ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeromorphicAt.smul
lemma smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f • g) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x)
Goal: MeromorphicAt (f • g) x
-- MeromorphicAt.mul
lemma mul {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f * g) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x)
Goal: MeromorphicAt (f * g) x
-- MeromorphicAt.neg
lemma neg {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) : MeromorphicAt (-f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x)
Goal: MeromorphicAt (-f) x
-- MeromorphicAt.neg_iff
@[simp]
lemma neg_iff {f : 𝕜 → E} {x : 𝕜} :
    MeromorphicAt (-f) x ↔ MeromorphicAt f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜}
Goal: MeromorphicAt (-f) x ↔ MeromorphicAt f x
-- MeromorphicAt.sub
lemma sub {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f - g) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x)
Goal: MeromorphicAt (f - g) x
-- MeromorphicAt.congr
lemma congr {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hfg : f =ᶠ[𝓝[≠] x] g) :
    MeromorphicAt g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hfg : f =ᶠ[𝓝[≠] x] g)
Goal: MeromorphicAt g x
-- MeromorphicAt.inv
lemma inv {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) : MeromorphicAt f⁻¹ x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x)
Goal: MeromorphicAt f⁻¹ x
-- MeromorphicAt.inv_iff
@[simp]
lemma inv_iff {f : 𝕜 → 𝕜} {x : 𝕜} :
    MeromorphicAt f⁻¹ x ↔ MeromorphicAt f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜}
Goal: MeromorphicAt f⁻¹ x ↔ MeromorphicAt f x
-- MeromorphicAt.div
lemma div {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
    MeromorphicAt (f / g) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x)
Goal: MeromorphicAt (f / g) x
-- MeromorphicAt.pow
lemma pow {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℕ) : MeromorphicAt (f ^ n) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℕ)
Goal: MeromorphicAt (f ^ n) x
-- MeromorphicAt.zpow
lemma zpow {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ) : MeromorphicAt (f ^ n) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ)
Goal: MeromorphicAt (f ^ n) x
-- MeromorphicAt.order_eq_top_iff
lemma order_eq_top_iff {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) :
    hf.order = ⊤ ↔ ∀ᶠ z in 𝓝[≠] x, f z = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x)
Goal: hf.order = ⊤ ↔ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
-- MeromorphicAt.order_eq_int_iff
lemma order_eq_int_iff {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ) : hf.order = n ↔
    ∃ g : 𝕜 → E, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ z in 𝓝[≠] x, f z = (z - x) ^ n • g z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ)
Goal: hf.order = ↑n ↔ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
-- AnalyticAt.meromorphicAt_order
lemma _root_.AnalyticAt.meromorphicAt_order {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜 f x) :
    hf.meromorphicAt.order = hf.order.map (↑) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜 f x)
Goal: ⋯.order = WithTop.map Nat.cast hf.order
-- MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt
lemma iff_eventuallyEq_zpow_smul_analyticAt {f : 𝕜 → E} {x : 𝕜} : MeromorphicAt f x ↔
    ∃ (n : ℤ) (g : 𝕜 → E), AnalyticAt 𝕜 g x ∧ ∀ᶠ z in 𝓝[≠] x, f z = (z - x) ^ n • g z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜}
Goal: MeromorphicAt f x ↔ ∃ n g, AnalyticAt 𝕜 g x ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z
-- AnalyticOn.meromorphicOn
lemma AnalyticOn.meromorphicOn {f : 𝕜 → E} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) :
    MeromorphicOn f U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U)
Goal: MeromorphicOn f U
-- MeromorphicOn.id
lemma id {U : Set 𝕜} : MeromorphicOn id U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {U : Set 𝕜}
Goal: MeromorphicOn id U
-- MeromorphicOn.const
lemma const (e : E) {U : Set 𝕜} : MeromorphicOn (fun _ ↦ e) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (e : E) {U : Set 𝕜}
Goal: MeromorphicOn (fun x => e) U
-- MeromorphicOn.mono_set
lemma mono_set {V : Set 𝕜} (hv : V ⊆ U) : MeromorphicOn f V 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {U : Set 𝕜} (hf : MeromorphicOn f U) {V : Set 𝕜} (hv : V ⊆ U)
Goal: MeromorphicOn f V
-- MeromorphicOn.add
lemma add : MeromorphicOn (f + g) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {U : Set 𝕜} (hf : MeromorphicOn f U) (hg : MeromorphicOn g U)
Goal: MeromorphicOn (f + g) U
-- MeromorphicOn.sub
lemma sub : MeromorphicOn (f - g) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {U : Set 𝕜} (hf : MeromorphicOn f U) (hg : MeromorphicOn g U)
Goal: MeromorphicOn (f - g) U
-- MeromorphicOn.neg
lemma neg : MeromorphicOn (-f) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {U : Set 𝕜} (hf : MeromorphicOn f U)
Goal: MeromorphicOn (-f) U
-- MeromorphicOn.neg_iff
@[simp] lemma neg_iff : MeromorphicOn (-f) U ↔ MeromorphicOn f U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {U : Set 𝕜}
Goal: MeromorphicOn (-f) U ↔ MeromorphicOn f U
-- MeromorphicOn.smul
lemma smul : MeromorphicOn (s • f) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : 𝕜 → 𝕜} {f : 𝕜 → E} {U : Set 𝕜} (hs : MeromorphicOn s U) (hf : MeromorphicOn f U)
Goal: MeromorphicOn (s • f) U
-- MeromorphicOn.mul
lemma mul : MeromorphicOn (s * t) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {s t : 𝕜 → 𝕜} {U : Set 𝕜} (hs : MeromorphicOn s U) (ht : MeromorphicOn t U)
Goal: MeromorphicOn (s * t) U
-- MeromorphicOn.inv
lemma inv : MeromorphicOn s⁻¹ U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {s : 𝕜 → 𝕜} {U : Set 𝕜} (hs : MeromorphicOn s U)
Goal: MeromorphicOn s⁻¹ U
-- MeromorphicOn.inv_iff
@[simp] lemma inv_iff : MeromorphicOn s⁻¹ U ↔ MeromorphicOn s U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {s : 𝕜 → 𝕜} {U : Set 𝕜}
Goal: MeromorphicOn s⁻¹ U ↔ MeromorphicOn s U
-- MeromorphicOn.div
lemma div : MeromorphicOn (s / t) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {s t : 𝕜 → 𝕜} {U : Set 𝕜} (hs : MeromorphicOn s U) (ht : MeromorphicOn t U)
Goal: MeromorphicOn (s / t) U
-- MeromorphicOn.pow
lemma pow (n : ℕ) : MeromorphicOn (s ^ n) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {s : 𝕜 → 𝕜} {U : Set 𝕜} (hs : MeromorphicOn s U) (n : ℕ)
Goal: MeromorphicOn (s ^ n) U
-- MeromorphicOn.zpow
lemma zpow (n : ℤ) : MeromorphicOn (s ^ n) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {s : 𝕜 → 𝕜} {U : Set 𝕜} (hs : MeromorphicOn s U) (n : ℤ)
Goal: MeromorphicOn (s ^ n) U
-- MeromorphicOn.congr
lemma congr (h_eq : Set.EqOn f g U) (hu : IsOpen U) : MeromorphicOn g U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : 𝕜 → E} {U : Set 𝕜} (hf : MeromorphicOn f U) (h_eq : Set.EqOn f g U) (hu : IsOpen U)
Goal: MeromorphicOn g U
-- MeasureTheory.hahn_decomposition
theorem hahn_decomposition [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    ∃ s,
      MeasurableSet s ∧
        (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
Goal: ∃ s,
  MeasurableSet s ∧
    (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t
-- FormalMultilinearSeries.radius_prod_eq_min
lemma FormalMultilinearSeries.radius_prod_eq_min
    (p : FormalMultilinearSeries 𝕜 E F) (q : FormalMultilinearSeries 𝕜 E G) :
    (p.prod q).radius = min p.radius q.radius 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (p : FormalMultilinearSeries 𝕜 E F) (q : FormalMultilinearSeries 𝕜 E G)
Goal: (p.prod q).radius = min p.radius q.radius
-- HasFPowerSeriesOnBall.prod
lemma HasFPowerSeriesOnBall.prod {e : E} {f : E → F} {g : E → G} {r s : ℝ≥0∞}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesOnBall f p e r) (hg : HasFPowerSeriesOnBall g q e s) :
    HasFPowerSeriesOnBall (fun x ↦ (f x, g x)) (p.prod q) e (min r s) 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {e : E} {f : E → F} {g : E → G} {r s : ℝ≥0∞} {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G} (hf : HasFPowerSeriesOnBall f p e r) (hg : HasFPowerSeriesOnBall g q e s)
Goal: HasFPowerSeriesOnBall (fun x => (f x, g x)) (p.prod q) e (min r s)
-- HasFPowerSeriesAt.prod
lemma HasFPowerSeriesAt.prod {e : E} {f : E → F} {g : E → G}
    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
    (hf : HasFPowerSeriesAt f p e) (hg : HasFPowerSeriesAt g q e) :
    HasFPowerSeriesAt (fun x ↦ (f x, g x)) (p.prod q) e 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {e : E} {f : E → F} {g : E → G} {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G} (hf : HasFPowerSeriesAt f p e) (hg : HasFPowerSeriesAt g q e)
Goal: HasFPowerSeriesAt (fun x => (f x, g x)) (p.prod q) e
-- AnalyticAt.prod
lemma AnalyticAt.prod {e : E} {f : E → F} {g : E → G}
    (hf : AnalyticAt 𝕜 f e) (hg : AnalyticAt 𝕜 g e) :
    AnalyticAt 𝕜 (fun x ↦ (f x, g x)) e 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {e : E} {f : E → F} {g : E → G} (hf : AnalyticAt 𝕜 f e) (hg : AnalyticAt 𝕜 g e)
Goal: AnalyticAt 𝕜 (fun x => (f x, g x)) e
-- AnalyticOn.prod
lemma AnalyticOn.prod {f : E → F} {g : E → G} {s : Set E}
    (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
    AnalyticOn 𝕜 (fun x ↦ (f x, g x)) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {g : E → G} {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s)
Goal: AnalyticOn 𝕜 (fun x => (f x, g x)) s
-- AnalyticAt.comp₂
theorem AnalyticAt.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {x : E}
    (ha : AnalyticAt 𝕜 h (f x, g x)) (fa : AnalyticAt 𝕜 f x)
    (ga : AnalyticAt 𝕜 g x) :
    AnalyticAt 𝕜 (fun x ↦ h (f x, g x)) x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} {H : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup H] [NormedSpace 𝕜 H] {h : F × G → H} {f : E → F} {g : E → G} {x : E} (ha : AnalyticAt 𝕜 h (f x, g x)) (fa : AnalyticAt 𝕜 f x) (ga : AnalyticAt 𝕜 g x)
Goal: AnalyticAt 𝕜 (fun x => h (f x, g x)) x
-- AnalyticOn.comp₂
theorem AnalyticOn.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {s : Set (F × G)} {t : Set E}
    (ha : AnalyticOn 𝕜 h s) (fa : AnalyticOn 𝕜 f t) (ga : AnalyticOn 𝕜 g t)
    (m : ∀ x, x ∈ t → (f x, g x) ∈ s) : AnalyticOn 𝕜 (fun x ↦ h (f x, g x)) t 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} {H : Type u_6} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup H] [NormedSpace 𝕜 H] {h : F × G → H} {f : E → F} {g : E → G} {s : Set (F × G)} {t : Set E} (ha : AnalyticOn 𝕜 h s) (fa : AnalyticOn 𝕜 f t) (ga : AnalyticOn 𝕜 g t) (m : ∀ x ∈ t, (f x, g x) ∈ s)
Goal: AnalyticOn 𝕜 (fun x => h (f x, g x)) t
-- AnalyticAt.curry_left
theorem AnalyticAt.curry_left {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
    AnalyticAt 𝕜 (fun x ↦ f (x, p.2)) p.1 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p)
Goal: AnalyticAt 𝕜 (fun x => f (x, p.2)) p.1
-- AnalyticAt.curry_right
theorem AnalyticAt.curry_right {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
    AnalyticAt 𝕜 (fun y ↦ f (p.1, y)) p.2 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p)
Goal: AnalyticAt 𝕜 (fun y => f (p.1, y)) p.2
-- AnalyticOn.curry_left
theorem AnalyticOn.curry_left {f : E × F → G} {s : Set (E × F)} {y : F} (fa : AnalyticOn 𝕜 f s) :
    AnalyticOn 𝕜 (fun x ↦ f (x, y)) {x | (x, y) ∈ s} 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} {s : Set (E × F)} {y : F} (fa : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (fun x => f (x, y)) {x | (x, y) ∈ s}
-- AnalyticOn.curry_right
theorem AnalyticOn.curry_right {f : E × F → G} {x : E} {s : Set (E × F)} (fa : AnalyticOn 𝕜 f s) :
    AnalyticOn 𝕜 (fun y ↦ f (x, y)) {y | (x, y) ∈ s} 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} {G : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E × F → G} {x : E} {s : Set (E × F)} (fa : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (fun y => f (x, y)) {y | (x, y) ∈ s}
-- analyticAt_smul
lemma analyticAt_smul [NormedSpace 𝕝 E] [IsScalarTower 𝕜 𝕝 E] (z : 𝕝 × E) :
    AnalyticAt 𝕜 (fun x : 𝕝 × E ↦ x.1 • x.2) z 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] [NormedSpace 𝕝 E] [IsScalarTower 𝕜 𝕝 E] (z : 𝕝 × E)
Goal: AnalyticAt 𝕜 (fun x => x.1 • x.2) z
-- analyticAt_mul
lemma analyticAt_mul (z : A × A) : AnalyticAt 𝕜 (fun x : A × A ↦ x.1 * x.2) z 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {A : Type u_8} [NormedRing A] [NormedAlgebra 𝕜 A] (z : A × A)
Goal: AnalyticAt 𝕜 (fun x => x.1 * x.2) z
-- AnalyticAt.smul
lemma AnalyticAt.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {z : E}
    (hf : AnalyticAt 𝕜 f z) (hg : AnalyticAt 𝕜 g z) :
    AnalyticAt 𝕜 (fun x ↦ f x • g x) z 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {z : E} (hf : AnalyticAt 𝕜 f z) (hg : AnalyticAt 𝕜 g z)
Goal: AnalyticAt 𝕜 (fun x => f x • g x) z
-- AnalyticOn.smul
lemma AnalyticOn.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {s : Set E}
    (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
    AnalyticOn 𝕜 (fun x ↦ f x • g x) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s)
Goal: AnalyticOn 𝕜 (fun x => f x • g x) s
-- AnalyticAt.mul
lemma AnalyticAt.mul {f g : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (hg : AnalyticAt 𝕜 g z) :
    AnalyticAt 𝕜 (fun x ↦ f x * g x) z 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_8} [NormedRing A] [NormedAlgebra 𝕜 A] {f g : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (hg : AnalyticAt 𝕜 g z)
Goal: AnalyticAt 𝕜 (fun x => f x * g x) z
-- AnalyticOn.mul
lemma AnalyticOn.mul {f g : E → A} {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
    AnalyticOn 𝕜 (fun x ↦ f x * g x) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_8} [NormedRing A] [NormedAlgebra 𝕜 A] {f g : E → A} {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s)
Goal: AnalyticOn 𝕜 (fun x => f x * g x) s
-- AnalyticAt.pow
lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_8} [NormedRing A] [NormedAlgebra 𝕜 A] {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ)
Goal: AnalyticAt 𝕜 (fun x => f x ^ n) z
-- AnalyticOn.pow
lemma AnalyticOn.pow {f : E → A} {s : Set E} (hf : AnalyticOn 𝕜 f s) (n : ℕ) :
    AnalyticOn 𝕜 (fun x ↦ f x ^ n) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {A : Type u_8} [NormedRing A] [NormedAlgebra 𝕜 A] {f : E → A} {s : Set E} (hf : AnalyticOn 𝕜 f s) (n : ℕ)
Goal: AnalyticOn 𝕜 (fun x => f x ^ n) s
-- formalMultilinearSeries_geometric_apply_norm
lemma formalMultilinearSeries_geometric_apply_norm (n : ℕ) :
    ‖formalMultilinearSeries_geometric 𝕜 A n‖ = 1 
Arguments: (𝕜 : Type u_9) (A : Type u_10) [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [NormOneClass A] (n : ℕ)
Goal: ‖formalMultilinearSeries_geometric 𝕜 A n‖ = 1
-- formalMultilinearSeries_geometric_radius
lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
    (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
    (formalMultilinearSeries_geometric 𝕜 A).radius = 1 
Arguments: (𝕜 : Type u_10) [NontriviallyNormedField 𝕜] (A : Type u_9) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A]
Goal: (formalMultilinearSeries_geometric 𝕜 A).radius = 1
-- hasFPowerSeriesOnBall_inv_one_sub
lemma hasFPowerSeriesOnBall_inv_one_sub
    (𝕜 𝕝 : Type*) [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] :
    HasFPowerSeriesOnBall (fun x : 𝕝 ↦ (1 - x)⁻¹) (formalMultilinearSeries_geometric 𝕜 𝕝) 0 1 
Arguments: (𝕜 : Type u_9) (𝕝 : Type u_10) [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝]
Goal: HasFPowerSeriesOnBall (fun x => (1 - x)⁻¹) (formalMultilinearSeries_geometric 𝕜 𝕝) 0 1
-- analyticAt_inv_one_sub
lemma analyticAt_inv_one_sub (𝕝 : Type*) [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] :
    AnalyticAt 𝕜 (fun x : 𝕝 ↦ (1 - x)⁻¹) 0 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] (𝕝 : Type u_9) [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝]
Goal: AnalyticAt 𝕜 (fun x => (1 - x)⁻¹) 0
-- analyticAt_inv
lemma analyticAt_inv {z : 𝕝} (hz : z ≠ 0) : AnalyticAt 𝕜 Inv.inv z 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] {z : 𝕝} (hz : z ≠ 0)
Goal: AnalyticAt 𝕜 Inv.inv z
-- analyticOn_inv
lemma analyticOn_inv : AnalyticOn 𝕜 (fun z ↦ z⁻¹) {z : 𝕝 | z ≠ 0} 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝]
Goal: AnalyticOn 𝕜 (fun z => z⁻¹) {z | z ≠ 0}
-- AnalyticAt.inv
theorem AnalyticAt.inv {f : E → 𝕝} {x : E} (fa : AnalyticAt 𝕜 f x) (f0 : f x ≠ 0) :
    AnalyticAt 𝕜 (fun x ↦ (f x)⁻¹) x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] {f : E → 𝕝} {x : E} (fa : AnalyticAt 𝕜 f x) (f0 : f x ≠ 0)
Goal: AnalyticAt 𝕜 (fun x => (f x)⁻¹) x
-- AnalyticOn.inv
theorem AnalyticOn.inv {f : E → 𝕝} {s : Set E} (fa : AnalyticOn 𝕜 f s) (f0 : ∀ x ∈ s, f x ≠ 0) :
    AnalyticOn 𝕜 (fun x ↦ (f x)⁻¹) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] {f : E → 𝕝} {s : Set E} (fa : AnalyticOn 𝕜 f s) (f0 : ∀ x ∈ s, f x ≠ 0)
Goal: AnalyticOn 𝕜 (fun x => (f x)⁻¹) s
-- AnalyticAt.div
theorem AnalyticAt.div {f g : E → 𝕝} {x : E}
    (fa : AnalyticAt 𝕜 f x) (ga : AnalyticAt 𝕜 g x) (g0 : g x ≠ 0) :
    AnalyticAt 𝕜 (fun x ↦ f x / g x) x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] {f g : E → 𝕝} {x : E} (fa : AnalyticAt 𝕜 f x) (ga : AnalyticAt 𝕜 g x) (g0 : g x ≠ 0)
Goal: AnalyticAt 𝕜 (fun x => f x / g x) x
-- AnalyticOn.div
theorem AnalyticOn.div {f g : E → 𝕝} {s : Set E}
    (fa : AnalyticOn 𝕜 f s) (ga : AnalyticOn 𝕜 g s) (g0 : ∀ x ∈ s, g x ≠ 0) :
    AnalyticOn 𝕜 (fun x ↦ f x / g x) s 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕝 : Type u_7} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] {f g : E → 𝕝} {s : Set E} (fa : AnalyticOn 𝕜 f s) (ga : AnalyticOn 𝕜 g s) (g0 : ∀ x ∈ s, g x ≠ 0)
Goal: AnalyticOn 𝕜 (fun x => f x / g x) s
-- Finset.analyticAt_sum
theorem Finset.analyticAt_sum {f : α → E → F} {c : E}
    (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
    AnalyticAt 𝕜 (fun z ↦ ∑ n ∈ N, f n z) c 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : α → E → F} {c : E} (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c)
Goal: AnalyticAt 𝕜 (fun z => ∑ n ∈ N, f n z) c
-- Finset.analyticOn_sum
theorem Finset.analyticOn_sum {f : α → E → F} {s : Set E}
    (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s) :
    AnalyticOn 𝕜 (fun z ↦ ∑ n ∈ N, f n z) s 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : α → E → F} {s : Set E} (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s)
Goal: AnalyticOn 𝕜 (fun z => ∑ n ∈ N, f n z) s