-pf) x n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f pf x n)
Goal: HasFiniteFPowerSeriesAt (-f) (-pf) x n
-- CPolynomialAt.neg
theorem CPolynomialAt.neg (hf : CPolynomialAt 𝕜 f x) : CPolynomialAt 𝕜 (-f) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (hf : CPolynomialAt 𝕜 f x)
Goal: CPolynomialAt 𝕜 (-f) x
-- HasFiniteFPowerSeriesOnBall.sub
theorem HasFiniteFPowerSeriesOnBall.sub (hf : HasFiniteFPowerSeriesOnBall f pf x n r)
    (hg : HasFiniteFPowerSeriesOnBall g pg x m r) :
    HasFiniteFPowerSeriesOnBall (f - g) (pf - pg) x (max n m) r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n m : ℕ} (hf : HasFiniteFPowerSeriesOnBall f pf x n r) (hg : HasFiniteFPowerSeriesOnBall g pg x m r)
Goal: HasFiniteFPowerSeriesOnBall (f - g) (pf - pg) x (max n m) r
-- HasFiniteFPowerSeriesAt.sub
theorem HasFiniteFPowerSeriesAt.sub (hf : HasFiniteFPowerSeriesAt f pf x n)
    (hg : HasFiniteFPowerSeriesAt g pg x m) :
    HasFiniteFPowerSeriesAt (f - g) (pf - pg) x (max n m) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {n m : ℕ} (hf : HasFiniteFPowerSeriesAt f pf x n) (hg : HasFiniteFPowerSeriesAt g pg x m)
Goal: HasFiniteFPowerSeriesAt (f - g) (pf - pg) x (max n m)
-- CPolynomialAt.sub
theorem CPolynomialAt.sub (hf : CPolynomialAt 𝕜 f x) (hg : CPolynomialAt 𝕜 g x) :
    CPolynomialAt 𝕜 (f - g) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {x : E} (hf : CPolynomialAt 𝕜 f x) (hg : CPolynomialAt 𝕜 g x)
Goal: CPolynomialAt 𝕜 (f - g) x
-- CPolynomialOn.mono
theorem CPolynomialOn.mono {s t : Set E} (hf : CPolynomialOn 𝕜 f t) (hst : s ⊆ t) :
    CPolynomialOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s t : Set E} (hf : CPolynomialOn 𝕜 f t) (hst : s ⊆ t)
Goal: CPolynomialOn 𝕜 f s
-- CPolynomialOn.congr'
theorem CPolynomialOn.congr' {s : Set E} (hf : CPolynomialOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
    CPolynomialOn 𝕜 g s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hf : CPolynomialOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g)
Goal: CPolynomialOn 𝕜 g s