ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- UniformSpace.Completion.coe_inv
@[norm_cast]
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) 
Arguments: {K : Type u_1} [Field K] [UniformSpace K] [TopologicalDivisionRing K] [CompletableTopField K] (x : K)
Goal: (↑K x)⁻¹ = ↑K x⁻¹
-- UniformSpace.Completion.mul_hatInv_cancel
theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 
Arguments: {K : Type u_1} [Field K] [UniformSpace K] [TopologicalDivisionRing K] [CompletableTopField K] [UniformAddGroup K] {x : hat K} (x_ne : x ≠ 0)
Goal: x * x.hatInv = 1
-- UniformSpace.Completion.coe_one
@[norm_cast]
theorem coe_one : ((1 : α) : Completion α) = 1 
Arguments: (α : Type u_1) [Ring α] [UniformSpace α]
Goal: ↑α 1 = 1
-- UniformSpace.Completion.coe_mul
@[norm_cast]
theorem coe_mul (a b : α) : ((a * b : α) : Completion α) = a * b 
Arguments: {α : Type u_1} [Ring α] [UniformSpace α] [TopologicalRing α] (a b : α)
Goal: ↑α (a * b) = ↑α a * ↑α b
-- UniformSpace.Completion.continuous_mul
theorem continuous_mul : Continuous fun p : Completion α × Completion α => p.1 * p.2 
Arguments: {α : Type u_1} [Ring α] [UniformSpace α] [TopologicalRing α] [UniformAddGroup α]
Goal: Continuous fun p => p.1 * p.2
-- UniformSpace.Completion.Continuous.mul
theorem Continuous.mul {β : Type*} [TopologicalSpace β] {f g : β → Completion α}
    (hf : Continuous f) (hg : Continuous g) : Continuous fun b => f b * g b 
Arguments: {α : Type u_1} [Ring α] [UniformSpace α] [TopologicalRing α] [UniformAddGroup α] {β : Type u_2} [TopologicalSpace β] {f g : β → UniformSpace.Completion α} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous fun b => f b * g b
-- UniformSpace.Completion.continuous_coeRingHom
theorem continuous_coeRingHom : Continuous (coeRingHom : α → Completion α) 
Arguments: {α : Type u_1} [Ring α] [UniformSpace α] [TopologicalRing α] [UniformAddGroup α]
Goal: Continuous ⇑UniformSpace.Completion.coeRingHom
-- UniformSpace.Completion.map_smul_eq_mul_coe
@[simp]
theorem map_smul_eq_mul_coe (r : R) :
    Completion.map (r • ·) = ((algebraMap R A r : Completion A) * ·) 
Arguments: (A : Type u_2) [Ring A] [UniformSpace A] [UniformAddGroup A] [TopologicalRing A] (R : Type u_3) [CommSemiring R] [Algebra R A] [UniformContinuousConstSMul R A] (r : R)
Goal: (UniformSpace.Completion.map fun x => r • x) = fun x => ↑A ((algebraMap R A) r) * x
-- UniformSpace.Completion.algebraMap_def
theorem algebraMap_def (r : R) :
    algebraMap R (Completion A) r = (algebraMap R A r : Completion A) 
Arguments: (A : Type u_2) [Ring A] [UniformSpace A] [UniformAddGroup A] [TopologicalRing A] (R : Type u_3) [CommSemiring R] [Algebra R A] [UniformContinuousConstSMul R A] (r : R)
Goal: (algebraMap R (UniformSpace.Completion A)) r = ↑A ((algebraMap R A) r)
-- UniformSpace.inseparableSetoid_ring
theorem inseparableSetoid_ring (α) [CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    inseparableSetoid α = Submodule.quotientRel (Ideal.closure ⊥) 
Arguments: (α : Type u_2) [CommRing α] [TopologicalSpace α] [TopologicalRing α]
Goal: inseparableSetoid α = Submodule.quotientRel ⊥.closure
-- UniformSpace.ring_sep_quot
@[deprecated UniformSpace.inseparableSetoid_ring (since := "2024-02-16")]
theorem ring_sep_quot (α : Type u) [r : CommRing α] [TopologicalSpace α] [TopologicalRing α] :
    SeparationQuotient α = (α ⧸ (⊥ : Ideal α).closure) 
Arguments: (α : Type u) [r : CommRing α] [TopologicalSpace α] [TopologicalRing α]
Goal: SeparationQuotient α = (α ⧸ ⊥.closure)
-- RingSubgroupsBasis.leftMul
structure RingSubgroupsBasis {A ι : Type*} [Ring A] (B : ι → AddSubgroup A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
    leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (x * ·) ⁻¹' B i
    rightMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (· * x) ⁻¹' B i
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (x : A) (i : ι)
Goal: ∃ j, ↑(B j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(B i)
-- RingSubgroupsBasis.inter
structure RingSubgroupsBasis {A ι : Type*} [Ring A] (B : ι → AddSubgroup A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
    leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (x * ·) ⁻¹' B i
    rightMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (· * x) ⁻¹' B i
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (i j : ι)
Goal: ∃ k, B k ≤ B i ⊓ B j
-- RingSubgroupsBasis.mul
structure RingSubgroupsBasis {A ι : Type*} [Ring A] (B : ι → AddSubgroup A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
    leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (x * ·) ⁻¹' B i
    rightMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (· * x) ⁻¹' B i
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (i : ι)
Goal: ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)
-- RingSubgroupsBasis.rightMul
structure RingSubgroupsBasis {A ι : Type*} [Ring A] (B : ι → AddSubgroup A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
    leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (x * ·) ⁻¹' B i
    rightMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (· * x) ⁻¹' B i
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] {B : ι → AddSubgroup A} (self : RingSubgroupsBasis B) (x : A) (i : ι)
Goal: ∃ j, ↑(B j) ⊆ (fun x_1 => x_1 * x) ⁻¹' ↑(B i)
-- RingSubgroupsBasis.of_comm
theorem of_comm {A ι : Type*} [CommRing A] (B : ι → AddSubgroup A)
    (inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j) (mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i)
    (leftMul : ∀ x : A, ∀ i, ∃ j, (B j : Set A) ⊆ (fun y : A => x * y) ⁻¹' B i) :
    RingSubgroupsBasis B 
Arguments: {A : Type u_3} {ι : Type u_4} [CommRing A] (B : ι → AddSubgroup A) (inter : ∀ (i j : ι), ∃ k, B k ≤ B i ⊓ B j) (mul : ∀ (i : ι), ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)) (leftMul : ∀ (x : A) (i : ι), ∃ j, ↑(B j) ⊆ (fun y => x * y) ⁻¹' ↑(B i))
Goal: RingSubgroupsBasis B
-- RingSubgroupsBasis.mem_addGroupFilterBasis_iff
theorem mem_addGroupFilterBasis_iff {V : Set A} :
    V ∈ hB.toRingFilterBasis.toAddGroupFilterBasis ↔ ∃ i, V = B i 
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) {V : Set A}
Goal: V ∈ RingFilterBasis.toAddGroupFilterBasis ↔ ∃ i, V = ↑(B i)
-- RingSubgroupsBasis.mem_addGroupFilterBasis
theorem mem_addGroupFilterBasis (i) : (B i : Set A) ∈ hB.toRingFilterBasis.toAddGroupFilterBasis 
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) (i : ι)
Goal: ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis
-- RingSubgroupsBasis.hasBasis_nhds_zero
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i 
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B)
Goal: (𝓝 0).HasBasis (fun x => True) fun i => ↑(B i)
-- RingSubgroupsBasis.hasBasis_nhds
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } 
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) (a : A)
Goal: (𝓝 a).HasBasis (fun x => True) fun i => {b | b - a ∈ B i}
-- RingSubgroupsBasis.nonarchimedean
theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology 
Arguments: {A : Type u_1} {ι : Type u_2} [Ring A] [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B)
Goal: NonarchimedeanRing A
-- SubmodulesRingBasis.leftMul
structure SubmodulesRingBasis (B : ι → Submodule R A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    leftMul : ∀ (a : A) (i), ∃ j, a • B j ≤ B i
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (self : SubmodulesRingBasis B) (a : A) (i : ι)
Goal: ∃ j, a • B j ≤ B i
-- SubmodulesRingBasis.inter
structure SubmodulesRingBasis (B : ι → Submodule R A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    leftMul : ∀ (a : A) (i), ∃ j, a • B j ≤ B i
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (self : SubmodulesRingBasis B) (i j : ι)
Goal: ∃ k, B k ≤ B i ⊓ B j
-- SubmodulesRingBasis.mul
structure SubmodulesRingBasis (B : ι → Submodule R A) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    leftMul : ∀ (a : A) (i), ∃ j, a • B j ≤ B i
    mul : ∀ i, ∃ j, (B j : Set A) * B j ⊆ B i
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (self : SubmodulesRingBasis B) (i : ι)
Goal: ∃ j, ↑(B j) * ↑(B j) ⊆ ↑(B i)
-- SubmodulesRingBasis.toRing_subgroups_basis
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] {B : ι → Submodule R A} (hB : SubmodulesRingBasis B)
Goal: RingSubgroupsBasis fun i => (B i).toAddSubgroup
-- SubmodulesBasis.inter
structure SubmodulesBasis [TopologicalSpace R] (B : ι → Submodule R M) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    smul : ∀ (m : M) (i : ι), ∀ᶠ a in 𝓝 (0 : R), a • m ∈ B i
Arguments: {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] {B : ι → Submodule R M} (self : SubmodulesBasis B) (i j : ι)
Goal: ∃ k, B k ≤ B i ⊓ B j
-- SubmodulesBasis.smul
structure SubmodulesBasis [TopologicalSpace R] (B : ι → Submodule R M) : Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    smul : ∀ (m : M) (i : ι), ∀ᶠ a in 𝓝 (0 : R), a • m ∈ B i
Arguments: {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] {B : ι → Submodule R M} (self : SubmodulesBasis B) (m : M) (i : ι)
Goal: ∀ᶠ (a : R) in 𝓝 0, a • m ∈ B i
-- SubmodulesBasis.nonarchimedean
theorem nonarchimedean (hB : SubmodulesBasis B) : @NonarchimedeanAddGroup M _ hB.topology 
Arguments: {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] [TopologicalSpace R] [Nonempty ι] {B : ι → Submodule R M} (hB : SubmodulesBasis B)
Goal: NonarchimedeanAddGroup M
-- SubmodulesRingBasis.toSubmodulesBasis
theorem SubmodulesRingBasis.toSubmodulesBasis : SubmodulesBasis B 
Arguments: {ι : Type u_1} {R : Type u_2} {A : Type u_3} [CommRing R] [CommRing A] [Algebra R A] [TopologicalSpace R] {B : ι → Submodule R A} (hB : SubmodulesRingBasis B) (hsmul : ∀ (m : A) (i : ι), ∀ᶠ (a : R) in 𝓝 0, a • m ∈ B i)
Goal: SubmodulesBasis B
-- RingFilterBasis.SubmodulesBasis.smul
structure RingFilterBasis.SubmodulesBasis (BR : RingFilterBasis R) (B : ι → Submodule R M) :
    Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    smul : ∀ (m : M) (i : ι), ∃ U ∈ BR, U ⊆ (· • m) ⁻¹' B i
Arguments: {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] {BR : RingFilterBasis R} {B : ι → Submodule R M} (self : BR.SubmodulesBasis B) (m : M) (i : ι)
Goal: ∃ U ∈ BR, U ⊆ (fun x => x • m) ⁻¹' ↑(B i)
-- RingFilterBasis.SubmodulesBasis.inter
structure RingFilterBasis.SubmodulesBasis (BR : RingFilterBasis R) (B : ι → Submodule R M) :
    Prop where
    inter : ∀ i j, ∃ k, B k ≤ B i ⊓ B j
    smul : ∀ (m : M) (i : ι), ∃ U ∈ BR, U ⊆ (· • m) ⁻¹' B i
Arguments: {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] {BR : RingFilterBasis R} {B : ι → Submodule R M} (self : BR.SubmodulesBasis B) (i j : ι)
Goal: ∃ k, B k ≤ B i ⊓ B j
-- RingFilterBasis.submodulesBasisIsBasis
theorem RingFilterBasis.submodulesBasisIsBasis (BR : RingFilterBasis R) {B : ι → Submodule R M}
    (hB : BR.SubmodulesBasis B) : @_root_.SubmodulesBasis ι R _ M _ _ BR.topology B 
Arguments: {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_4} [AddCommGroup M] [Module R M] (BR : RingFilterBasis R) {B : ι → Submodule R M} (hB : BR.SubmodulesBasis B)
Goal: SubmodulesBasis B
-- equicontinuous_of_equicontinuousAt_zero
@[to_additive]
theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
    [UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
    (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    Equicontinuous ((↑) ∘ F) 
Arguments: {ι : Type u_1} {G : Type u_2} {M : Type u_3} {hom : Type u_4} [TopologicalSpace G] [UniformSpace M] [AddGroup G] [AddGroup M] [TopologicalAddGroup G] [UniformAddGroup M] [FunLike hom G M] [AddMonoidHomClass hom G M] (F : ι → hom) (hf : EquicontinuousAt (DFunLike.coe ∘ F) 0)
Goal: Equicontinuous (DFunLike.coe ∘ F)
-- equicontinuous_of_equicontinuousAt_one
@[to_additive]
theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
    [UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
    (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    Equicontinuous ((↑) ∘ F) 
Arguments: {ι : Type u_1} {G : Type u_2} {M : Type u_3} {hom : Type u_4} [TopologicalSpace G] [UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M] [FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom) (hf : EquicontinuousAt (DFunLike.coe ∘ F) 1)
Goal: Equicontinuous (DFunLike.coe ∘ F)
-- uniformEquicontinuous_of_equicontinuousAt_zero
@[to_additive]
theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
    [UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M]
    (F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    UniformEquicontinuous ((↑) ∘ F) 
Arguments: {ι : Type u_1} {G : Type u_2} {M : Type u_3} {hom : Type u_4} [UniformSpace G] [UniformSpace M] [AddGroup G] [AddGroup M] [UniformAddGroup G] [UniformAddGroup M] [FunLike hom G M] [AddMonoidHomClass hom G M] (F : ι → hom) (hf : EquicontinuousAt (DFunLike.coe ∘ F) 0)
Goal: UniformEquicontinuous (DFunLike.coe ∘ F)
-- uniformEquicontinuous_of_equicontinuousAt_one
@[to_additive]
theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
    [UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
    [FunLike hom G M] [MonoidHomClass hom G M]
    (F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
    UniformEquicontinuous ((↑) ∘ F) 
Arguments: {ι : Type u_1} {G : Type u_2} {M : Type u_3} {hom : Type u_4} [UniformSpace G] [UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M] [FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom) (hf : EquicontinuousAt (DFunLike.coe ∘ F) 1)
Goal: UniformEquicontinuous (DFunLike.coe ∘ F)
-- AddGroupFilterBasis.uniformAddGroup
protected theorem uniformAddGroup : @UniformAddGroup G B.uniformSpace _ 
Arguments: {G : Type u_1} [AddCommGroup G] (B : AddGroupFilterBasis G)
Goal: UniformAddGroup G
-- AddGroupFilterBasis.cauchy_iff
theorem cauchy_iff {F : Filter G} :
    @Cauchy G B.uniformSpace F ↔
      F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U 
Arguments: {G : Type u_1} [AddCommGroup G] (B : AddGroupFilterBasis G) {F : Filter G}
Goal: Cauchy F ↔ F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
-- TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_addGroup
@[to_additive
  "Every topological additive group
  in which there exists a compact set with nonempty interior is locally compact."]
theorem TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
    (K : PositiveCompacts G) : LocallyCompactSpace G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (K : TopologicalSpace.PositiveCompacts G)
Goal: LocallyCompactSpace G
-- TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
@[to_additive
  "Every topological additive group
  in which there exists a compact set with nonempty interior is locally compact."]
theorem TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
    (K : PositiveCompacts G) : LocallyCompactSpace G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (K : TopologicalSpace.PositiveCompacts G)
Goal: LocallyCompactSpace G
-- Polynomial.continuous_eval₂
@[continuity, fun_prop]
protected theorem continuous_eval₂ [Semiring S] (p : S[X]) (f : S →+* R) :
    Continuous fun x => p.eval₂ f x 
Arguments: {R : Type u_1} {S : Type u_2} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] [Semiring S] (p : Polynomial S) (f : S →+* R)
Goal: Continuous fun x => Polynomial.eval₂ f x p
-- Polynomial.continuous
@[continuity, fun_prop]
protected theorem continuous : Continuous fun x => p.eval x 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R)
Goal: Continuous fun x => Polynomial.eval x p
-- Polynomial.continuousAt
@[fun_prop]
protected theorem continuousAt {a : R} : ContinuousAt (fun x => p.eval x) a 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) {a : R}
Goal: ContinuousAt (fun x => Polynomial.eval x p) a
-- Polynomial.continuousWithinAt
@[fun_prop]
protected theorem continuousWithinAt {s a} : ContinuousWithinAt (fun x => p.eval x) s a 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) {s : Set R} {a : R}
Goal: ContinuousWithinAt (fun x => Polynomial.eval x p) s a
-- Polynomial.continuousOn
@[fun_prop]
protected theorem continuousOn {s} : ContinuousOn (fun x => p.eval x) s 
Arguments: {R : Type u_1} [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (p : Polynomial R) {s : Set R}
Goal: ContinuousOn (fun x => Polynomial.eval x p) s
-- Polynomial.continuous_aeval
@[continuity, fun_prop]
protected theorem continuous_aeval : Continuous fun x : A => aeval x p 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R)
Goal: Continuous fun x => (Polynomial.aeval x) p
-- Polynomial.continuousAt_aeval
@[fun_prop]
protected theorem continuousAt_aeval {a : A} : ContinuousAt (fun x : A => aeval x p) a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) {a : A}
Goal: ContinuousAt (fun x => (Polynomial.aeval x) p) a
-- Polynomial.continuousWithinAt_aeval
@[fun_prop]
protected theorem continuousWithinAt_aeval {s a} :
    ContinuousWithinAt (fun x : A => aeval x p) s a 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) {s : Set A} {a : A}
Goal: ContinuousWithinAt (fun x => (Polynomial.aeval x) p) s a
-- Polynomial.continuousOn_aeval
@[fun_prop]
protected theorem continuousOn_aeval {s} : ContinuousOn (fun x : A => aeval x p) s 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] [TopologicalSpace A] [TopologicalSemiring A] (p : Polynomial R) {s : Set A}
Goal: ContinuousOn (fun x => (Polynomial.aeval x) p) s
-- Polynomial.tendsto_abv_eval₂_atTop
theorem tendsto_abv_eval₂_atTop {R S k α : Type*} [Semiring R] [Ring S] [LinearOrderedField k]
    (f : R →+* S) (abv : S → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
    (hf : f p.leadingCoeff ≠ 0) {l : Filter α} {z : α → S} (hz : Tendsto (abv ∘ z) l atTop) :
    Tendsto (fun x => abv (p.eval₂ f (z x))) l atTop 
Arguments: {R : Type u_1} {S : Type u_2} {k : Type u_3} {α : Type u_4} [Semiring R] [Ring S] [LinearOrderedField k] (f : R →+* S) (abv : S → k) [IsAbsoluteValue abv] (p : Polynomial R) (hd : 0 < p.degree) (hf : f p.leadingCoeff ≠ 0) {l : Filter α} {z : α → S} (hz : Filter.Tendsto (abv ∘ z) l Filter.atTop)
Goal: Filter.Tendsto (fun x => abv (Polynomial.eval₂ f (z x) p)) l Filter.atTop
-- Polynomial.tendsto_abv_atTop
theorem tendsto_abv_atTop {R k α : Type*} [Ring R] [LinearOrderedField k] (abv : R → k)
    [IsAbsoluteValue abv] (p : R[X]) (h : 0 < degree p) {l : Filter α} {z : α → R}
    (hz : Tendsto (abv ∘ z) l atTop) : Tendsto (fun x => abv (p.eval (z x))) l atTop 
Arguments: {R : Type u_1} {k : Type u_2} {α : Type u_3} [Ring R] [LinearOrderedField k] (abv : R → k) [IsAbsoluteValue abv] (p : Polynomial R) (h : 0 < p.degree) {l : Filter α} {z : α → R} (hz : Filter.Tendsto (abv ∘ z) l Filter.atTop)
Goal: Filter.Tendsto (fun x => abv (Polynomial.eval (z x) p)) l Filter.atTop
-- Polynomial.tendsto_abv_aeval_atTop
theorem tendsto_abv_aeval_atTop {R A k α : Type*} [CommSemiring R] [Ring A] [Algebra R A]
    [LinearOrderedField k] (abv : A → k) [IsAbsoluteValue abv] (p : R[X]) (hd : 0 < degree p)
    (h₀ : algebraMap R A p.leadingCoeff ≠ 0) {l : Filter α} {z : α → A}
    (hz : Tendsto (abv ∘ z) l atTop) : Tendsto (fun x => abv (aeval (z x) p)) l atTop 
Arguments: {R : Type u_1} {A : Type u_2} {k : Type u_3} {α : Type u_4} [CommSemiring R] [Ring A] [Algebra R A] [LinearOrderedField k] (abv : A → k) [IsAbsoluteValue abv] (p : Polynomial R) (hd : 0 < p.degree) (h₀ : (algebraMap R A) p.leadingCoeff ≠ 0) {l : Filter α} {z : α → A} (hz : Filter.Tendsto (abv ∘ z) l Filter.atTop)
Goal: Filter.Tendsto (fun x => abv ((Polynomial.aeval (z x)) p)) l Filter.atTop
-- Polynomial.tendsto_norm_atTop
theorem tendsto_norm_atTop (p : R[X]) (h : 0 < degree p) {l : Filter α} {z : α → R}
    (hz : Tendsto (fun x => ‖z x‖) l atTop) : Tendsto (fun x => ‖p.eval (z x)‖) l atTop 
Arguments: {α : Type u_1} {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] (p : Polynomial R) (h : 0 < p.degree) {l : Filter α} {z : α → R} (hz : Filter.Tendsto (fun x => ‖z x‖) l Filter.atTop)
Goal: Filter.Tendsto (fun x => ‖Polynomial.eval (z x) p‖) l Filter.atTop
-- Polynomial.exists_forall_norm_le
theorem exists_forall_norm_le [ProperSpace R] (p : R[X]) : ∃ x, ∀ y, ‖p.eval x‖ ≤ ‖p.eval y‖ 
Arguments: {R : Type u_2} [NormedRing R] [IsAbsoluteValue norm] [ProperSpace R] (p : Polynomial R)
Goal: ∃ x, ∀ (y : R), ‖Polynomial.eval x p‖ ≤ ‖Polynomial.eval y p‖
-- Polynomial.eq_one_of_roots_le
theorem eq_one_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (hB : B < 0) (h1 : p.Monic)
    (h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) : p = 1 
Arguments: {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {p : Polynomial F} {f : F →+* K} {B : ℝ} (hB : B < 0) (h1 : p.Monic) (h2 : Polynomial.Splits f p) (h3 : ∀ z ∈ (Polynomial.map f p).roots, ‖z‖ ≤ B)
Goal: p = 1
-- Polynomial.coeff_le_of_roots_le
theorem coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic)
    (h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
    ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * p.natDegree.choose i 
Arguments: {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {p : Polynomial F} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic) (h2 : Polynomial.Splits f p) (h3 : ∀ z ∈ (Polynomial.map f p).roots, ‖z‖ ≤ B)
Goal: ‖(Polynomial.map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)
-- Polynomial.coeff_bdd_of_roots_le
theorem coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]} (h1 : p.Monic)
    (h2 : Splits f p) (h3 : p.natDegree ≤ d) (h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) (i : ℕ) :
    ‖(map f p).coeff i‖ ≤ max B 1 ^ d * d.choose (d / 2) 
Arguments: {F : Type u_3} {K : Type u_4} [CommRing F] [NormedField K] {B : ℝ} {d : ℕ} (f : F →+* K) {p : Polynomial F} (h1 : p.Monic) (h2 : Polynomial.Splits f p) (h3 : p.natDegree ≤ d) (h4 : ∀ z ∈ (Polynomial.map f p).roots, ‖z‖ ≤ B) (i : ℕ)
Goal: ‖(Polynomial.map f p).coeff i‖ ≤ max B 1 ^ d * ↑(d.choose (d / 2))
-- ContinuousMultilinearMap.cont
structure ContinuousMultilinearMap (R : Type u) {ι : Type v} (M₁ : ι → Type w₁) (M₂ : Type w₂)
  [Semiring R] [∀ i, AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [∀ i, Module R (M₁ i)] [Module R M₂]
  [∀ i, TopologicalSpace (M₁ i)] [TopologicalSpace M₂] extends MultilinearMap R M₁ M₂ where
  cont : Continuous toFun
Arguments: {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [(i : ι) → TopologicalSpace (M₁ i)] [TopologicalSpace M₂] (self : ContinuousMultilinearMap R M₁ M₂)
Goal: Continuous self.toFun