ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DoubleCentralizer.norm_def'
theorem norm_def' (a : 𝓜(𝕜, A)) : ‖a‖ = ‖toProdMulOppositeHom a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: ‖a‖ = ‖DoubleCentralizer.toProdMulOppositeHom a‖
-- DoubleCentralizer.nnnorm_def'
theorem nnnorm_def' (a : 𝓜(𝕜, A)) : ‖a‖₊ = ‖toProdMulOppositeHom a‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] (a : 𝓜(𝕜, A))
Goal: ‖a‖₊ = ‖DoubleCentralizer.toProdMulOppositeHom a‖₊
-- DoubleCentralizer.uniformEmbedding_toProdMulOpposite
theorem uniformEmbedding_toProdMulOpposite : UniformEmbedding (@toProdMulOpposite 𝕜 A _ _ _ _ _) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A]
Goal: UniformEmbedding DoubleCentralizer.toProdMulOpposite
-- DoubleCentralizer.norm_fst_eq_snd
theorem norm_fst_eq_snd (a : 𝓜(𝕜, A)) : ‖a.fst‖ = ‖a.snd‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing A] [CstarRing A] (a : 𝓜(𝕜, A))
Goal: ‖a.toProd.1‖ = ‖a.toProd.2‖
-- DoubleCentralizer.nnnorm_fst_eq_snd
theorem nnnorm_fst_eq_snd (a : 𝓜(𝕜, A)) : ‖a.fst‖₊ = ‖a.snd‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing A] [CstarRing A] (a : 𝓜(𝕜, A))
Goal: ‖a.toProd.1‖₊ = ‖a.toProd.2‖₊
-- DoubleCentralizer.norm_fst
@[simp]
theorem norm_fst (a : 𝓜(𝕜, A)) : ‖a.fst‖ = ‖a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing A] [CstarRing A] (a : 𝓜(𝕜, A))
Goal: ‖a.toProd.1‖ = ‖a‖
-- DoubleCentralizer.norm_snd
@[simp]
theorem norm_snd (a : 𝓜(𝕜, A)) : ‖a.snd‖ = ‖a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing A] [CstarRing A] (a : 𝓜(𝕜, A))
Goal: ‖a.toProd.2‖ = ‖a‖
-- DoubleCentralizer.nnnorm_fst
@[simp]
theorem nnnorm_fst (a : 𝓜(𝕜, A)) : ‖a.fst‖₊ = ‖a‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing A] [CstarRing A] (a : 𝓜(𝕜, A))
Goal: ‖a.toProd.1‖₊ = ‖a‖₊
-- DoubleCentralizer.nnnorm_snd
@[simp]
theorem nnnorm_snd (a : 𝓜(𝕜, A)) : ‖a.snd‖₊ = ‖a‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [SMulCommClass 𝕜 A A] [IsScalarTower 𝕜 A A] [StarRing A] [CstarRing A] (a : 𝓜(𝕜, A))
Goal: ‖a.toProd.2‖₊ = ‖a‖₊
-- ConvexOn.slope_mono_adjacent
theorem ConvexOn.slope_mono_adjacent (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)
-- ConcaveOn.slope_anti_adjacent
theorem ConcaveOn.slope_anti_adjacent (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x < y) (hyz : y < z) : (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
-- StrictConvexOn.slope_strict_mono_adjacent
theorem StrictConvexOn.slope_strict_mono_adjacent (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜}
    (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
    (f y - f x) / (y - x) < (f z - f y) / (z - y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f y - f x) / (y - x) < (f z - f y) / (z - y)
-- StrictConcaveOn.slope_anti_adjacent
theorem StrictConcaveOn.slope_anti_adjacent (hf : StrictConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
    (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f z - f y) / (z - y) < (f y - f x) / (y - x) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConcaveOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f z - f y) / (z - y) < (f y - f x) / (y - x)
-- convexOn_of_slope_mono_adjacent
theorem convexOn_of_slope_mono_adjacent (hs : Convex 𝕜 s)
    (hf :
      ∀ {x y z : 𝕜},
        x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :
    ConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y))
Goal: ConvexOn 𝕜 s f
-- concaveOn_of_slope_anti_adjacent
theorem concaveOn_of_slope_anti_adjacent (hs : Convex 𝕜 s)
    (hf :
      ∀ {x y z : 𝕜},
        x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) :
    ConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x))
Goal: ConcaveOn 𝕜 s f
-- strictConvexOn_of_slope_strict_mono_adjacent
theorem strictConvexOn_of_slope_strict_mono_adjacent (hs : Convex 𝕜 s)
    (hf :
      ∀ {x y z : 𝕜},
        x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)) :
    StrictConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y))
Goal: StrictConvexOn 𝕜 s f
-- strictConcaveOn_of_slope_strict_anti_adjacent
theorem strictConcaveOn_of_slope_strict_anti_adjacent (hs : Convex 𝕜 s)
    (hf :
      ∀ {x y z : 𝕜},
        x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)) :
    StrictConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hs : Convex 𝕜 s) (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x))
Goal: StrictConcaveOn 𝕜 s f
-- convexOn_iff_slope_mono_adjacent
theorem convexOn_iff_slope_mono_adjacent :
    ConvexOn 𝕜 s f ↔
      Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄,
          x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜}
Goal: ConvexOn 𝕜 s f ↔
  Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)
-- concaveOn_iff_slope_anti_adjacent
theorem concaveOn_iff_slope_anti_adjacent :
    ConcaveOn 𝕜 s f ↔
      Convex 𝕜 s ∧
        ∀ ⦃x y z : 𝕜⦄,
          x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜}
Goal: ConcaveOn 𝕜 s f ↔
  Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
-- strictConvexOn_iff_slope_strict_mono_adjacent
theorem strictConvexOn_iff_slope_strict_mono_adjacent :
    StrictConvexOn 𝕜 s f ↔
      Convex 𝕜 s ∧
        ∀ ⦃x y z : 𝕜⦄,
          x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜}
Goal: StrictConvexOn 𝕜 s f ↔
  Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)
-- strictConcaveOn_iff_slope_strict_anti_adjacent
theorem strictConcaveOn_iff_slope_strict_anti_adjacent :
    StrictConcaveOn 𝕜 s f ↔
      Convex 𝕜 s ∧
        ∀ ⦃x y z : 𝕜⦄,
          x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜}
Goal: StrictConcaveOn 𝕜 s f ↔
  Convex 𝕜 s ∧ ∀ ⦃x y z : 𝕜⦄, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)
-- ConvexOn.secant_mono_aux1
theorem ConvexOn.secant_mono_aux1 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x < y) (hyz : y < z) : (z - x) * f y ≤ (z - y) * f x + (y - x) * f z 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (z - x) * f y ≤ (z - y) * f x + (y - x) * f z
-- ConvexOn.secant_mono_aux2
theorem ConvexOn.secant_mono_aux2 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f x) / (z - x) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f y - f x) / (y - x) ≤ (f z - f x) / (z - x)
-- ConvexOn.secant_mono_aux3
theorem ConvexOn.secant_mono_aux3 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
    (hxy : x < y) (hyz : y < z) : (f z - f x) / (z - x) ≤ (f z - f y) / (z - y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f z - f x) / (z - x) ≤ (f z - f y) / (z - y)
-- ConvexOn.secant_mono
theorem ConvexOn.secant_mono (hf : ConvexOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s)
    (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x ≤ y) :
    (f x - f a) / (x - a) ≤ (f y - f a) / (y - a) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConvexOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x ≤ y)
Goal: (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)
-- StrictConvexOn.secant_strict_mono_aux1
theorem StrictConvexOn.secant_strict_mono_aux1 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
    (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (z - x) * f y < (z - y) * f x + (y - x) * f z 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (z - x) * f y < (z - y) * f x + (y - x) * f z
-- StrictConvexOn.secant_strict_mono_aux2
theorem StrictConvexOn.secant_strict_mono_aux2 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
    (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) < (f z - f x) / (z - x) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f y - f x) / (y - x) < (f z - f x) / (z - x)
-- StrictConvexOn.secant_strict_mono_aux3
theorem StrictConvexOn.secant_strict_mono_aux3 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
    (hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f z - f x) / (z - x) < (f z - f y) / (z - y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z)
Goal: (f z - f x) / (z - x) < (f z - f y) / (z - y)
-- StrictConvexOn.secant_strict_mono
theorem StrictConvexOn.secant_strict_mono (hf : StrictConvexOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s)
    (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y) :
    (f x - f a) / (x - a) < (f y - f a) / (y - a) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConvexOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y)
Goal: (f x - f a) / (x - a) < (f y - f a) / (y - a)
-- StrictConcaveOn.secant_strict_mono
theorem StrictConcaveOn.secant_strict_mono (hf : StrictConcaveOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s)
    (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y) :
    (f y - f a) / (y - a) < (f x - f a) / (x - a) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : StrictConcaveOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s) (hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y)
Goal: (f y - f a) / (y - a) < (f x - f a) / (x - a)
-- ConvexOn.strict_mono_of_lt
theorem ConvexOn.strict_mono_of_lt (hf : ConvexOn 𝕜 s f) {x y : 𝕜} (hx : x ∈ s) (hxy : x < y)
    (hxy' : f x < f y) : StrictMonoOn f (s ∩ Set.Ici y) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} (hf : ConvexOn 𝕜 s f) {x y : 𝕜} (hx : x ∈ s) (hxy : x < y) (hxy' : f x < f y)
Goal: StrictMonoOn f (s ∩ Set.Ici y)
-- FormalMultilinearSeries.partialSum_continuous
theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    Continuous (p.partialSum n) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [Ring 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 E] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ)
Goal: Continuous (p.partialSum n)
-- FormalMultilinearSeries.le_radius_of_bound
theorem le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
    (r : 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] (p : FormalMultilinearSeries 𝕜 E F) (C : ℝ) {r : ℝ≥0} (h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.le_radius_of_bound_nnreal
theorem le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ n : ℕ, ‖p n‖₊ * r ^ n ≤ C) :
    (r : 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] (p : FormalMultilinearSeries 𝕜 E F) (C : ℝ≥0) {r : ℝ≥0} (h : ∀ (n : ℕ), ‖p n‖₊ * r ^ n ≤ C)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.le_radius_of_isBigO
theorem le_radius_of_isBigO (h : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) :
    ↑r ≤ 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) {r : ℝ≥0} (h : (fun n => ‖p n‖ * ↑r ^ n) =O[Filter.atTop] fun x => 1)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.le_radius_of_eventually_le
theorem le_radius_of_eventually_le (C) (h : ∀ᶠ n in atTop, ‖p n‖ * (r : ℝ) ^ n ≤ C) :
    ↑r ≤ 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) {r : ℝ≥0} (C : ℝ) (h : ∀ᶠ (n : ℕ) in Filter.atTop, ‖p n‖ * ↑r ^ n ≤ C)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.le_radius_of_summable_nnnorm
theorem le_radius_of_summable_nnnorm (h : Summable fun n => ‖p n‖₊ * r ^ n) : ↑r ≤ 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) {r : ℝ≥0} (h : Summable fun n => ‖p n‖₊ * r ^ n)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.le_radius_of_summable
theorem le_radius_of_summable (h : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ 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) {r : ℝ≥0} (h : Summable fun n => ‖p n‖ * ↑r ^ n)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.radius_eq_top_of_forall_nnreal_isBigO
theorem radius_eq_top_of_forall_nnreal_isBigO
    (h : ∀ r : ℝ≥0, (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] fun _ => (1 : ℝ)) : 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) (h : ∀ (r : ℝ≥0), (fun n => ‖p n‖ * ↑r ^ n) =O[Filter.atTop] fun x => 1)
Goal: p.radius = ⊤
-- FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero
theorem radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in atTop, p n = 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) (h : ∀ᶠ (n : ℕ) in Filter.atTop, p n = 0)
Goal: p.radius = ⊤
-- FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
theorem radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 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) (n : ℕ) (hn : ∀ (m : ℕ), p (m + n) = 0)
Goal: p.radius = ⊤
-- FormalMultilinearSeries.constFormalMultilinearSeries_radius
@[simp]
theorem constFormalMultilinearSeries_radius {v : F} :
    (constFormalMultilinearSeries 𝕜 E v).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] {v : F}
Goal: (constFormalMultilinearSeries 𝕜 E v).radius = ⊤
-- FormalMultilinearSeries.zero_radius
@[simp] lemma zero_radius : (0 : FormalMultilinearSeries 𝕜 E F).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]
Goal: FormalMultilinearSeries.radius 0 = ⊤
-- FormalMultilinearSeries.isLittleO_of_lt_radius
theorem isLittleO_of_lt_radius (h : ↑r < p.radius) :
    ∃ a ∈ Ioo (0 : ℝ) 1, (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (a ^ ·) 
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) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: ∃ a ∈ Set.Ioo 0 1, (fun n => ‖p n‖ * ↑r ^ n) =o[Filter.atTop] fun x => a ^ x
-- FormalMultilinearSeries.isLittleO_one_of_lt_radius
theorem isLittleO_one_of_lt_radius (h : ↑r < p.radius) :
    (fun n => ‖p n‖ * (r : ℝ) ^ n) =o[atTop] (fun _ => 1 : 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] (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: (fun n => ‖p n‖ * ↑r ^ n) =o[Filter.atTop] fun x => 1
-- FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius
theorem norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
    ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ 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] (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: ∃ a ∈ Set.Ioo 0 1, ∃ C > 0, ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C * a ^ n
-- FormalMultilinearSeries.lt_radius_of_isBigO
theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
    (hp : (fun n => ‖p n‖ * (r : ℝ) ^ n) =O[atTop] (a ^ ·)) : ↑r < 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) {r : ℝ≥0} (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Set.Ioo (-1) 1) (hp : (fun n => ‖p n‖ * ↑r ^ n) =O[Filter.atTop] fun x => a ^ x)
Goal: ↑r < p.radius
-- FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius
theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
    (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C 
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) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: ∃ C > 0, ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
-- FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius
theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
    (h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / (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) {r : ℝ≥0} (h0 : 0 < r) (h : ↑r < p.radius)
Goal: ∃ C > 0, ∀ (n : ℕ), ‖p n‖ ≤ C / ↑r ^ n
-- FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
    (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r ^ n ≤ C 
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) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: ∃ C > 0, ∀ (n : ℕ), ‖p n‖₊ * r ^ n ≤ C
-- FormalMultilinearSeries.le_radius_of_tendsto
theorem le_radius_of_tendsto (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ}
    (h : Tendsto (fun n => ‖p n‖ * (r : ℝ) ^ n) atTop (𝓝 l)) : ↑r ≤ 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] {r : ℝ≥0} (p : FormalMultilinearSeries 𝕜 E F) {l : ℝ} (h : Filter.Tendsto (fun n => ‖p n‖ * ↑r ^ n) Filter.atTop (𝓝 l))
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.le_radius_of_summable_norm
theorem le_radius_of_summable_norm (p : FormalMultilinearSeries 𝕜 E F)
    (hs : Summable fun n => ‖p n‖ * (r : ℝ) ^ n) : ↑r ≤ 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] {r : ℝ≥0} (p : FormalMultilinearSeries 𝕜 E F) (hs : Summable fun n => ‖p n‖ * ↑r ^ n)
Goal: ↑r ≤ p.radius
-- FormalMultilinearSeries.not_summable_norm_of_radius_lt_nnnorm
theorem not_summable_norm_of_radius_lt_nnnorm (p : FormalMultilinearSeries 𝕜 E F) {x : E}
    (h : p.radius < ‖x‖₊) : ¬Summable fun n => ‖p n‖ * ‖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] (p : FormalMultilinearSeries 𝕜 E F) {x : E} (h : p.radius < ↑‖x‖₊)
Goal: ¬Summable fun n => ‖p n‖ * ‖x‖ ^ n
-- FormalMultilinearSeries.summable_norm_mul_pow
theorem summable_norm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
    Summable fun n : 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) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: Summable fun n => ‖p n‖ * ↑r ^ n
-- FormalMultilinearSeries.summable_norm_apply
theorem summable_norm_apply (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] (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.summable_nnnorm_mul_pow
theorem summable_nnnorm_mul_pow (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (h : ↑r < p.radius) :
    Summable fun n : ℕ => ‖p n‖₊ * 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) {r : ℝ≥0} (h : ↑r < p.radius)
Goal: Summable fun n => ‖p n‖₊ * r ^ n