ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ZeroAtInftyContinuousMap.coe_sub
@[simp]
theorem coe_sub : ⇑(f - g) = f - g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (f g : α →C₀ β)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- ZeroAtInftyContinuousMap.sub_apply
theorem sub_apply : (f - g) x = f x - g x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddGroup β] [TopologicalAddGroup β] (f g : α →C₀ β)
Goal: (f - g) x = f x - g x
-- ZeroAtInftyContinuousMap.uniformContinuous
theorem uniformContinuous (f : F) : UniformContinuous (f : β → γ) 
Arguments: {F : Type u_1} {β : Type v} {γ : Type w} [UniformSpace β] [UniformSpace γ] [Zero γ] [FunLike F β γ] [ZeroAtInftyContinuousMapClass F β γ] (f : F)
Goal: UniformContinuous ⇑f
-- ZeroAtInftyContinuousMap.bounded
protected theorem bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C 
Arguments: {F : Type u_1} {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] [FunLike F α β] [ZeroAtInftyContinuousMapClass F α β] (f : F)
Goal: ∃ C, ∀ (x y : α), dist (f x) (f y) ≤ C
-- ZeroAtInftyContinuousMap.isBounded_range
theorem isBounded_range (f : C₀(α, β)) : IsBounded (range f) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : α →C₀ β)
Goal: Bornology.IsBounded (Set.range ⇑f)
-- ZeroAtInftyContinuousMap.isBounded_image
theorem isBounded_image (f : C₀(α, β)) (s : Set α) : IsBounded (f '' s) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : α →C₀ β) (s : Set α)
Goal: Bornology.IsBounded (⇑f '' s)
-- ZeroAtInftyContinuousMap.toBCF_apply
@[simps!]
def toBCF (f : C₀(α, β)) : α →ᵇ β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : α →C₀ β) (a : α)
Goal: f.toBCF a = f a
-- ZeroAtInftyContinuousMap.toBCF_toFun
@[simps!]
def toBCF (f : C₀(α, β)) : α →ᵇ β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] (f : α →C₀ β) (a : α)
Goal: f.toBCF a = f a
-- ZeroAtInftyContinuousMap.toBCF_injective
theorem toBCF_injective : Function.Injective (toBCF : C₀(α, β) → α →ᵇ β) 
Arguments: (α : Type u) (β : Type v) [TopologicalSpace α] [PseudoMetricSpace β] [Zero β]
Goal: Function.Injective ZeroAtInftyContinuousMap.toBCF
-- ZeroAtInftyContinuousMap.dist_toBCF_eq_dist
@[simp]
theorem dist_toBCF_eq_dist {f g : C₀(α, β)} : dist f.toBCF g.toBCF = dist f g 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] {f g : α →C₀ β}
Goal: dist f.toBCF g.toBCF = dist f g
-- ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly
theorem tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : Filter ι} :
    Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β] {ι : Type u_2} {F : ι → α →C₀ β} {f : α →C₀ β} {l : Filter ι}
Goal: Filter.Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => ⇑(F i)) (⇑f) l
-- ZeroAtInftyContinuousMap.isometry_toBCF
theorem isometry_toBCF : Isometry (toBCF : C₀(α, β) → α →ᵇ β) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β]
Goal: Isometry ZeroAtInftyContinuousMap.toBCF
-- ZeroAtInftyContinuousMap.isClosed_range_toBCF
theorem isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →ᵇ β)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [PseudoMetricSpace β] [Zero β]
Goal: IsClosed (Set.range ZeroAtInftyContinuousMap.toBCF)
-- ZeroAtInftyContinuousMap.norm_toBCF_eq_norm
@[simp]
theorem norm_toBCF_eq_norm {f : C₀(α, β)} : ‖f.toBCF‖ = ‖f‖ 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [SeminormedAddCommGroup β] {f : α →C₀ β}
Goal: ‖f.toBCF‖ = ‖f‖
-- ZeroAtInftyContinuousMap.coe_star
@[simp]
theorem coe_star (f : C₀(α, β)) : ⇑(star f) = star (⇑f) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [StarAddMonoid β] [ContinuousStar β] (f : α →C₀ β)
Goal: ⇑(star f) = star ⇑f
-- ZeroAtInftyContinuousMap.star_apply
theorem star_apply (f : C₀(α, β)) (x : α) : (star f) x = star (f x) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [StarAddMonoid β] [ContinuousStar β] (f : α →C₀ β) (x : α)
Goal: (star f) x = star (f x)
-- ZeroAtInftyContinuousMap.coe_comp_to_continuous_fun
@[simp]
theorem coe_comp_to_continuous_fun (f : C₀(γ, δ)) (g : β →co γ) : ((f.comp g) : β → δ) = f ∘ g 
Arguments: {β : Type v} {γ : Type w} {δ : Type u_2} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : γ →C₀ δ) (g : CocompactMap β γ)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- ZeroAtInftyContinuousMap.comp_id
@[simp]
theorem comp_id (f : C₀(γ, δ)) : f.comp (CocompactMap.id γ) = f 
Arguments: {γ : Type w} {δ : Type u_2} [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : γ →C₀ δ)
Goal: f.comp (CocompactMap.id γ) = f
-- ZeroAtInftyContinuousMap.comp_assoc
@[simp]
theorem comp_assoc (f : C₀(γ, δ)) (g : β →co γ) (h : α →co β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] {δ : Type u_2} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : γ →C₀ δ) (g : CocompactMap β γ) (h : CocompactMap α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- ZeroAtInftyContinuousMap.zero_comp
@[simp]
theorem zero_comp (g : β →co γ) : (0 : C₀(γ, δ)).comp g = 0 
Arguments: {β : Type v} {γ : Type w} {δ : Type u_2} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (g : CocompactMap β γ)
Goal: ZeroAtInftyContinuousMap.comp 0 g = 0
-- exists_Ioo_extr_on_Icc
theorem exists_Ioo_extr_on_Icc (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
    ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c 
Arguments: {X : Type u_1} {Y : Type u_2} [ConditionallyCompleteLinearOrder X] [DenselyOrdered X] [TopologicalSpace X] [OrderTopology X] [LinearOrder Y] [TopologicalSpace Y] [OrderTopology Y] {f : X → Y} {a b : X} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfI : f a = f b)
Goal: ∃ c ∈ Set.Ioo a b, IsExtrOn f (Set.Icc a b) c
-- exists_isLocalExtr_Ioo
theorem exists_isLocalExtr_Ioo (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
    ∃ c ∈ Ioo a b, IsLocalExtr f c 
Arguments: {X : Type u_1} {Y : Type u_2} [ConditionallyCompleteLinearOrder X] [DenselyOrdered X] [TopologicalSpace X] [OrderTopology X] [LinearOrder Y] [TopologicalSpace Y] [OrderTopology Y] {f : X → Y} {a b : X} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfI : f a = f b)
Goal: ∃ c ∈ Set.Ioo a b, IsLocalExtr f c
-- exists_isExtrOn_Ioo_of_tendsto
lemma exists_isExtrOn_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))
    (ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) :
    ∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c 
Arguments: {X : Type u_1} {Y : Type u_2} [ConditionallyCompleteLinearOrder X] [DenselyOrdered X] [TopologicalSpace X] [OrderTopology X] [LinearOrder Y] [TopologicalSpace Y] [OrderTopology Y] {f : X → Y} {a b : X} {l : Y} (hab : a < b) (hfc : ContinuousOn f (Set.Ioo a b)) (ha : Filter.Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Filter.Tendsto f (𝓝[<] b) (𝓝 l))
Goal: ∃ c ∈ Set.Ioo a b, IsExtrOn f (Set.Ioo a b) c
-- exists_isLocalExtr_Ioo_of_tendsto
lemma exists_isLocalExtr_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))
    (ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) :
    ∃ c ∈ Ioo a b, IsLocalExtr f c 
Arguments: {X : Type u_1} {Y : Type u_2} [ConditionallyCompleteLinearOrder X] [DenselyOrdered X] [TopologicalSpace X] [OrderTopology X] [LinearOrder Y] [TopologicalSpace Y] [OrderTopology Y] {f : X → Y} {a b : X} {l : Y} (hab : a < b) (hfc : ContinuousOn f (Set.Ioo a b)) (ha : Filter.Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Filter.Tendsto f (𝓝[<] b) (𝓝 l))
Goal: ∃ c ∈ Set.Ioo a b, IsLocalExtr f c
-- NormedField.valuation_apply
theorem valuation_apply (x : K) : valuation h x = ‖x‖₊ 
Arguments: {K : Type u_1} [hK : NormedField K] (h : IsNonarchimedean norm) (x : K)
Goal: (NormedField.valuation h) x = ‖x‖₊
-- Valued.norm_nonneg
theorem norm_nonneg (x : L) : 0 ≤ norm x 
Arguments: {L : Type u_2} [Field L] {Γ₀ : Type u_3} [LinearOrderedCommGroupWithZero Γ₀] [val : Valued L Γ₀] [hv : Valued.v.RankOne] (x : L)
Goal: 0 ≤ Valued.norm x
-- Valued.norm_add_le
theorem norm_add_le (x y : L) : norm (x + y) ≤ max (norm x) (norm y) 
Arguments: {L : Type u_2} [Field L] {Γ₀ : Type u_3} [LinearOrderedCommGroupWithZero Γ₀] [val : Valued L Γ₀] [hv : Valued.v.RankOne] (x y : L)
Goal: Valued.norm (x + y) ≤ max (Valued.norm x) (Valued.norm y)
-- Valued.norm_eq_zero
theorem norm_eq_zero {x : L} (hx : norm x = 0) : x = 0 
Arguments: {L : Type u_2} [Field L] {Γ₀ : Type u_3} [LinearOrderedCommGroupWithZero Γ₀] [val : Valued L Γ₀] [hv : Valued.v.RankOne] {x : L} (hx : Valued.norm x = 0)
Goal: x = 0
-- Rat.denseRange_cast
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
    [Archimedean 𝕜] : DenseRange ((↑) : ℚ → 𝕜) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] [Archimedean 𝕜]
Goal: DenseRange Rat.cast
-- AddSubgroup.dense_of_not_isolated_zero
theorem dense_of_not_isolated_zero (S : AddSubgroup G) (hS : ∀ ε > 0, ∃ g ∈ S, g ∈ Ioo 0 ε) :
    Dense (S : Set G) 
Arguments: {G : Type u_1} [LinearOrderedAddCommGroup G] [TopologicalSpace G] [OrderTopology G] [Archimedean G] (S : AddSubgroup G) (hS : ∀ ε > 0, ∃ g ∈ S, g ∈ Set.Ioo 0 ε)
Goal: Dense ↑S
-- AddSubgroup.dense_of_no_min
theorem dense_of_no_min (S : AddSubgroup G) (hbot : S ≠ ⊥)
    (H : ¬∃ a : G, IsLeast { g : G | g ∈ S ∧ 0 < g } a) : Dense (S : Set G) 
Arguments: {G : Type u_1} [LinearOrderedAddCommGroup G] [TopologicalSpace G] [OrderTopology G] [Archimedean G] (S : AddSubgroup G) (hbot : S ≠ ⊥) (H : ¬∃ a, IsLeast {g | g ∈ S ∧ 0 < g} a)
Goal: Dense ↑S
-- AddSubgroup.dense_or_cyclic
theorem dense_or_cyclic (S : AddSubgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} 
Arguments: {G : Type u_1} [LinearOrderedAddCommGroup G] [TopologicalSpace G] [OrderTopology G] [Archimedean G] (S : AddSubgroup G)
Goal: Dense ↑S ∨ ∃ a, S = AddSubgroup.closure {a}
-- TopCat.Presheaf.isGluing_iff_pairwise
theorem isGluing_iff_pairwise {sf s} : IsGluing F U sf s ↔
    ∀ i, (F.mapCone (Pairwise.cocone U).op).π.app i s = objPairwiseOfFamily sf i 
Arguments: {X : TopCat} {F : TopCat.Presheaf (Type u) X} {ι : Type x} {U : ι → TopologicalSpace.Opens ↑X} {sf : (i : ι) → (CategoryTheory.forget (Type u)).obj (F.obj (Opposite.op (U i)))} {s : (CategoryTheory.forget (Type u)).obj (F.obj (Opposite.op (iSup U)))}
Goal: F.IsGluing U sf s ↔
  ∀ (i : (CategoryTheory.Pairwise ι)ᵒᵖ),
    (CategoryTheory.Functor.mapCone F (CategoryTheory.Pairwise.cocone U).op).π.app i s =
      TopCat.Presheaf.objPairwiseOfFamily sf i
-- TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing_types
theorem isSheaf_iff_isSheafUniqueGluing_types : F.IsSheaf ↔ F.IsSheafUniqueGluing 
Arguments: {X : TopCat} (F : TopCat.Presheaf (Type u) X)
Goal: F.IsSheaf ↔ F.IsSheafUniqueGluing
-- TopCat.Presheaf.isSheaf_of_isSheafUniqueGluing_types
theorem isSheaf_of_isSheafUniqueGluing_types (Fsh : F.IsSheafUniqueGluing) : F.IsSheaf 
Arguments: {X : TopCat} (F : TopCat.Presheaf (Type u) X) (Fsh : F.IsSheafUniqueGluing)
Goal: F.IsSheaf
-- TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing
theorem isSheaf_iff_isSheafUniqueGluing : F.IsSheaf ↔ F.IsSheafUniqueGluing 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasLimits C] [(CategoryTheory.forget C).ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheaf ↔ F.IsSheafUniqueGluing
-- TopCat.Sheaf.existsUnique_gluing
theorem existsUnique_gluing (sf : ∀ i : ι, F.1.obj (op (U i))) (h : IsCompatible F.1 U sf) :
    ∃! s : F.1.obj (op (iSup U)), IsGluing F.1 U sf s 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.ConcreteCategory.forget.ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits CategoryTheory.ConcreteCategory.forget] {X : TopCat} (F : TopCat.Sheaf C X) {ι : Type v} (U : ι → TopologicalSpace.Opens ↑X) (sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (Opposite.op (U i)))) (h : TopCat.Presheaf.IsCompatible F.val U sf)
Goal: ∃! s, TopCat.Presheaf.IsGluing F.val U sf s
-- TopCat.Sheaf.existsUnique_gluing'
theorem existsUnique_gluing' (V : Opens X) (iUV : ∀ i : ι, U i ⟶ V) (hcover : V ≤ iSup U)
    (sf : ∀ i : ι, F.1.obj (op (U i))) (h : IsCompatible F.1 U sf) :
    ∃! s : F.1.obj (op V), ∀ i : ι, F.1.map (iUV i).op s = sf i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.ConcreteCategory.forget.ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits CategoryTheory.ConcreteCategory.forget] {X : TopCat} (F : TopCat.Sheaf C X) {ι : Type v} (U : ι → TopologicalSpace.Opens ↑X) (V : TopologicalSpace.Opens ↑X) (iUV : (i : ι) → U i ⟶ V) (hcover : V ≤ iSup U) (sf : (i : ι) → (CategoryTheory.forget C).obj (F.val.obj (Opposite.op (U i)))) (h : TopCat.Presheaf.IsCompatible F.val U sf)
Goal: ∃! s, ∀ (i : ι), (F.val.map (iUV i).op) s = sf i
-- TopCat.Sheaf.eq_of_locally_eq
@[ext]
theorem eq_of_locally_eq (s t : F.1.obj (op (iSup U)))
    (h : ∀ i, F.1.map (Opens.leSupr U i).op s = F.1.map (Opens.leSupr U i).op t) : s = t 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.ConcreteCategory.forget.ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits CategoryTheory.ConcreteCategory.forget] {X : TopCat} (F : TopCat.Sheaf C X) {ι : Type v} (U : ι → TopologicalSpace.Opens ↑X) (s t : (CategoryTheory.forget C).obj (F.val.obj (Opposite.op (iSup U)))) (h :
  ∀ (i : ι),
    (F.val.map (TopologicalSpace.Opens.leSupr U i).op) s = (F.val.map (TopologicalSpace.Opens.leSupr U i).op) t)
Goal: s = t
-- TopCat.Sheaf.eq_of_locally_eq'
theorem eq_of_locally_eq' (V : Opens X) (iUV : ∀ i : ι, U i ⟶ V) (hcover : V ≤ iSup U)
    (s t : F.1.obj (op V)) (h : ∀ i, F.1.map (iUV i).op s = F.1.map (iUV i).op t) : s = t 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.ConcreteCategory.forget.ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits CategoryTheory.ConcreteCategory.forget] {X : TopCat} (F : TopCat.Sheaf C X) {ι : Type v} (U : ι → TopologicalSpace.Opens ↑X) (V : TopologicalSpace.Opens ↑X) (iUV : (i : ι) → U i ⟶ V) (hcover : V ≤ iSup U) (s t : (CategoryTheory.forget C).obj (F.val.obj (Opposite.op V))) (h : ∀ (i : ι), (F.val.map (iUV i).op) s = (F.val.map (iUV i).op) t)
Goal: s = t
-- TopCat.Sheaf.eq_of_locally_eq₂
theorem eq_of_locally_eq₂ {U₁ U₂ V : Opens X} (i₁ : U₁ ⟶ V) (i₂ : U₂ ⟶ V) (hcover : V ≤ U₁ ⊔ U₂)
    (s t : F.1.obj (op V)) (h₁ : F.1.map i₁.op s = F.1.map i₁.op t)
    (h₂ : F.1.map i₂.op s = F.1.map i₂.op t) : s = t 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.Limits.HasLimits C] [CategoryTheory.ConcreteCategory.forget.ReflectsIsomorphisms] [CategoryTheory.Limits.PreservesLimits CategoryTheory.ConcreteCategory.forget] {X : TopCat} (F : TopCat.Sheaf C X) {U₁ U₂ V : TopologicalSpace.Opens ↑X} (i₁ : U₁ ⟶ V) (i₂ : U₂ ⟶ V) (hcover : V ≤ U₁ ⊔ U₂) (s t : (CategoryTheory.forget C).obj (F.val.obj (Opposite.op V))) (h₁ : (F.val.map i₁.op) s = (F.val.map i₁.op) t) (h₂ : (F.val.map i₂.op) s = (F.val.map i₂.op) t)
Goal: s = t
-- TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff
theorem objSupIsoProdEqLocus_inv_eq_iff {X : TopCat.{u}} (F : X.Sheaf CommRingCat.{u})
    {U V W UW VW : Opens X} (e : W ≤ U ⊔ V) (x) (y : F.1.obj (op W))
    (h₁ : UW = U ⊓ W) (h₂ : VW = V ⊓ W) :
    F.1.map (homOfLE e).op ((F.objSupIsoProdEqLocus U V).inv x) = y ↔
    F.1.map (homOfLE (h₁ ▸ inf_le_left : UW ≤ U)).op x.1.1 =
      F.1.map (homOfLE <| h₁ ▸ inf_le_right).op y ∧
    F.1.map (homOfLE (h₂ ▸ inf_le_left : VW ≤ V)).op x.1.2 =
      F.1.map (homOfLE <| h₂ ▸ inf_le_right).op y 
Arguments: {X : TopCat} (F : TopCat.Sheaf CommRingCat X) {U V W UW VW : TopologicalSpace.Opens ↑X} (e : W ≤ U ⊔ V) (x :
  ↥((RingHom.comp (F.val.map (CategoryTheory.homOfLE ⋯).op)
          (RingHom.fst ↑(F.val.obj (Opposite.op U)) ↑(F.val.obj (Opposite.op V)))).eqLocus
      (RingHom.comp (F.val.map (CategoryTheory.homOfLE ⋯).op)
        (RingHom.snd ↑(F.val.obj (Opposite.op U)) ↑(F.val.obj (Opposite.op V)))))) (y : ↑(F.val.obj (Opposite.op W))) (h₁ : UW = U ⊓ W) (h₂ : VW = V ⊓ W)
Goal: (F.val.map (CategoryTheory.homOfLE e).op) ((F.objSupIsoProdEqLocus U V).inv x) = y ↔
  (F.val.map (CategoryTheory.homOfLE ⋯).op) (↑x).1 = (F.val.map (CategoryTheory.homOfLE ⋯).op) y ∧
    (F.val.map (CategoryTheory.homOfLE ⋯).op) (↑x).2 = (F.val.map (CategoryTheory.homOfLE ⋯).op) y
-- TopologicalRing.of_norm
theorem TopologicalRing.of_norm {R 𝕜 : Type*} [NonUnitalNonAssocRing R] [LinearOrderedField 𝕜]
    [TopologicalSpace R] [TopologicalAddGroup R] (norm : R → 𝕜)
    (norm_nonneg : ∀ x, 0 ≤ norm x) (norm_mul_le : ∀ x y, norm (x * y) ≤ norm x * norm y)
    (nhds_basis : (𝓝 (0 : R)).HasBasis ((0 : 𝕜) < ·) (fun ε ↦ { x | norm x < ε })) :
    TopologicalRing R 
Arguments: {R : Type u_1} {𝕜 : Type u_2} [NonUnitalNonAssocRing R] [LinearOrderedField 𝕜] [TopologicalSpace R] [TopologicalAddGroup R] (norm : R → 𝕜) (norm_nonneg : ∀ (x : R), 0 ≤ norm x) (norm_mul_le : ∀ (x y : R), norm (x * y) ≤ norm x * norm y) (nhds_basis : (𝓝 0).HasBasis (fun x => 0 < x) fun ε => {x | norm x < ε})
Goal: TopologicalRing R
-- Filter.Tendsto.atTop_mul
theorem Filter.Tendsto.atTop_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop)
    (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l (𝓝 C))
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atTop
-- Filter.Tendsto.mul_atTop
theorem Filter.Tendsto.mul_atTop {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
    (hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l (𝓝 C)) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atTop
-- Filter.Tendsto.atTop_mul_neg
theorem Filter.Tendsto.atTop_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atTop)
    (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l Filter.atTop) (hg : Filter.Tendsto g l (𝓝 C))
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atBot
-- Filter.Tendsto.neg_mul_atTop
theorem Filter.Tendsto.neg_mul_atTop {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
    (hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atBot 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l (𝓝 C)) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atBot
-- Filter.Tendsto.atBot_mul
theorem Filter.Tendsto.atBot_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot)
    (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l (𝓝 C))
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atBot
-- Filter.Tendsto.atBot_mul_neg
theorem Filter.Tendsto.atBot_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atBot)
    (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l Filter.atBot) (hg : Filter.Tendsto g l (𝓝 C))
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atTop
-- Filter.Tendsto.mul_atBot
theorem Filter.Tendsto.mul_atBot {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C))
    (hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atBot 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : 0 < C) (hf : Filter.Tendsto f l (𝓝 C)) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atBot
-- Filter.Tendsto.neg_mul_atBot
theorem Filter.Tendsto.neg_mul_atBot {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
    (hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {C : 𝕜} (hC : C < 0) (hf : Filter.Tendsto f l (𝓝 C)) (hg : Filter.Tendsto g l Filter.atBot)
Goal: Filter.Tendsto (fun x => f x * g x) l Filter.atTop
-- inv_atTop₀
@[simp]
lemma inv_atTop₀ : (atTop : Filter 𝕜)⁻¹ = 𝓝[>] 0 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
Goal: Filter.atTop⁻¹ = 𝓝[>] 0
-- inv_nhdsWithin_Ioi_zero
@[simp] lemma inv_nhdsWithin_Ioi_zero : (𝓝[>] (0 : 𝕜))⁻¹ = atTop 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
Goal: (𝓝[>] 0)⁻¹ = Filter.atTop
-- tendsto_inv_zero_atTop
theorem tendsto_inv_zero_atTop : Tendsto (fun x : 𝕜 => x⁻¹) (𝓝[>] (0 : 𝕜)) atTop 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
Goal: Filter.Tendsto (fun x => x⁻¹) (𝓝[>] 0) Filter.atTop
-- tendsto_inv_atTop_zero'
theorem tendsto_inv_atTop_zero' : Tendsto (fun r : 𝕜 => r⁻¹) atTop (𝓝[>] (0 : 𝕜)) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
Goal: Filter.Tendsto (fun r => r⁻¹) Filter.atTop (𝓝[>] 0)
-- tendsto_inv_atTop_zero
theorem tendsto_inv_atTop_zero : Tendsto (fun r : 𝕜 => r⁻¹) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
Goal: Filter.Tendsto (fun r => r⁻¹) Filter.atTop (𝓝 0)
-- Filter.Tendsto.div_atTop
theorem Filter.Tendsto.div_atTop {a : 𝕜} (h : Tendsto f l (𝓝 a)) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x / g x) l (𝓝 0) 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f g : α → 𝕜} {a : 𝕜} (h : Filter.Tendsto f l (𝓝 a)) (hg : Filter.Tendsto g l Filter.atTop)
Goal: Filter.Tendsto (fun x => f x / g x) l (𝓝 0)
-- Filter.Tendsto.inv_tendsto_atTop
theorem Filter.Tendsto.inv_tendsto_atTop (h : Tendsto f l atTop) : Tendsto f⁻¹ l (𝓝 0) 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} (h : Filter.Tendsto f l Filter.atTop)
Goal: Filter.Tendsto f⁻¹ l (𝓝 0)
-- Filter.Tendsto.inv_tendsto_zero
theorem Filter.Tendsto.inv_tendsto_zero (h : Tendsto f l (𝓝[>] 0)) : Tendsto f⁻¹ l atTop 
Arguments: {𝕜 : Type u_1} {α : Type u_2} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {l : Filter α} {f : α → 𝕜} (h : Filter.Tendsto f l (𝓝[>] 0))
Goal: Filter.Tendsto f⁻¹ l Filter.atTop
-- tendsto_pow_neg_atTop
theorem tendsto_pow_neg_atTop {n : ℕ} (hn : n ≠ 0) :
    Tendsto (fun x : 𝕜 => x ^ (-(n : ℤ))) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℕ} (hn : n ≠ 0)
Goal: Filter.Tendsto (fun x => x ^ (-↑n)) Filter.atTop (𝓝 0)
-- tendsto_zpow_atTop_zero
theorem tendsto_zpow_atTop_zero {n : ℤ} (hn : n < 0) :
    Tendsto (fun x : 𝕜 => x ^ n) atTop (𝓝 0) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {n : ℤ} (hn : n < 0)
Goal: Filter.Tendsto (fun x => x ^ n) Filter.atTop (𝓝 0)