ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- continuous_selfAdjointPart
theorem continuous_selfAdjointPart [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    Continuous (selfAdjointPart R (A := A)) 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A]
Goal: Continuous ⇑(selfAdjointPart R)
-- continuous_skewAdjointPart
theorem continuous_skewAdjointPart [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    Continuous (skewAdjointPart R (A := A)) 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A]
Goal: Continuous ⇑(skewAdjointPart R)
-- continuous_decomposeProdAdjoint
theorem continuous_decomposeProdAdjoint [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : Continuous (StarModule.decomposeProdAdjoint R A) 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [TopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A]
Goal: Continuous ⇑(StarModule.decomposeProdAdjoint R A)
-- continuous_decomposeProdAdjoint_symm
theorem continuous_decomposeProdAdjoint_symm [TopologicalAddGroup A] :
    Continuous (StarModule.decomposeProdAdjoint R A).symm 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [TopologicalAddGroup A]
Goal: Continuous ⇑(StarModule.decomposeProdAdjoint R A).symm
-- selfAdjointPartL_toFun_coe
@[simps!]
def selfAdjointPartL [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] selfAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] (x : A)
Goal: ↑((selfAdjointPartL R A) x) = ⅟2 • x + ⅟2 • star x
-- selfAdjointPartL_apply_coe
@[simps!]
def selfAdjointPartL [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] selfAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousAdd A] [ContinuousStar A] [ContinuousConstSMul R A] (x : A)
Goal: ↑((selfAdjointPartL R A) x) = ⅟2 • x + ⅟2 • star x
-- skewAdjointPartL_toFun_coe
@[simps!]
def skewAdjointPartL [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] skewAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] (x : A)
Goal: ↑((skewAdjointPartL R A) x) = ⅟2 • (x - star x)
-- skewAdjointPartL_apply_coe
@[simps!]
def skewAdjointPartL [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] :
    A →L[R] skewAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [ContinuousSub A] [ContinuousStar A] [ContinuousConstSMul R A] (x : A)
Goal: ↑((skewAdjointPartL R A) x) = ⅟2 • (x - star x)
-- StarModule.decomposeProdAdjointL_symm_apply
@[simps!]
def StarModule.decomposeProdAdjointL [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : A ≃L[R] selfAdjoint A × skewAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [TopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A] (a : ↥(selfAdjoint A) × ↥(skewAdjoint A))
Goal: (StarModule.decomposeProdAdjointL R A).symm a =
  (selfAdjoint.submodule R A).subtype a.1 + (skewAdjoint.submodule R A).subtype a.2
-- StarModule.decomposeProdAdjointL_apply
@[simps!]
def StarModule.decomposeProdAdjointL [TopologicalAddGroup A] [ContinuousStar A]
    [ContinuousConstSMul R A] : A ≃L[R] selfAdjoint A × skewAdjoint A 
Arguments: (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] [TopologicalSpace A] [TopologicalAddGroup A] [ContinuousStar A] [ContinuousConstSMul R A] (i : A)
Goal: (StarModule.decomposeProdAdjointL R A) i = ((selfAdjointPart R) i, (skewAdjointPart R) i)
-- LocallyConvexSpace.convex_basis
class LocallyConvexSpace (𝕜 E : Type*) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E]
    [TopologicalSpace E] : Prop where
  convex_basis : ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [self : LocallyConvexSpace 𝕜 E] (x : E)
Goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ Convex 𝕜 s) id
-- locallyConvexSpace_iff
theorem locallyConvexSpace_iff :
    LocallyConvexSpace 𝕜 E ↔ ∀ x : E, (𝓝 x).HasBasis (fun s : Set E => s ∈ 𝓝 x ∧ Convex 𝕜 s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
Goal: LocallyConvexSpace 𝕜 E ↔ ∀ (x : E), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ Convex 𝕜 s) id
-- LocallyConvexSpace.ofBases
theorem LocallyConvexSpace.ofBases {ι : Type*} (b : E → ι → Set E) (p : E → ι → Prop)
    (hbasis : ∀ x : E, (𝓝 x).HasBasis (p x) (b x)) (hconvex : ∀ x i, p x i → Convex 𝕜 (b x i)) :
    LocallyConvexSpace 𝕜 E 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] {ι : Type u_3} (b : E → ι → Set E) (p : E → ι → Prop) (hbasis : ∀ (x : E), (𝓝 x).HasBasis (p x) (b x)) (hconvex : ∀ (x : E) (i : ι), p x i → Convex 𝕜 (b x i))
Goal: LocallyConvexSpace 𝕜 E
-- LocallyConvexSpace.convex_basis_zero
theorem LocallyConvexSpace.convex_basis_zero [LocallyConvexSpace 𝕜 E] :
    (𝓝 0 : Filter E).HasBasis (fun s => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [LocallyConvexSpace 𝕜 E]
Goal: (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Convex 𝕜 s) id
-- locallyConvexSpace_iff_exists_convex_subset
theorem locallyConvexSpace_iff_exists_convex_subset :
    LocallyConvexSpace 𝕜 E ↔ ∀ x : E, ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, Convex 𝕜 S ∧ S ⊆ U 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
Goal: LocallyConvexSpace 𝕜 E ↔ ∀ (x : E), ∀ U ∈ 𝓝 x, ∃ S ∈ 𝓝 x, Convex 𝕜 S ∧ S ⊆ U
-- LocallyConvexSpace.ofBasisZero
theorem LocallyConvexSpace.ofBasisZero {ι : Type*} (b : ι → Set E) (p : ι → Prop)
    (hbasis : (𝓝 0).HasBasis p b) (hconvex : ∀ i, p i → Convex 𝕜 (b i)) :
    LocallyConvexSpace 𝕜 E 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] {ι : Type u_3} (b : ι → Set E) (p : ι → Prop) (hbasis : (𝓝 0).HasBasis p b) (hconvex : ∀ (i : ι), p i → Convex 𝕜 (b i))
Goal: LocallyConvexSpace 𝕜 E
-- locallyConvexSpace_iff_zero
theorem locallyConvexSpace_iff_zero : LocallyConvexSpace 𝕜 E ↔
    (𝓝 0 : Filter E).HasBasis (fun s : Set E => s ∈ (𝓝 0 : Filter E) ∧ Convex 𝕜 s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E]
Goal: LocallyConvexSpace 𝕜 E ↔ (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Convex 𝕜 s) id
-- locallyConvexSpace_iff_exists_convex_subset_zero
theorem locallyConvexSpace_iff_exists_convex_subset_zero :
    LocallyConvexSpace 𝕜 E ↔ ∀ U ∈ (𝓝 0 : Filter E), ∃ S ∈ (𝓝 0 : Filter E), Convex 𝕜 S ∧ S ⊆ U 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [OrderedSemiring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E]
Goal: LocallyConvexSpace 𝕜 E ↔ ∀ U ∈ 𝓝 0, ∃ S ∈ 𝓝 0, Convex 𝕜 S ∧ S ⊆ U
-- LocallyConvexSpace.convex_open_basis_zero
theorem LocallyConvexSpace.convex_open_basis_zero [LocallyConvexSpace 𝕜 E] :
    (𝓝 0 : Filter E).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) id 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [LocallyConvexSpace 𝕜 E]
Goal: (𝓝 0).HasBasis (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) id
-- Disjoint.exists_open_convexes
theorem Disjoint.exists_open_convexes [LocallyConvexSpace 𝕜 E] {s t : Set E} (disj : Disjoint s t)
    (hs₁ : Convex 𝕜 s) (hs₂ : IsCompact s) (ht₁ : Convex 𝕜 t) (ht₂ : IsClosed t) :
    ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [LocallyConvexSpace 𝕜 E] {s t : Set E} (disj : Disjoint s t) (hs₁ : Convex 𝕜 s) (hs₂ : IsCompact s) (ht₁ : Convex 𝕜 t) (ht₂ : IsClosed t)
Goal: ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
-- locallyConvexSpace_sInf
theorem locallyConvexSpace_sInf {ts : Set (TopologicalSpace E)}
    (h : ∀ t ∈ ts, @LocallyConvexSpace 𝕜 E _ _ _ t) : @LocallyConvexSpace 𝕜 E _ _ _ (sInf ts) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {ts : Set (TopologicalSpace E)} (h : ∀ t ∈ ts, LocallyConvexSpace 𝕜 E)
Goal: LocallyConvexSpace 𝕜 E
-- locallyConvexSpace_iInf
theorem locallyConvexSpace_iInf {ts' : ι → TopologicalSpace E}
    (h' : ∀ i, @LocallyConvexSpace 𝕜 E _ _ _ (ts' i)) :
    @LocallyConvexSpace 𝕜 E _ _ _ (⨅ i, ts' i) 
Arguments: {ι : Sort u_1} {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {ts' : ι → TopologicalSpace E} (h' : ∀ (i : ι), LocallyConvexSpace 𝕜 E)
Goal: LocallyConvexSpace 𝕜 E
-- locallyConvexSpace_inf
theorem locallyConvexSpace_inf {t₁ t₂ : TopologicalSpace E} (h₁ : @LocallyConvexSpace 𝕜 E _ _ _ t₁)
    (h₂ : @LocallyConvexSpace 𝕜 E _ _ _ t₂) : @LocallyConvexSpace 𝕜 E _ _ _ (t₁ ⊓ t₂) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {t₁ t₂ : TopologicalSpace E} (h₁ : LocallyConvexSpace 𝕜 E) (h₂ : LocallyConvexSpace 𝕜 E)
Goal: LocallyConvexSpace 𝕜 E
-- locallyConvexSpace_induced
theorem locallyConvexSpace_induced {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F]
    (f : E →ₗ[𝕜] F) : @LocallyConvexSpace 𝕜 E _ _ _ (t.induced f) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] {t : TopologicalSpace F} [LocallyConvexSpace 𝕜 F] (f : E →ₗ[𝕜] F)
Goal: LocallyConvexSpace 𝕜 E
-- NonarchimedeanAddGroup.is_nonarchimedean
class NonarchimedeanAddGroup (G : Type*) [AddGroup G] [TopologicalSpace G] extends
  TopologicalAddGroup G : Prop where
  is_nonarchimedean : ∀ U ∈ 𝓝 (0 : G), ∃ V : OpenAddSubgroup G, (V : Set G) ⊆ U
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [self : NonarchimedeanAddGroup G] (U : Set G)
Goal: U ∈ 𝓝 0 → ∃ V, ↑V ⊆ U
-- NonarchimedeanGroup.is_nonarchimedean
@[to_additive]
class NonarchimedeanGroup (G : Type*) [Group G] [TopologicalSpace G] extends TopologicalGroup G :
  Prop where
  is_nonarchimedean : ∀ U ∈ 𝓝 (1 : G), ∃ V : OpenSubgroup G, (V : Set G) ⊆ U
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [self : NonarchimedeanGroup G] (U : Set G)
Goal: U ∈ 𝓝 1 → ∃ V, ↑V ⊆ U
-- NonarchimedeanRing.is_nonarchimedean
class NonarchimedeanRing (R : Type*) [Ring R] [TopologicalSpace R] extends TopologicalRing R :
  Prop where
  is_nonarchimedean : ∀ U ∈ 𝓝 (0 : R), ∃ V : OpenAddSubgroup R, (V : Set R) ⊆ U
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [self : NonarchimedeanRing R] (U : Set R)
Goal: U ∈ 𝓝 0 → ∃ V, ↑V ⊆ U
-- NonarchimedeanAddGroup.nonarchimedean_of_emb
@[to_additive]
theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [NonarchimedeanAddGroup G] {H : Type u_2} [AddGroup H] [TopologicalSpace H] [TopologicalAddGroup H] (f : G →+ H) (emb : OpenEmbedding ⇑f)
Goal: NonarchimedeanAddGroup H
-- NonarchimedeanGroup.nonarchimedean_of_emb
@[to_additive]
theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [NonarchimedeanGroup G] {H : Type u_2} [Group H] [TopologicalSpace H] [TopologicalGroup H] (f : G →* H) (emb : OpenEmbedding ⇑f)
Goal: NonarchimedeanGroup H
-- NonarchimedeanGroup.prod_subset
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains the cartesian product of
an open neighborhood in each group."]
theorem prod_subset {U} (hU : U ∈ 𝓝 (1 : G × K)) :
    ∃ (V : OpenSubgroup G) (W : OpenSubgroup K), (V : Set G) ×ˢ (W : Set K) ⊆ U 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [NonarchimedeanGroup G] {K : Type u_3} [Group K] [TopologicalSpace K] [NonarchimedeanGroup K] {U : Set (G × K)} (hU : U ∈ 𝓝 1)
Goal: ∃ V W, ↑V ×ˢ ↑W ⊆ U
-- NonarchimedeanAddGroup.prod_subset
@[to_additive NonarchimedeanAddGroup.prod_subset "An open neighborhood of the identity in
the cartesian product of two nonarchimedean groups contains the cartesian product of
an open neighborhood in each group."]
theorem prod_subset {U} (hU : U ∈ 𝓝 (1 : G × K)) :
    ∃ (V : OpenSubgroup G) (W : OpenSubgroup K), (V : Set G) ×ˢ (W : Set K) ⊆ U 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [NonarchimedeanAddGroup G] {K : Type u_3} [AddGroup K] [TopologicalSpace K] [NonarchimedeanAddGroup K] {U : Set (G × K)} (hU : U ∈ 𝓝 0)
Goal: ∃ V W, ↑V ×ˢ ↑W ⊆ U
-- NonarchimedeanGroup.prod_self_subset
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the cartesian square of
an open neighborhood in the group."]
theorem prod_self_subset {U} (hU : U ∈ 𝓝 (1 : G × G)) :
    ∃ V : OpenSubgroup G, (V : Set G) ×ˢ (V : Set G) ⊆ U 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [NonarchimedeanGroup G] {U : Set (G × G)} (hU : U ∈ 𝓝 1)
Goal: ∃ V, ↑V ×ˢ ↑V ⊆ U
-- NonarchimedeanAddGroup.prod_self_subset
@[to_additive NonarchimedeanAddGroup.prod_self_subset "An open neighborhood of the identity in
the cartesian square of a nonarchimedean group contains the cartesian square of
an open neighborhood in the group."]
theorem prod_self_subset {U} (hU : U ∈ 𝓝 (1 : G × G)) :
    ∃ V : OpenSubgroup G, (V : Set G) ×ˢ (V : Set G) ⊆ U 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [NonarchimedeanAddGroup G] {U : Set (G × G)} (hU : U ∈ 𝓝 0)
Goal: ∃ V, ↑V ×ˢ ↑V ⊆ U
-- NonarchimedeanRing.left_mul_subset
theorem left_mul_subset (U : OpenAddSubgroup R) (r : R) :
    ∃ V : OpenAddSubgroup R, r • (V : Set R) ⊆ U 
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [NonarchimedeanRing R] (U : OpenAddSubgroup R) (r : R)
Goal: ∃ V, r • ↑V ⊆ ↑U
-- NonarchimedeanRing.mul_subset
theorem mul_subset (U : OpenAddSubgroup R) : ∃ V : OpenAddSubgroup R, (V : Set R) * V ⊆ U 
Arguments: {R : Type u_1} [Ring R] [TopologicalSpace R] [NonarchimedeanRing R] (U : OpenAddSubgroup R)
Goal: ∃ V, ↑V * ↑V ⊆ ↑U
-- WeakBilin.coeFn_continuous
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y 
Arguments: {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: Continuous fun x y => (B x) y
-- WeakBilin.eval_continuous
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y 
Arguments: {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (y : F)
Goal: Continuous fun x => (B x) y
-- WeakBilin.continuous_of_continuous_eval
theorem continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakBilin B}
    (h : ∀ y, Continuous fun a => B (g a) y) : Continuous g 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [TopologicalSpace α] {g : α → WeakBilin B} (h : ∀ (y : F), Continuous fun a => (B (g a)) y)
Goal: Continuous g
-- WeakBilin.embedding
theorem embedding {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
    Embedding fun (x : WeakBilin B) y => B x y 
Arguments: {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective ⇑B)
Goal: Embedding fun x y => (B x) y
-- WeakBilin.tendsto_iff_forall_eval_tendsto
theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
    (hB : Function.Injective B) :
    Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B} (hB : Function.Injective ⇑B)
Goal: Filter.Tendsto f l (𝓝 x) ↔ ∀ (y : F), Filter.Tendsto (fun i => (B (f i)) y) l (𝓝 ((B x) y))
-- topDualPairing_apply
theorem topDualPairing_apply (v : E →L[𝕜] 𝕜) (x : E) : topDualPairing 𝕜 E v x = v x 
Arguments: {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (v : E →L[𝕜] 𝕜) (x : E)
Goal: ((topDualPairing 𝕜 E) v) x = v x
-- WeakDual.coeFn_continuous
theorem coeFn_continuous : Continuous fun (x : WeakDual 𝕜 E) y => x y 
Arguments: {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
Goal: Continuous fun x y => x y
-- WeakDual.eval_continuous
theorem eval_continuous (y : E) : Continuous fun x : WeakDual 𝕜 E => x y 
Arguments: {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (y : E)
Goal: Continuous fun x => x y
-- WeakDual.continuous_of_continuous_eval
theorem continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakDual 𝕜 E}
    (h : ∀ y, Continuous fun a => (g a) y) : Continuous g 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalSpace α] {g : α → WeakDual 𝕜 E} (h : ∀ (y : E), Continuous fun a => (g a) y)
Goal: Continuous g
-- WeakSpace.map_apply
theorem map_apply (f : E →L[𝕜] F) (x : E) : WeakSpace.map f x = f x 
Arguments: {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] (f : E →L[𝕜] F) (x : E)
Goal: (WeakSpace.map f) x = f x
-- WeakSpace.coe_map
@[simp]
theorem coe_map (f : E →L[𝕜] F) : (WeakSpace.map f : E → F) = f 
Arguments: {𝕜 : Type u_2} {E : Type u_5} {F : Type u_6} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] (f : E →L[𝕜] F)
Goal: ⇑(WeakSpace.map f) = ⇑f
-- continuousLinearMapToWeakSpace_eq_toWeakSpace
@[simp]
theorem continuousLinearMapToWeakSpace_eq_toWeakSpace (x : E) :
    continuousLinearMapToWeakSpace 𝕜 E x = toWeakSpace 𝕜 E x 
Arguments: (𝕜 : Type u_2) (E : Type u_5) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (x : E)
Goal: (continuousLinearMapToWeakSpace 𝕜 E) x = (toWeakSpace 𝕜 E) x
-- continuousLinearMapToWeakSpace_bijective
theorem continuousLinearMapToWeakSpace_bijective :
    Function.Bijective (continuousLinearMapToWeakSpace 𝕜 E) 
Arguments: {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
Goal: Function.Bijective ⇑(continuousLinearMapToWeakSpace 𝕜 E)
-- isOpenMap_toWeakSpace_symm
theorem isOpenMap_toWeakSpace_symm : IsOpenMap (toWeakSpace 𝕜 E).symm 
Arguments: {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
Goal: IsOpenMap ⇑(toWeakSpace 𝕜 E).symm
-- WeakSpace.isOpen_of_isOpen
theorem WeakSpace.isOpen_of_isOpen (V : Set E)
    (hV : IsOpen ((continuousLinearMapToWeakSpace 𝕜 E) '' V : Set (WeakSpace 𝕜 E))) : IsOpen V 
Arguments: {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (V : Set E) (hV : IsOpen (⇑(continuousLinearMapToWeakSpace 𝕜 E) '' V))
Goal: IsOpen V
-- tendsto_iff_forall_eval_tendsto_topDualPairing
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter α} {f : α → WeakDual 𝕜 E}
    {x : WeakDual 𝕜 E} :
    Tendsto f l (𝓝 x) ↔
      ∀ y, Tendsto (fun i => topDualPairing 𝕜 E (f i) y) l (𝓝 (topDualPairing 𝕜 E x y)) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] {l : Filter α} {f : α → WeakDual 𝕜 E} {x : WeakDual 𝕜 E}
Goal: Filter.Tendsto f l (𝓝 x) ↔
  ∀ (y : E), Filter.Tendsto (fun i => ((topDualPairing 𝕜 E) (f i)) y) l (𝓝 (((topDualPairing 𝕜 E) x) y))
-- continuum_le_cardinal_of_nontriviallyNormedField
theorem continuum_le_cardinal_of_nontriviallyNormedField
    (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] : 𝔠 ≤ #𝕜 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
Goal: 𝔠 ≤ #𝕜
-- continuum_le_cardinal_of_module
theorem continuum_le_cardinal_of_module
    (𝕜 : Type u) (E : Type v) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
    [AddCommGroup E] [Module 𝕜 E] [Nontrivial E] : 𝔠 ≤ #E 
Arguments: (𝕜 : Type u) (E : Type v) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [AddCommGroup E] [Module 𝕜 E] [Nontrivial E]
Goal: 𝔠 ≤ #E
-- cardinal_eq_of_mem_nhds_zero
lemma cardinal_eq_of_mem_nhds_zero
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : s ∈ 𝓝 (0 : E)) : #s = #E 
Arguments: {E : Type u_1} (𝕜 : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : s ∈ 𝓝 0)
Goal: #↑s = #E
-- cardinal_eq_of_mem_nhds
theorem cardinal_eq_of_mem_nhds
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
    {s : Set E} {x : E} (hs : s ∈ 𝓝 x) : #s = #E 
Arguments: {E : Type u_1} (𝕜 : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E} {x : E} (hs : s ∈ 𝓝 x)
Goal: #↑s = #E
-- cardinal_eq_of_isOpen
theorem cardinal_eq_of_isOpen
    {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
    [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E}
    (hs : IsOpen s) (h's : s.Nonempty) : #s = #E 
Arguments: {E : Type u_1} (𝕜 : Type u_2) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : IsOpen s) (h's : s.Nonempty)
Goal: #↑s = #E