ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AnalyticOn.congr
theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
    AnalyticOn 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} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : Set.EqOn f g s)
Goal: AnalyticOn 𝕜 g s
-- analyticOn_congr
theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
    AnalyticOn 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} (hs : IsOpen s) (h : Set.EqOn f g s)
Goal: AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s
-- AnalyticOn.add
theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
    AnalyticOn 𝕜 (f + 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 : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s)
Goal: AnalyticOn 𝕜 (f + g) s
-- AnalyticOn.neg
theorem AnalyticOn.neg {s : Set E} (hf : AnalyticOn 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 : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (-f) s
-- AnalyticOn.sub
theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
    AnalyticOn 𝕜 (f - 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 : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s)
Goal: AnalyticOn 𝕜 (f - g) s
-- HasFPowerSeriesOnBall.coeff_zero
theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
    pf 0 v = 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} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E)
Goal: (pf 0) v = f x
-- HasFPowerSeriesAt.coeff_zero
theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
    pf 0 v = 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} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E)
Goal: (pf 0) v = f x
-- ContinuousLinearMap.comp_hasFPowerSeriesOnBall
theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
    (h : HasFPowerSeriesOnBall f p x r) :
    HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (g : F →L[𝕜] G) (h : HasFPowerSeriesOnBall f p x r)
Goal: HasFPowerSeriesOnBall (⇑g ∘ f) (g.compFormalMultilinearSeries p) x r
-- ContinuousLinearMap.comp_analyticOn
theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
    AnalyticOn 𝕜 (g ∘ f) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (⇑g ∘ f) s
-- HasFPowerSeriesOnBall.uniform_geometric_approx'
theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
    (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
    ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
      ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ 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∞} {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r) (h : ↑r' < r)
Goal: ∃ a ∈ Set.Ioo 0 1,
  ∃ C > 0, ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / ↑r')) ^ n
-- HasFPowerSeriesOnBall.uniform_geometric_approx
theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
    (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
    ∃ a ∈ Ioo (0 : ℝ) 1,
      ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ 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∞} {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r) (h : ↑r' < r)
Goal: ∃ a ∈ Set.Ioo 0 1, ∃ C > 0, ∀ y ∈ Metric.ball 0 ↑r', ∀ (n : ℕ), ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n
-- HasFPowerSeriesAt.isBigO_sub_partialSum_pow
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
    (fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ 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} (hf : HasFPowerSeriesAt f p x) (n : ℕ)
Goal: (fun y => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n
-- HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
    (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
    (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
      fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ 
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∞} (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r)
Goal: (fun y => f y.1 - f y.2 - (p 1) fun x => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖
-- HasFPowerSeriesOnBall.image_sub_sub_deriv_le
theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
    (hr : r' < r) :
    ∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
      ‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ 
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∞} (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r)
Goal: ∃ C,
  ∀ y ∈ EMetric.ball x r',
    ∀ z ∈ EMetric.ball x r', ‖f y - f z - (p 1) fun x => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖
-- HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub
theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
    (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
      ‖y - (x, x)‖ * ‖y.1 - y.2‖ 
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} (hf : HasFPowerSeriesAt f p x)
Goal: (fun y => f y.1 - f y.2 - (p 1) fun x => y.1 - y.2) =O[𝓝 (x, x)] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖
-- HasFPowerSeriesOnBall.tendstoUniformlyOn
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
    (h : (r' : ℝ≥0∞) < r) :
    TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
      (Metric.ball (0 : E) 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∞} {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r) (h : ↑r' < r)
Goal: TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) Filter.atTop (Metric.ball 0 ↑r')
-- HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
    TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
      (EMetric.ball (0 : E) 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∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) Filter.atTop (EMetric.ball 0 r)
-- HasFPowerSeriesOnBall.tendstoUniformlyOn'
theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
    (h : (r' : ℝ≥0∞) < r) :
    TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) 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∞} {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r) (h : ↑r' < r)
Goal: TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f Filter.atTop (Metric.ball x ↑r')
-- HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn'
theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
    TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
      (EMetric.ball (x : E) 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∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f Filter.atTop (EMetric.ball x r)
-- HasFPowerSeriesOnBall.continuousOn
protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
    ContinuousOn f (EMetric.ball x 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∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: ContinuousOn f (EMetric.ball x r)
-- HasFPowerSeriesAt.continuousAt
protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
    ContinuousAt 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} (hf : HasFPowerSeriesAt f p x)
Goal: ContinuousAt f x
-- AnalyticAt.continuousAt
protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt 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 : AnalyticAt 𝕜 f x)
Goal: ContinuousAt f x
-- AnalyticOn.continuousOn
protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn 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 : AnalyticOn 𝕜 f s)
Goal: ContinuousOn f s
-- AnalyticOn.continuous
theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f 
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} (fa : AnalyticOn 𝕜 f Set.univ)
Goal: Continuous f
-- FormalMultilinearSeries.hasFPowerSeriesOnBall
protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
    (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
    HasFPowerSeriesOnBall p.sum p 0 p.radius 
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] [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius)
Goal: HasFPowerSeriesOnBall p.sum p 0 p.radius
-- HasFPowerSeriesOnBall.sum
theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
    (hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y 
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∞} (h : HasFPowerSeriesOnBall f p x r) {y : E} (hy : y ∈ EMetric.ball 0 r)
Goal: f (x + y) = p.sum y
-- FormalMultilinearSeries.continuousOn
protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
    ContinuousOn p.sum (EMetric.ball 0 p.radius) 
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] {p : FormalMultilinearSeries 𝕜 E F} [CompleteSpace F]
Goal: ContinuousOn p.sum (EMetric.ball 0 p.radius)
-- Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero
theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
    (h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 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] {n : ℕ} {p : ContinuousMultilinearMap 𝕜 (fun i => E) F} (h : (fun y => p fun x => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E)
Goal: (p fun x => y) = 0
-- HasFPowerSeriesAt.apply_eq_zero
theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
    (h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 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] {p : FormalMultilinearSeries 𝕜 E F} {x : E} (h : HasFPowerSeriesAt 0 p x) (n : ℕ) (y : E)
Goal: ((p n) fun x => y) = 0
-- HasFPowerSeriesAt.eq_zero
theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
    (h : HasFPowerSeriesAt 0 p x) : p = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜} (h : HasFPowerSeriesAt 0 p x)
Goal: p = 0
-- HasFPowerSeriesAt.eq_formalMultilinearSeries
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
    {f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x)
Goal: p₁ = p₂
-- HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
    {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
    (hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = g z)
Goal: p = q
-- HasFPowerSeriesAt.eq_zero_of_eventually
theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
    {x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0)
Goal: p = 0
-- HasFPowerSeriesOnBall.exchange_radius
theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
    {r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
    (h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁) (h₂ : HasFPowerSeriesOnBall f p₂ x r₂)
Goal: HasFPowerSeriesOnBall f p₁ x r₂
-- HasFPowerSeriesOnBall.r_eq_top_of_exists
theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
    {p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
    (h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
      HasFPowerSeriesOnBall f p' x r') :
    HasFPowerSeriesOnBall f p x ∞ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜} {p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r) (h' : ∀ (r' : ℝ≥0), 0 < r' → ∃ p', HasFPowerSeriesOnBall f p' x ↑r')
Goal: HasFPowerSeriesOnBall f p x ⊤
-- FormalMultilinearSeries.changeOriginSeriesTerm_apply
theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
    (x y : E) :
    (p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
      p (k + l) (s.piecewise (fun _ => x) fun _ => y) 
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] (p : FormalMultilinearSeries 𝕜 E F) (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) (x y : E)
Goal: (((p.changeOriginSeriesTerm k l s hs) fun x_1 => x) fun x => y) = (p (k + l)) (s.piecewise (fun x_1 => x) fun x => y)
-- FormalMultilinearSeries.norm_changeOriginSeriesTerm
@[simp]
theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
    ‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ 
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] (p : FormalMultilinearSeries 𝕜 E F) (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
Goal: ‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖
-- FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm
@[simp]
theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
    ‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ 
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] (p : FormalMultilinearSeries 𝕜 E F) (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
Goal: ‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊
-- FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
    (hs : s.card = l) (x y : E) :
    ‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
      ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k 
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] (p : FormalMultilinearSeries 𝕜 E F) (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) (x y : E)
Goal: ‖((p.changeOriginSeriesTerm k l s hs) fun x_1 => x) fun x => y‖₊ ≤ ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k
-- FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum
theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
    ‖p.changeOriginSeries k l‖₊ ≤
      ∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ 
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] (p : FormalMultilinearSeries 𝕜 E F) (k l : ℕ)
Goal: ‖p.changeOriginSeries k l‖₊ ≤ ∑' (x : { s // s.card = l }), ‖p (k + l)‖₊
-- FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum
theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
    ‖p.changeOriginSeries k l fun _ => x‖₊ ≤
      ∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l 
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] (p : FormalMultilinearSeries 𝕜 E F) (k l : ℕ) (x : E)
Goal: ‖(p.changeOriginSeries k l) fun x_1 => x‖₊ ≤ ∑' (x_1 : { s // s.card = l }), ‖p (k + l)‖₊ * ‖x‖₊ ^ l
-- FormalMultilinearSeries.changeOriginIndexEquiv_symm_apply_fst
@[simps]
def changeOriginIndexEquiv :
    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) 
Arguments: (s : (n : ℕ) × Finset (Fin n))
Goal: (FormalMultilinearSeries.changeOriginIndexEquiv.symm s).fst = s.fst - s.snd.card
-- FormalMultilinearSeries.changeOriginIndexEquiv_symm_apply_snd_snd_coe
@[simps]
def changeOriginIndexEquiv :
    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) 
Arguments: (s : (n : ℕ) × Finset (Fin n))
Goal: ↑(FormalMultilinearSeries.changeOriginIndexEquiv.symm s).snd.snd = Finset.map (finCongr ⋯).toEmbedding s.snd
-- FormalMultilinearSeries.changeOriginIndexEquiv_symm_apply_snd_fst
@[simps]
def changeOriginIndexEquiv :
    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) 
Arguments: (s : (n : ℕ) × Finset (Fin n))
Goal: (FormalMultilinearSeries.changeOriginIndexEquiv.symm s).snd.fst = s.snd.card
-- FormalMultilinearSeries.changeOriginIndexEquiv_apply_snd
@[simps]
def changeOriginIndexEquiv :
    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) 
Arguments: (s : (k : ℕ) × (l : ℕ) × { s // s.card = l })
Goal: (FormalMultilinearSeries.changeOriginIndexEquiv s).snd = ↑s.snd.snd
-- FormalMultilinearSeries.changeOriginIndexEquiv_apply_fst
@[simps]
def changeOriginIndexEquiv :
    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) 
Arguments: (s : (k : ℕ) × (l : ℕ) × { s // s.card = l })
Goal: (FormalMultilinearSeries.changeOriginIndexEquiv s).fst = s.fst + s.snd.fst
-- FormalMultilinearSeries.changeOriginSeriesTerm_changeOriginIndexEquiv_symm
lemma changeOriginSeriesTerm_changeOriginIndexEquiv_symm (n t) :
    let s := changeOriginIndexEquiv.symm ⟨n, t⟩
    p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ ↦ x) (fun _ ↦ y) =
    p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) 
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] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} (n : ℕ) (t : Finset (Fin n))
Goal: let s := FormalMultilinearSeries.changeOriginIndexEquiv.symm ⟨n, t⟩;
(((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y) =
  (p n) (t.piecewise (fun x_1 => x) fun x => y)