ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FormalMultilinearSeries.summable
protected theorem summable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
    (hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => p n fun _ => 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] [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E} (hx : x ∈ EMetric.ball 0 p.radius)
Goal: Summable fun n => (p n) fun x_1 => x
-- FormalMultilinearSeries.radius_eq_top_of_summable_norm
theorem radius_eq_top_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
    (hs : ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : 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) (hs : ∀ (r : ℝ≥0), Summable fun n => ‖p n‖ * ↑r ^ n)
Goal: p.radius = ⊤
-- FormalMultilinearSeries.radius_eq_top_iff_summable_norm
theorem radius_eq_top_iff_summable_norm (p : FormalMultilinearSeries 𝕜 E F) :
    p.radius = ∞ ↔ ∀ r : ℝ≥0, Summable fun n => ‖p n‖ * (r : 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] (p : FormalMultilinearSeries 𝕜 E F)
Goal: p.radius = ⊤ ↔ ∀ (r : ℝ≥0), Summable fun n => ‖p n‖ * ↑r ^ n
-- FormalMultilinearSeries.le_mul_pow_of_radius_pos
theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius) :
    ∃ (C r : _) (hC : 0 < C) (_ : 0 < r), ∀ n, ‖p n‖ ≤ C * 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] (p : FormalMultilinearSeries 𝕜 E F) (h : 0 < p.radius)
Goal: ∃ C r, ∃ (_ : 0 < C) (_ : 0 < r), ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
-- FormalMultilinearSeries.min_radius_le_radius_add
theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
    min p.radius q.radius ≤ (p + q).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 q : FormalMultilinearSeries 𝕜 E F)
Goal: min p.radius q.radius ≤ (p + q).radius
-- FormalMultilinearSeries.radius_neg
@[simp]
theorem radius_neg (p : FormalMultilinearSeries 𝕜 E F) : (-p).radius = 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)
Goal: (-p).radius = p.radius
-- FormalMultilinearSeries.hasSum
protected theorem hasSum [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
    (hx : x ∈ EMetric.ball (0 : E) p.radius) : HasSum (fun n : ℕ => p n fun _ => x) (p.sum 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] [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x : E} (hx : x ∈ EMetric.ball 0 p.radius)
Goal: HasSum (fun n => (p n) fun x_1 => x) (p.sum x)
-- FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp
theorem radius_le_radius_continuousLinearMap_comp (p : FormalMultilinearSeries 𝕜 E F)
    (f : F →L[𝕜] G) : p.radius ≤ (f.compFormalMultilinearSeries p).radius 
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] (p : FormalMultilinearSeries 𝕜 E F) (f : F →L[𝕜] G)
Goal: p.radius ≤ (f.compFormalMultilinearSeries p).radius
-- HasFPowerSeriesOnBall.r_le
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
    Prop where
  r_le : r ≤ p.radius
  r_pos : 0 < r
  hasSum :
    ∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + 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∞} (self : HasFPowerSeriesOnBall f p x r)
Goal: r ≤ p.radius
-- HasFPowerSeriesOnBall.r_pos
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
    Prop where
  r_le : r ≤ p.radius
  r_pos : 0 < r
  hasSum :
    ∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + 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∞} (self : HasFPowerSeriesOnBall f p x r)
Goal: 0 < r
-- HasFPowerSeriesOnBall.hasSum
structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) (r : ℝ≥0∞) :
    Prop where
  r_le : r ≤ p.radius
  r_pos : 0 < r
  hasSum :
    ∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + 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∞} (self : HasFPowerSeriesOnBall f p x r) {y : E}
Goal: y ∈ EMetric.ball 0 r → HasSum (fun n => (p n) fun x => y) (f (x + y))
-- HasFPowerSeriesWithinOnBall.continuousWithinAt
structure HasFPowerSeriesWithinOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (s : Set E)
    (x : E) (r : ℝ≥0∞) : Prop where
    r_le : r ≤ p.radius
    r_pos : 0 < r
    hasSum : ∀ {y}, x + y ∈ s → y ∈ EMetric.ball (0 : E) r →
    HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
    continuousWithinAt : ContinuousWithinAt f s 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} {s : Set E} {x : E} {r : ℝ≥0∞} (self : HasFPowerSeriesWithinOnBall f p s x r)
Goal: ContinuousWithinAt f s x
-- HasFPowerSeriesWithinOnBall.r_le
structure HasFPowerSeriesWithinOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (s : Set E)
    (x : E) (r : ℝ≥0∞) : Prop where
    r_le : r ≤ p.radius
    r_pos : 0 < r
    hasSum : ∀ {y}, x + y ∈ s → y ∈ EMetric.ball (0 : E) r →
    HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
    continuousWithinAt : ContinuousWithinAt f s 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} {s : Set E} {x : E} {r : ℝ≥0∞} (self : HasFPowerSeriesWithinOnBall f p s x r)
Goal: r ≤ p.radius
-- HasFPowerSeriesWithinOnBall.hasSum
structure HasFPowerSeriesWithinOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (s : Set E)
    (x : E) (r : ℝ≥0∞) : Prop where
    r_le : r ≤ p.radius
    r_pos : 0 < r
    hasSum : ∀ {y}, x + y ∈ s → y ∈ EMetric.ball (0 : E) r →
    HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
    continuousWithinAt : ContinuousWithinAt f s 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} {s : Set E} {x : E} {r : ℝ≥0∞} (self : HasFPowerSeriesWithinOnBall f p s x r) {y : E}
Goal: x + y ∈ s → y ∈ EMetric.ball 0 r → HasSum (fun n => (p n) fun x => y) (f (x + y))
-- HasFPowerSeriesWithinOnBall.r_pos
structure HasFPowerSeriesWithinOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (s : Set E)
    (x : E) (r : ℝ≥0∞) : Prop where
    r_le : r ≤ p.radius
    r_pos : 0 < r
    hasSum : ∀ {y}, x + y ∈ s → y ∈ EMetric.ball (0 : E) r →
    HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
    continuousWithinAt : ContinuousWithinAt f s 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} {s : Set E} {x : E} {r : ℝ≥0∞} (self : HasFPowerSeriesWithinOnBall f p s x r)
Goal: 0 < r
-- HasFPowerSeriesOnBall.hasFPowerSeriesAt
theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
    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} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: HasFPowerSeriesAt f p x
-- HasFPowerSeriesAt.analyticAt
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p 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} {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFPowerSeriesAt f p x)
Goal: AnalyticAt 𝕜 f x
-- HasFPowerSeriesOnBall.analyticAt
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : 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} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: AnalyticAt 𝕜 f x
-- HasFPowerSeriesOnBall.congr
theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
    (hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p 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 g : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r) (hg : Set.EqOn f g (EMetric.ball x r))
Goal: HasFPowerSeriesOnBall g p x r
-- HasFPowerSeriesOnBall.comp_sub
theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y : E) :
    HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) 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) (y : E)
Goal: HasFPowerSeriesOnBall (fun z => f (z - y)) p (x + y) r
-- HasFPowerSeriesOnBall.hasSum_sub
theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
    (hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f 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∞} (hf : HasFPowerSeriesOnBall f p x r) {y : E} (hy : y ∈ EMetric.ball x r)
Goal: HasSum (fun n => (p n) fun x_1 => y - x) (f y)
-- HasFPowerSeriesOnBall.radius_pos
theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 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] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: 0 < p.radius
-- HasFPowerSeriesAt.radius_pos
theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 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] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFPowerSeriesAt f p x)
Goal: 0 < p.radius
-- HasFPowerSeriesOnBall.mono
theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
    (hr : r' ≤ r) : HasFPowerSeriesOnBall f p 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' : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r') (hr : r' ≤ r)
Goal: HasFPowerSeriesOnBall f p x r'
-- HasFPowerSeriesAt.congr
theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
    HasFPowerSeriesAt g 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 g : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g)
Goal: HasFPowerSeriesAt g p x
-- HasFPowerSeriesAt.eventually
protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
    ∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p 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} (hf : HasFPowerSeriesAt f p x)
Goal: ∀ᶠ (r : ℝ≥0∞) in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r
-- HasFPowerSeriesOnBall.eventually_hasSum
theorem HasFPowerSeriesOnBall.eventually_hasSum (hf : HasFPowerSeriesOnBall f p x r) :
    ∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + 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∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: ∀ᶠ (y : E) in 𝓝 0, HasSum (fun n => (p n) fun x => y) (f (x + y))
-- HasFPowerSeriesAt.eventually_hasSum
theorem HasFPowerSeriesAt.eventually_hasSum (hf : HasFPowerSeriesAt f p x) :
    ∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + 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} (hf : HasFPowerSeriesAt f p x)
Goal: ∀ᶠ (y : E) in 𝓝 0, HasSum (fun n => (p n) fun x => y) (f (x + y))
-- HasFPowerSeriesOnBall.eventually_hasSum_sub
theorem HasFPowerSeriesOnBall.eventually_hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) :
    ∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f 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∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: ∀ᶠ (y : E) in 𝓝 x, HasSum (fun n => (p n) fun x_1 => y - x) (f y)
-- HasFPowerSeriesAt.eventually_hasSum_sub
theorem HasFPowerSeriesAt.eventually_hasSum_sub (hf : HasFPowerSeriesAt f p x) :
    ∀ᶠ y in 𝓝 x, HasSum (fun n : ℕ => p n fun _ : Fin n => y - x) (f 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} (hf : HasFPowerSeriesAt f p x)
Goal: ∀ᶠ (y : E) in 𝓝 x, HasSum (fun n => (p n) fun x_1 => y - x) (f y)
-- HasFPowerSeriesOnBall.eventually_eq_zero
theorem HasFPowerSeriesOnBall.eventually_eq_zero
    (hf : HasFPowerSeriesOnBall f (0 : FormalMultilinearSeries 𝕜 E F) x r) :
    ∀ᶠ z in 𝓝 x, f z = 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} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f 0 x r)
Goal: ∀ᶠ (z : E) in 𝓝 x, f z = 0
-- HasFPowerSeriesAt.eventually_eq_zero
theorem HasFPowerSeriesAt.eventually_eq_zero
    (hf : HasFPowerSeriesAt f (0 : FormalMultilinearSeries 𝕜 E F) x) : ∀ᶠ z in 𝓝 x, f z = 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} {x : E} (hf : HasFPowerSeriesAt f 0 x)
Goal: ∀ᶠ (z : E) in 𝓝 x, f z = 0
-- hasFPowerSeriesOnBall_const
theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
    HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ 
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: HasFPowerSeriesOnBall (fun x => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤
-- hasFPowerSeriesAt_const
theorem hasFPowerSeriesAt_const {c : F} {e : E} :
    HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e 
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: HasFPowerSeriesAt (fun x => c) (constFormalMultilinearSeries 𝕜 E c) e
-- analyticAt_const
theorem analyticAt_const {v : F} : AnalyticAt 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: AnalyticAt 𝕜 (fun x => v) x
-- analyticOn_const
theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 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: AnalyticOn 𝕜 (fun x => v) s
-- HasFPowerSeriesOnBall.add
theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
    (hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) 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 g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f pf x r) (hg : HasFPowerSeriesOnBall g pg x r)
Goal: HasFPowerSeriesOnBall (f + g) (pf + pg) x r
-- HasFPowerSeriesAt.add
theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
    HasFPowerSeriesAt (f + g) (pf + pg) 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} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x)
Goal: HasFPowerSeriesAt (f + g) (pf + pg) x
-- AnalyticAt.congr
theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 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 : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g)
Goal: AnalyticAt 𝕜 g x
-- analyticAt_congr
theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 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: AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x
-- AnalyticAt.add
theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 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 : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x)
Goal: AnalyticAt 𝕜 (f + g) x
-- HasFPowerSeriesOnBall.neg
theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
    HasFPowerSeriesOnBall (-f) (-pf) 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} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f pf x r)
Goal: HasFPowerSeriesOnBall (-f) (-pf) x r
-- HasFPowerSeriesAt.neg
theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) 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)
Goal: HasFPowerSeriesAt (-f) (-pf) x
-- AnalyticAt.neg
theorem AnalyticAt.neg (hf : AnalyticAt 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 : AnalyticAt 𝕜 f x)
Goal: AnalyticAt 𝕜 (-f) x
-- HasFPowerSeriesOnBall.sub
theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
    (hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) 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 g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f pf x r) (hg : HasFPowerSeriesOnBall g pg x r)
Goal: HasFPowerSeriesOnBall (f - g) (pf - pg) x r
-- HasFPowerSeriesAt.sub
theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
    HasFPowerSeriesAt (f - g) (pf - pg) 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} {pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x)
Goal: HasFPowerSeriesAt (f - g) (pf - pg) x
-- AnalyticAt.sub
theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
    AnalyticAt 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 : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x)
Goal: AnalyticAt 𝕜 (f - g) x
-- AnalyticOn.mono
theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : 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 t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t)
Goal: AnalyticOn 𝕜 f s
-- AnalyticOn.congr'
theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] 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} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g)
Goal: AnalyticOn 𝕜 g s
-- analyticOn_congr'
theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] 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} (h : f =ᶠ[𝓝ˢ s] g)
Goal: AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s