ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Asymptotics.IsBigO.natCast_atTop
theorem IsBigO.natCast_atTop {R : Type*} [StrictOrderedSemiring R] [Archimedean R]
    {f : R → E} {g : R → F} (h : f =O[atTop] g) :
    (fun (n : ℕ) => f n) =O[atTop] (fun n => g n) 
Arguments: {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {R : Type u_17} [StrictOrderedSemiring R] [Archimedean R] {f : R → E} {g : R → F} (h : f =O[Filter.atTop] g)
Goal: (fun n => f ↑n) =O[Filter.atTop] fun n => g ↑n
-- Asymptotics.IsLittleO.natCast_atTop
theorem IsLittleO.natCast_atTop {R : Type*} [StrictOrderedSemiring R] [Archimedean R]
    {f : R → E} {g : R → F} (h : f =o[atTop] g) :
    (fun (n : ℕ) => f n) =o[atTop] (fun n => g n) 
Arguments: {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {R : Type u_17} [StrictOrderedSemiring R] [Archimedean R] {f : R → E} {g : R → F} (h : f =o[Filter.atTop] g)
Goal: (fun n => f ↑n) =o[Filter.atTop] fun n => g ↑n
-- Asymptotics.isBigO_atTop_iff_eventually_exists
theorem isBigO_atTop_iff_eventually_exists {α : Type*} [SemilatticeSup α] [Nonempty α]
    {f : α → E} {g : α → F} : f =O[atTop] g ↔ ∀ᶠ n₀ in atTop, ∃ c, ∀ n ≥ n₀, ‖f n‖ ≤ c * ‖g n‖ 
Arguments: {E : Type u_3} {F : Type u_4} [Norm E] [Norm F] {α : Type u_17} [SemilatticeSup α] [Nonempty α] {f : α → E} {g : α → F}
Goal: f =O[Filter.atTop] g ↔ ∀ᶠ (n₀ : α) in Filter.atTop, ∃ c, ∀ n ≥ n₀, ‖f n‖ ≤ c * ‖g n‖
-- Asymptotics.isBigO_atTop_iff_eventually_exists_pos
theorem isBigO_atTop_iff_eventually_exists_pos {α : Type*}
    [SemilatticeSup α] [Nonempty α] {f : α → G} {g : α → G'} :
    f =O[atTop] g ↔ ∀ᶠ n₀ in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖f n‖ ≤ ‖g n‖ 
Arguments: {G : Type u_5} {G' : Type u_8} [Norm G] [SeminormedAddCommGroup G'] {α : Type u_17} [SemilatticeSup α] [Nonempty α] {f : α → G} {g : α → G'}
Goal: f =O[Filter.atTop] g ↔ ∀ᶠ (n₀ : α) in Filter.atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖f n‖ ≤ ‖g n‖
-- summable_of_isBigO
theorem summable_of_isBigO {ι E} [SeminormedAddCommGroup E] [CompleteSpace E]
    {f : ι → E} {g : ι → ℝ} (hg : Summable g) (h : f =O[cofinite] g) : Summable f 
Arguments: {ι : Type u_1} {E : Type u_2} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ι → E} {g : ι → ℝ} (hg : Summable g) (h : f =O[Filter.cofinite] g)
Goal: Summable f
-- summable_of_isBigO_nat
theorem summable_of_isBigO_nat {E} [SeminormedAddCommGroup E] [CompleteSpace E]
    {f : ℕ → E} {g : ℕ → ℝ} (hg : Summable g) (h : f =O[atTop] g) : Summable f 
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [CompleteSpace E] {f : ℕ → E} {g : ℕ → ℝ} (hg : Summable g) (h : f =O[Filter.atTop] g)
Goal: Summable f
-- Asymptotics.IsBigO.comp_summable_norm
lemma Asymptotics.IsBigO.comp_summable_norm {ι E F : Type*}
    [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] {f : E → F} {g : ι → E}
    (hf : f =O[𝓝 0] id) (hg : Summable (‖g ·‖)) : Summable (‖f <| g ·‖) 
Arguments: {ι : Type u_1} {E : Type u_2} {F : Type u_3} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] {f : E → F} {g : ι → E} (hf : f =O[𝓝 0] id) (hg : Summable fun x => ‖g x‖)
Goal: Summable fun x => ‖f (g x)‖
-- PartialHomeomorph.isBigOWith_congr
theorem isBigOWith_congr (e : PartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E}
    {g : β → F} {C : ℝ} : IsBigOWith C (𝓝 b) f g ↔ IsBigOWith C (𝓝 (e.symm b)) (f ∘ e) (g ∘ e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : PartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E} {g : β → F} {C : ℝ}
Goal: Asymptotics.IsBigOWith C (𝓝 b) f g ↔ Asymptotics.IsBigOWith C (𝓝 (↑e.symm b)) (f ∘ ↑e) (g ∘ ↑e)
-- PartialHomeomorph.isBigO_congr
theorem isBigO_congr (e : PartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E}
    {g : β → F} : f =O[𝓝 b] g ↔ (f ∘ e) =O[𝓝 (e.symm b)] (g ∘ e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : PartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E} {g : β → F}
Goal: f =O[𝓝 b] g ↔ (f ∘ ↑e) =O[𝓝 (↑e.symm b)] (g ∘ ↑e)
-- PartialHomeomorph.isLittleO_congr
theorem isLittleO_congr (e : PartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E}
    {g : β → F} : f =o[𝓝 b] g ↔ (f ∘ e) =o[𝓝 (e.symm b)] (g ∘ e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : PartialHomeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E} {g : β → F}
Goal: f =o[𝓝 b] g ↔ (f ∘ ↑e) =o[𝓝 (↑e.symm b)] (g ∘ ↑e)
-- Homeomorph.isBigOWith_congr
theorem isBigOWith_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} {C : ℝ} :
    IsBigOWith C (𝓝 b) f g ↔ IsBigOWith C (𝓝 (e.symm b)) (f ∘ e) (g ∘ e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} {C : ℝ}
Goal: Asymptotics.IsBigOWith C (𝓝 b) f g ↔ Asymptotics.IsBigOWith C (𝓝 (e.symm b)) (f ∘ ⇑e) (g ∘ ⇑e)
-- Homeomorph.isBigO_congr
theorem isBigO_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :
    f =O[𝓝 b] g ↔ (f ∘ e) =O[𝓝 (e.symm b)] (g ∘ e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F}
Goal: f =O[𝓝 b] g ↔ (f ∘ ⇑e) =O[𝓝 (e.symm b)] (g ∘ ⇑e)
-- Homeomorph.isLittleO_congr
theorem isLittleO_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :
    f =o[𝓝 b] g ↔ (f ∘ e) =o[𝓝 (e.symm b)] (g ∘ e) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {E : Type u_3} [Norm E] {F : Type u_4} [Norm F] (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F}
Goal: f =o[𝓝 b] g ↔ (f ∘ ⇑e) =o[𝓝 (e.symm b)] (g ∘ ⇑e)
-- WithLp.unitization_norm_def
lemma unitization_norm_def (x : WithLp 1 (Unitization 𝕜 A)) :
    ‖x‖ = ‖(WithLp.equiv 1 _ x).fst‖ + ‖(WithLp.equiv 1 _ x).snd‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] (x : WithLp 1 (Unitization 𝕜 A))
Goal: ‖x‖ = ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).fst‖ + ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).snd‖
-- WithLp.unitization_nnnorm_def
lemma unitization_nnnorm_def (x : WithLp 1 (Unitization 𝕜 A)) :
    ‖x‖₊ = ‖(WithLp.equiv 1 _ x).fst‖₊ + ‖(WithLp.equiv 1 _ x).snd‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] (x : WithLp 1 (Unitization 𝕜 A))
Goal: ‖x‖₊ = ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).fst‖₊ + ‖((WithLp.equiv 1 (Unitization 𝕜 A)) x).snd‖₊
-- WithLp.unitization_norm_inr
lemma unitization_norm_inr (x : A) : ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm x‖ = ‖x‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] (x : A)
Goal: ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm ↑x‖ = ‖x‖
-- WithLp.unitization_nnnorm_inr
lemma unitization_nnnorm_inr (x : A) : ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm x‖₊ = ‖x‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] (x : A)
Goal: ‖(WithLp.equiv 1 (Unitization 𝕜 A)).symm ↑x‖₊ = ‖x‖₊
-- WithLp.unitization_isometry_inr
lemma unitization_isometry_inr :
    Isometry (fun x : A ↦ (WithLp.equiv 1 (Unitization 𝕜 A)).symm x) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A]
Goal: Isometry fun x => (WithLp.equiv 1 (Unitization 𝕜 A)).symm ↑x
-- WithLp.unitization_mul
@[simp]
lemma unitization_mul (x y : WithLp 1 (Unitization 𝕜 A)) :
    WithLp.equiv 1 _ (x * y) = (WithLp.equiv 1 _ x) * (WithLp.equiv 1 _ y) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] (x y : WithLp 1 (Unitization 𝕜 A))
Goal: (WithLp.equiv 1 (Unitization 𝕜 A)) (x * y) = (WithLp.equiv 1 (Unitization 𝕜 A)) x * (WithLp.equiv 1 (Unitization 𝕜 A)) y
-- WithLp.unitization_algebraMap
@[simp]
lemma unitization_algebraMap (r : 𝕜) :
    WithLp.equiv 1 _ (algebraMap 𝕜 (WithLp 1 (Unitization 𝕜 A)) r) =
      algebraMap 𝕜 (Unitization 𝕜 A) r 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] (r : 𝕜)
Goal: (WithLp.equiv 1 (Unitization 𝕜 A)) ((algebraMap 𝕜 (WithLp 1 (Unitization 𝕜 A))) r) = (algebraMap 𝕜 (Unitization 𝕜 A)) r
-- WithLp.unitizationAlgEquiv_apply
@[simps!]
def unitizationAlgEquiv (R : Type*) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A]
    [IsScalarTower R 𝕜 A] : WithLp 1 (Unitization 𝕜 A) ≃ₐ[R] Unitization 𝕜 A 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] (R : Type u_3) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A] [IsScalarTower R 𝕜 A] (a : WithLp 1 (Unitization 𝕜 A))
Goal: (WithLp.unitizationAlgEquiv R) a = a
-- WithLp.unitizationAlgEquiv_symm_apply
@[simps!]
def unitizationAlgEquiv (R : Type*) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A]
    [IsScalarTower R 𝕜 A] : WithLp 1 (Unitization 𝕜 A) ≃ₐ[R] Unitization 𝕜 A 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] (R : Type u_3) [CommSemiring R] [Algebra R 𝕜] [DistribMulAction R A] [IsScalarTower R 𝕜 A] (a : WithLp 1 (Unitization 𝕜 A))
Goal: (WithLp.unitizationAlgEquiv R).symm a = a
-- RingSeminorm.mul_le'
structure RingSeminorm (R : Type*) [NonUnitalNonAssocRing R] extends AddGroupSeminorm R where
    mul_le' : ∀ x y : R, toFun (x * y) ≤ toFun x * toFun y
Arguments: {R : Type u_4} [NonUnitalNonAssocRing R] (self : RingSeminorm R) (x y : R)
Goal: self.toFun (x * y) ≤ self.toFun x * self.toFun y
-- RingSeminorm.toFun_eq_coe
@[simp]
theorem toFun_eq_coe (p : RingSeminorm R) : (p.toAddGroupSeminorm : R → ℝ) = p 
Arguments: {R : Type u_2} [NonUnitalRing R] (p : RingSeminorm R)
Goal: ⇑p.toAddGroupSeminorm = ⇑p
-- RingSeminorm.ext
@[ext]
theorem ext {p q : RingSeminorm R} : (∀ x, p x = q x) → p = q 
Arguments: {R : Type u_2} [NonUnitalRing R] {p q : RingSeminorm R}
Goal: (∀ (x : R), p x = q x) → p = q
-- RingSeminorm.eq_zero_iff
theorem eq_zero_iff {p : RingSeminorm R} : p = 0 ↔ ∀ x, p x = 0 
Arguments: {R : Type u_2} [NonUnitalRing R] {p : RingSeminorm R}
Goal: p = 0 ↔ ∀ (x : R), p x = 0
-- RingSeminorm.ne_zero_iff
theorem ne_zero_iff {p : RingSeminorm R} : p ≠ 0 ↔ ∃ x, p x ≠ 0 
Arguments: {R : Type u_2} [NonUnitalRing R] {p : RingSeminorm R}
Goal: p ≠ 0 ↔ ∃ x, p x ≠ 0
-- RingSeminorm.apply_one
@[simp]
theorem apply_one [DecidableEq R] (x : R) : (1 : RingSeminorm R) x = if x = 0 then 0 else 1 
Arguments: {R : Type u_2} [NonUnitalRing R] [DecidableEq R] (x : R)
Goal: 1 x = if x = 0 then 0 else 1
-- RingSeminorm.seminorm_one_eq_one_iff_ne_zero
theorem seminorm_one_eq_one_iff_ne_zero (hp : p 1 ≤ 1) : p 1 = 1 ↔ p ≠ 0 
Arguments: {R : Type u_2} [Ring R] (p : RingSeminorm R) (hp : p 1 ≤ 1)
Goal: p 1 = 1 ↔ p ≠ 0
-- RingNorm.toFun_eq_coe
theorem toFun_eq_coe (p : RingNorm R) : p.toFun = p 
Arguments: {R : Type u_2} [NonUnitalRing R] (p : RingNorm R)
Goal: p.toFun = ⇑p
-- RingNorm.ext
@[ext]
theorem ext {p q : RingNorm R} : (∀ x, p x = q x) → p = q 
Arguments: {R : Type u_2} [NonUnitalRing R] {p q : RingNorm R}
Goal: (∀ (x : R), p x = q x) → p = q
-- RingNorm.apply_one
@[simp]
theorem apply_one [DecidableEq R] (x : R) : (1 : RingNorm R) x = if x = 0 then 0 else 1 
Arguments: (R : Type u_2) [NonUnitalRing R] [DecidableEq R] (x : R)
Goal: 1 x = if x = 0 then 0 else 1
-- MulRingSeminorm.toFun_eq_coe
@[simp]
theorem toFun_eq_coe (p : MulRingSeminorm R) : (p.toAddGroupSeminorm : R → ℝ) = p 
Arguments: {R : Type u_2} [NonAssocRing R] (p : MulRingSeminorm R)
Goal: ⇑p.toAddGroupSeminorm = ⇑p
-- MulRingSeminorm.ext
@[ext]
theorem ext {p q : MulRingSeminorm R} : (∀ x, p x = q x) → p = q 
Arguments: {R : Type u_2} [NonAssocRing R] {p q : MulRingSeminorm R}
Goal: (∀ (x : R), p x = q x) → p = q
-- MulRingSeminorm.apply_one
@[simp]
theorem apply_one (x : R) : (1 : MulRingSeminorm R) x = if x = 0 then 0 else 1 
Arguments: {R : Type u_2} [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] (x : R)
Goal: 1 x = if x = 0 then 0 else 1
-- MulRingNorm.toFun_eq_coe
theorem toFun_eq_coe (p : MulRingNorm R) : p.toFun = p 
Arguments: {R : Type u_2} [NonAssocRing R] (p : MulRingNorm R)
Goal: p.toFun = ⇑p
-- MulRingNorm.ext
@[ext]
theorem ext {p q : MulRingNorm R} : (∀ x, p x = q x) → p = q 
Arguments: {R : Type u_2} [NonAssocRing R] {p q : MulRingNorm R}
Goal: (∀ (x : R), p x = q x) → p = q
-- MulRingNorm.apply_one
@[simp]
theorem apply_one (x : R) : (1 : MulRingNorm R) x = if x = 0 then 0 else 1 
Arguments: (R : Type u_2) [NonAssocRing R] [DecidableEq R] [NoZeroDivisors R] [Nontrivial R] (x : R)
Goal: 1 x = if x = 0 then 0 else 1
-- MulRingNorm.equiv_refl
lemma equiv_refl (f : MulRingNorm R) : equiv f f 
Arguments: {R : Type u_4} [Ring R] (f : MulRingNorm R)
Goal: f.equiv f
-- MulRingNorm.equiv_symm
lemma equiv_symm {f g : MulRingNorm R} (hfg : equiv f g) : equiv g f 
Arguments: {R : Type u_4} [Ring R] {f g : MulRingNorm R} (hfg : f.equiv g)
Goal: g.equiv f
-- MulRingNorm.equiv_trans
lemma equiv_trans {f g k : MulRingNorm R} (hfg : equiv f g) (hgk : equiv g k) :
    equiv f k 
Arguments: {R : Type u_4} [Ring R] {f g k : MulRingNorm R} (hfg : f.equiv g) (hgk : g.equiv k)
Goal: f.equiv k
-- normRingNorm_toFun
@[simps!]
def normRingNorm (R : Type*) [NonUnitalNormedRing R] : RingNorm R 
Arguments: (R : Type u_4) [NonUnitalNormedRing R]
Goal: ∀ (a : R), (normRingNorm R).toFun a = ‖a‖
-- MulRingNorm_nat_le_nat
lemma MulRingNorm_nat_le_nat {R : Type*} [Ring R] (n : ℕ) (f : MulRingNorm R) : f n ≤ n 
Arguments: {R : Type u_4} [Ring R] (n : ℕ) (f : MulRingNorm R)
Goal: f ↑n ≤ ↑n
-- MulRingNorm.apply_natAbs_eq
lemma MulRingNorm.apply_natAbs_eq {R : Type*} [Ring R] (x : ℤ) (f : MulRingNorm R) : f (natAbs x) =
    f x 
Arguments: {R : Type u_4} [Ring R] (x : ℤ) (f : MulRingNorm R)
Goal: f ↑x.natAbs = f ↑x
-- contDiffOn_clm_apply
theorem contDiffOn_clm_apply {n : ℕ∞} {f : E → F →L[𝕜] G} {s : Set E} [FiniteDimensional 𝕜 F] :
    ContDiffOn 𝕜 n f s ↔ ∀ y, ContDiffOn 𝕜 n (fun x => f x y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace 𝕜] {n : ℕ∞} {f : E → F →L[𝕜] G} {s : Set E} [FiniteDimensional 𝕜 F]
Goal: ContDiffOn 𝕜 n f s ↔ ∀ (y : F), ContDiffOn 𝕜 n (fun x => (f x) y) s
-- contDiff_clm_apply_iff
theorem contDiff_clm_apply_iff {n : ℕ∞} {f : E → F →L[𝕜] G} [FiniteDimensional 𝕜 F] :
    ContDiff 𝕜 n f ↔ ∀ y, ContDiff 𝕜 n fun x => f x y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace 𝕜] {n : ℕ∞} {f : E → F →L[𝕜] G} [FiniteDimensional 𝕜 F]
Goal: ContDiff 𝕜 n f ↔ ∀ (y : F), ContDiff 𝕜 n fun x => (f x) y
-- contDiff_succ_iff_fderiv_apply
theorem contDiff_succ_iff_fderiv_apply [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} :
    ContDiff 𝕜 (n + 1 : ℕ) f ↔ Differentiable 𝕜 f ∧ ∀ y, ContDiff 𝕜 n fun x => fderiv 𝕜 f x y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F}
Goal: ContDiff 𝕜 (↑(n + 1)) f ↔ Differentiable 𝕜 f ∧ ∀ (y : E), ContDiff 𝕜 ↑n fun x => (fderiv 𝕜 f x) y
-- contDiffOn_succ_of_fderiv_apply
theorem contDiffOn_succ_of_fderiv_apply [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} {s : Set E}
    (hf : DifferentiableOn 𝕜 f s) (h : ∀ y, ContDiffOn 𝕜 n (fun x => fderivWithin 𝕜 f s x y) s) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} {s : Set E} (hf : DifferentiableOn 𝕜 f s) (h : ∀ (y : E), ContDiffOn 𝕜 (↑n) (fun x => (fderivWithin 𝕜 f s x) y) s)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s
-- contDiffOn_succ_iff_fderiv_apply
theorem contDiffOn_succ_iff_fderiv_apply [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} {s : Set E}
    (hs : UniqueDiffOn 𝕜 s) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
      DifferentiableOn 𝕜 f s ∧ ∀ y, ContDiffOn 𝕜 n (fun x => fderivWithin 𝕜 f s x y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} {s : Set E} (hs : UniqueDiffOn 𝕜 s)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s ↔
  DifferentiableOn 𝕜 f s ∧ ∀ (y : E), ContDiffOn 𝕜 (↑n) (fun x => (fderivWithin 𝕜 f s x) y) s
-- Units.val_oneSub
@[simps val]
def oneSub (t : R) (h : ‖t‖ < 1) : Rˣ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (t : R) (h : ‖t‖ < 1)
Goal: ↑(Units.oneSub t h) = 1 - t
-- Units.val_add
@[simps! val]
def add (x : Rˣ) (t : R) (h : ‖t‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (t : R) (h : ‖t‖ < ‖↑x⁻¹‖⁻¹)
Goal: ↑(x.add t h) = ↑x + t
-- Units.val_ofNearby
@[simps! val]
def ofNearby (x : Rˣ) (y : R) (h : ‖y - x‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (y : R) (h : ‖y - ↑x‖ < ‖↑x⁻¹‖⁻¹)
Goal: ↑(x.ofNearby y h) = y
-- Units.isOpen
protected theorem isOpen : IsOpen { x : R | IsUnit x } 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R]
Goal: IsOpen {x | IsUnit x}
-- Units.nhds
protected theorem nhds (x : Rˣ) : { x : R | IsUnit x } ∈ 𝓝 (x : R) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ)
Goal: {x | IsUnit x} ∈ 𝓝 ↑x
-- nonunits.subset_compl_ball
theorem subset_compl_ball : nonunits R ⊆ (Metric.ball (1 : R) 1)ᶜ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R]
Goal: nonunits R ⊆ (Metric.ball 1 1)ᶜ
-- nonunits.isClosed
protected theorem isClosed : IsClosed (nonunits R) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R]
Goal: IsClosed (nonunits R)
-- NormedRing.inverse_one_sub
theorem inverse_one_sub (t : R) (h : ‖t‖ < 1) : inverse (1 - t) = ↑(Units.oneSub t h)⁻¹ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (t : R) (h : ‖t‖ < 1)
Goal: Ring.inverse (1 - t) = ↑(Units.oneSub t h)⁻¹
-- NormedRing.inverse_add
theorem inverse_add (x : Rˣ) :
    ∀ᶠ t in 𝓝 0, inverse ((x : R) + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ)
Goal: ∀ᶠ (t : R) in 𝓝 0, Ring.inverse (↑x + t) = Ring.inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹
-- NormedRing.inverse_one_sub_nth_order'
theorem inverse_one_sub_nth_order' (n : ℕ) {t : R} (ht : ‖t‖ < 1) :
    inverse ((1 : R) - t) = (∑ i ∈ range n, t ^ i) + t ^ n * inverse (1 - t) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (n : ℕ) {t : R} (ht : ‖t‖ < 1)
Goal: Ring.inverse (1 - t) = ∑ i ∈ Finset.range n, t ^ i + t ^ n * Ring.inverse (1 - t)
-- NormedRing.inverse_one_sub_nth_order
theorem inverse_one_sub_nth_order (n : ℕ) :
    ∀ᶠ t in 𝓝 0, inverse ((1 : R) - t) = (∑ i ∈ range n, t ^ i) + t ^ n * inverse (1 - t) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (n : ℕ)
Goal: ∀ᶠ (t : R) in 𝓝 0, Ring.inverse (1 - t) = ∑ i ∈ Finset.range n, t ^ i + t ^ n * Ring.inverse (1 - t)
-- NormedRing.inverse_add_nth_order
theorem inverse_add_nth_order (x : Rˣ) (n : ℕ) :
    ∀ᶠ t in 𝓝 0, inverse ((x : R) + t) =
      (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * inverse (x + t) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (n : ℕ)
Goal: ∀ᶠ (t : R) in 𝓝 0,
  Ring.inverse (↑x + t) = (∑ i ∈ Finset.range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * Ring.inverse (↑x + t)
-- NormedRing.inverse_one_sub_norm
theorem inverse_one_sub_norm : (fun t : R => inverse (1 - t)) =O[𝓝 0] (fun _t => 1 : R → ℝ) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R]
Goal: (fun t => Ring.inverse (1 - t)) =O[𝓝 0] fun _t => 1
-- NormedRing.inverse_add_norm
theorem inverse_add_norm (x : Rˣ) : (fun t : R => inverse (↑x + t)) =O[𝓝 0] fun _t => (1 : ℝ) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ)
Goal: (fun t => Ring.inverse (↑x + t)) =O[𝓝 0] fun _t => 1
-- NormedRing.inverse_add_norm_diff_nth_order
theorem inverse_add_norm_diff_nth_order (x : Rˣ) (n : ℕ) :
    (fun t : R => inverse (↑x + t) - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 (0 : R)]
      fun t => ‖t‖ ^ n 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ) (n : ℕ)
Goal: (fun t => Ring.inverse (↑x + t) - (∑ i ∈ Finset.range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ n
-- NormedRing.inverse_add_norm_diff_first_order
theorem inverse_add_norm_diff_first_order (x : Rˣ) :
    (fun t : R => inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ)
Goal: (fun t => Ring.inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖
-- NormedRing.inverse_add_norm_diff_second_order
theorem inverse_add_norm_diff_second_order (x : Rˣ) :
    (fun t : R => inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ 2 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ)
Goal: (fun t => Ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ 2
-- NormedRing.inverse_continuousAt
theorem inverse_continuousAt (x : Rˣ) : ContinuousAt inverse (x : R) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (x : Rˣ)
Goal: ContinuousAt Ring.inverse ↑x
-- Units.openEmbedding_val
theorem openEmbedding_val : OpenEmbedding (val : Rˣ → R) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R]
Goal: OpenEmbedding Units.val
-- Units.isOpenMap_val
theorem isOpenMap_val : IsOpenMap (val : Rˣ → R) 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R]
Goal: IsOpenMap Units.val
-- Ideal.eq_top_of_norm_lt_one
theorem eq_top_of_norm_lt_one (I : Ideal R) {x : R} (hxI : x ∈ I) (hx : ‖1 - x‖ < 1) : I = ⊤ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (I : Ideal R) {x : R} (hxI : x ∈ I) (hx : ‖1 - x‖ < 1)
Goal: I = ⊤
-- Ideal.closure_ne_top
theorem closure_ne_top (I : Ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤ 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] (I : Ideal R) (hI : I ≠ ⊤)
Goal: I.closure ≠ ⊤
-- Ideal.IsMaximal.closure_eq
theorem IsMaximal.closure_eq {I : Ideal R} (hI : I.IsMaximal) : I.closure = I 
Arguments: {R : Type u_1} [NormedRing R] [CompleteSpace R] {I : Ideal R} (hI : I.IsMaximal)
Goal: I.closure = I
-- seminormFromConst_seq_def
lemma seminormFromConst_seq_def (x : R) :
    seminormFromConst_seq c f x = fun n ↦ f (x * c ^ n) / f c ^ n 
Arguments: {R : Type u_1} [CommRing R] (c : R) (f : RingSeminorm R) (x : R)
Goal: seminormFromConst_seq c f x = fun n => f (x * c ^ n) / f c ^ n
-- seminormFromConst_seq_nonneg
theorem seminormFromConst_seq_nonneg (x : R) : 0 ≤ seminormFromConst_seq c f x 
Arguments: {R : Type u_1} [CommRing R] (c : R) (f : RingSeminorm R) (x : R)
Goal: 0 ≤ seminormFromConst_seq c f x
-- seminormFromConst_bddBelow
theorem seminormFromConst_bddBelow (x : R) :
    BddBelow (Set.range (seminormFromConst_seq c f x)) 
Arguments: {R : Type u_1} [CommRing R] (c : R) (f : RingSeminorm R) (x : R)
Goal: BddBelow (Set.range (seminormFromConst_seq c f x))
-- seminormFromConst_seq_zero
theorem seminormFromConst_seq_zero (hf : f 0 = 0) : seminormFromConst_seq c f 0 = 0 
Arguments: {R : Type u_1} [CommRing R] (c : R) {f : RingSeminorm R} (hf : f 0 = 0)
Goal: seminormFromConst_seq c f 0 = 0
-- seminormFromConst_seq_one
theorem seminormFromConst_seq_one (n : ℕ) (hn : 1 ≤ n) : seminormFromConst_seq c f 1 n = 1 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (n : ℕ) (hn : 1 ≤ n)
Goal: seminormFromConst_seq c f 1 n = 1
-- seminormFromConst_seq_antitone
theorem seminormFromConst_seq_antitone (x : R) : Antitone (seminormFromConst_seq c f x) 
Arguments: {R : Type u_1} [CommRing R] {c : R} {f : RingSeminorm R} (hf1 : f 1 ≤ 1) (hc : f c ≠ 0) (hpm : IsPowMul ⇑f) (x : R)
Goal: Antitone (seminormFromConst_seq c f x)