ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SkyscraperPresheafFunctor.map'_id
theorem SkyscraperPresheafFunctor.map'_id {a : C} :
    SkyscraperPresheafFunctor.map' p₀ (𝟙 a) = 𝟙 _ 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{w, v} C] [CategoryTheory.Limits.HasTerminal C] {a : C}
Goal: SkyscraperPresheafFunctor.map' p₀ (𝟙 a) = 𝟙 (skyscraperPresheaf p₀ a)
-- SkyscraperPresheafFunctor.map'_comp
theorem SkyscraperPresheafFunctor.map'_comp {a b c : C} (f : a ⟶ b) (g : b ⟶ c) :
    SkyscraperPresheafFunctor.map' p₀ (f ≫ g) =
      SkyscraperPresheafFunctor.map' p₀ f ≫ SkyscraperPresheafFunctor.map' p₀ g 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{w, v} C] [CategoryTheory.Limits.HasTerminal C] {a b c : C} (f : a ⟶ b) (g : b ⟶ c)
Goal: SkyscraperPresheafFunctor.map' p₀ (f ≫ g) = SkyscraperPresheafFunctor.map' p₀ f ≫ SkyscraperPresheafFunctor.map' p₀ g
-- skyscraperPresheafFunctor_obj
@[simps]
def skyscraperPresheafFunctor : C ⥤ Presheaf C X 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{w, v} C] [CategoryTheory.Limits.HasTerminal C] (A : C)
Goal: (skyscraperPresheafFunctor p₀).obj A = skyscraperPresheaf p₀ A
-- skyscraperPresheafFunctor_map
@[simps]
def skyscraperPresheafFunctor : C ⥤ Presheaf C X 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{w, v} C] [CategoryTheory.Limits.HasTerminal C]
Goal: ∀ {X_1 Y : C} (f : X_1 ⟶ Y), (skyscraperPresheafFunctor p₀).map f = SkyscraperPresheafFunctor.map' p₀ f
-- skyscraperPresheafCoconeOfSpecializes_ι_app
@[simps]
def skyscraperPresheafCoconeOfSpecializes {y : X} (h : p₀ ⤳ y) :
    Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A) 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] (A : C) [CategoryTheory.Limits.HasTerminal C] {y : ↑X} (h : p₀ ⤳ y) (U : (TopologicalSpace.OpenNhds y)ᵒᵖ)
Goal: (skyscraperPresheafCoconeOfSpecializes p₀ A h).ι.app U = CategoryTheory.eqToHom ⋯
-- skyscraperPresheafCoconeOfSpecializes_pt
@[simps]
def skyscraperPresheafCoconeOfSpecializes {y : X} (h : p₀ ⤳ y) :
    Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A) 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] (A : C) [CategoryTheory.Limits.HasTerminal C] {y : ↑X} (h : p₀ ⤳ y)
Goal: (skyscraperPresheafCoconeOfSpecializes p₀ A h).pt = A
-- skyscraperPresheafCocone_ι_app
@[simps]
def skyscraperPresheafCocone (y : X) :
    Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A) 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] (A : C) [CategoryTheory.Limits.HasTerminal C] (y : ↑X)
Goal: ∀ (x : (TopologicalSpace.OpenNhds y)ᵒᵖ),
  (skyscraperPresheafCocone p₀ A y).ι.app x =
    CategoryTheory.Limits.terminal.from (((TopologicalSpace.OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A).obj x)
-- skyscraperPresheafCocone_pt
@[simps]
def skyscraperPresheafCocone (y : X) :
    Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A) 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] (A : C) [CategoryTheory.Limits.HasTerminal C] (y : ↑X)
Goal: (skyscraperPresheafCocone p₀ A y).pt = ⊤_ C
-- skyscraperPresheaf_isSheaf
theorem skyscraperPresheaf_isSheaf : (skyscraperPresheaf p₀ A).IsSheaf 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] (A : C) [CategoryTheory.Limits.HasTerminal C]
Goal: (skyscraperPresheaf p₀ A).IsSheaf
-- StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf_app
@[simps]
def toSkyscraperPresheaf {𝓕 : Presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c) :
    𝓕 ⟶ skyscraperPresheaf p₀ c 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasColimits C] {𝓕 : TopCat.Presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c) (U : (TopologicalSpace.Opens ↑X)ᵒᵖ)
Goal: (StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf p₀ f).app U =
  if h : p₀ ∈ Opposite.unop U then 𝓕.germ ⟨p₀, h⟩ ≫ f ≫ CategoryTheory.eqToHom ⋯
  else (⋯ ▸ CategoryTheory.Limits.terminalIsTerminal).from (𝓕.obj U)
-- StalkSkyscraperPresheafAdjunctionAuxs.to_skyscraper_fromStalk
theorem to_skyscraper_fromStalk {𝓕 : Presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraperPresheaf p₀ c) :
    toSkyscraperPresheaf p₀ (fromStalk _ f) = f 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasColimits C] {𝓕 : TopCat.Presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraperPresheaf p₀ c)
Goal: StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf p₀ (StalkSkyscraperPresheafAdjunctionAuxs.fromStalk p₀ f) = f
-- StalkSkyscraperPresheafAdjunctionAuxs.fromStalk_to_skyscraper
theorem fromStalk_to_skyscraper {𝓕 : Presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c) :
    fromStalk p₀ (toSkyscraperPresheaf _ f) = f 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasColimits C] {𝓕 : TopCat.Presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c)
Goal: StalkSkyscraperPresheafAdjunctionAuxs.fromStalk p₀ (StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf p₀ f) = f
-- StalkSkyscraperPresheafAdjunctionAuxs.unit_app
@[simps]
protected def unit :
    𝟭 (Presheaf C X) ⟶ Presheaf.stalkFunctor C p₀ ⋙ skyscraperPresheafFunctor p₀ 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasColimits C] (𝓕 : TopCat.Presheaf C X)
Goal: (StalkSkyscraperPresheafAdjunctionAuxs.unit p₀).app 𝓕 =
  StalkSkyscraperPresheafAdjunctionAuxs.toSkyscraperPresheaf p₀ (𝟙 (((𝟭 (TopCat.Presheaf C X)).obj 𝓕).stalk p₀))
-- StalkSkyscraperPresheafAdjunctionAuxs.counit_app
@[simps]
protected def counit :
    skyscraperPresheafFunctor p₀ ⋙ (Presheaf.stalkFunctor C p₀ : Presheaf C X ⥤ C) ⟶ 𝟭 C 
Arguments: {X : TopCat} (p₀ : ↑X) [(U : TopologicalSpace.Opens ↑X) → Decidable (p₀ ∈ U)] {C : Type v} [CategoryTheory.Category.{u, v} C] [CategoryTheory.Limits.HasTerminal C] [CategoryTheory.Limits.HasColimits C] (c : C)
Goal: (StalkSkyscraperPresheafAdjunctionAuxs.counit p₀).app c = (skyscraperPresheafStalkOfSpecializes p₀ c ⋯).hom
-- CompactlySupportedContinuousMap.hasCompactSupport'
structure CompactlySupportedContinuousMap (α β : Type*) [TopologicalSpace α] [Zero β]
    [TopologicalSpace β] extends ContinuousMap α β where
    hasCompactSupport' : HasCompactSupport toFun
Arguments: {α : Type u_5} {β : Type u_6} [TopologicalSpace α] [Zero β] [TopologicalSpace β] (self : α →C_c β)
Goal: HasCompactSupport self.toFun
-- CompactlySupportedContinuousMapClass.hasCompactSupport
class CompactlySupportedContinuousMapClass (F : Type*) (α β : outParam <| Type*)
    [TopologicalSpace α] [Zero β] [TopologicalSpace β] [FunLike F α β]
    extends ContinuousMapClass F α β : Prop where
    hasCompactSupport (f : F) : HasCompactSupport f
Arguments: {F : Type u_5} {α : outParam (Type u_6)} {β : outParam (Type u_7)} [TopologicalSpace α] [Zero β] [TopologicalSpace β] [FunLike F α β] [self : CompactlySupportedContinuousMapClass F α β] (f : F)
Goal: HasCompactSupport ⇑f
-- CompactlySupportedContinuousMap.hasCompactSupport
protected lemma hasCompactSupport (f : C_c(α, β)) : HasCompactSupport f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C_c β)
Goal: HasCompactSupport ⇑f
-- CompactlySupportedContinuousMap.coe_toContinuousMap
@[simp]
theorem coe_toContinuousMap (f : C_c(α, β)) : (f.toContinuousMap : α → β) = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C_c β)
Goal: ⇑f.toContinuousMap = ⇑f
-- CompactlySupportedContinuousMap.ext
@[ext]
theorem ext {f g : C_c(α, β)} (h : ∀ x, f x = g x) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {f g : α →C_c β} (h : ∀ (x : α), f x = g x)
Goal: f = g
-- CompactlySupportedContinuousMap.coe_mk
@[simp]
theorem coe_mk (f : C(α, β)) (h : HasCompactSupport f) : ⇑(⟨f, h⟩ : C_c(α, β)) = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : C(α, β)) (h : HasCompactSupport ⇑f)
Goal: ⇑{ toContinuousMap := f, hasCompactSupport' := h } = ⇑f
-- CompactlySupportedContinuousMap.coe_copy
@[simp]
theorem coe_copy (f : C_c(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C_c β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- CompactlySupportedContinuousMap.copy_eq
theorem copy_eq (f : C_c(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (f : α →C_c β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- CompactlySupportedContinuousMap.eq_of_empty
theorem eq_of_empty [IsEmpty α] (f g : C_c(α, β)) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [IsEmpty α] (f g : α →C_c β)
Goal: f = g
-- CompactlySupportedContinuousMap.ContinuousMap.liftCompactlySupported_symm_apply
@[simps]
def ContinuousMap.liftCompactlySupported [CompactSpace α] : C(α, β) ≃ C_c(α, β) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [CompactSpace α] (f : α →C_c β)
Goal: CompactlySupportedContinuousMap.ContinuousMap.liftCompactlySupported.symm f = ↑f
-- CompactlySupportedContinuousMap.ContinuousMap.liftCompactlySupported_apply_toFun
@[simps]
def ContinuousMap.liftCompactlySupported [CompactSpace α] : C(α, β) ≃ C_c(α, β) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [CompactSpace α] (f : C(α, β)) (a : α)
Goal: (CompactlySupportedContinuousMap.ContinuousMap.liftCompactlySupported f) a = f a
-- CompactlySupportedContinuousMap.coe_zero
@[simp]
theorem coe_zero [Zero β] : ⇑(0 : C_c(α, β)) = 0 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β]
Goal: ⇑0 = 0
-- CompactlySupportedContinuousMap.zero_apply
theorem zero_apply [Zero β] : (0 : C_c(α, β)) x = 0 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (x : α) [Zero β]
Goal: 0 x = 0
-- CompactlySupportedContinuousMap.coe_mul
@[simp]
theorem coe_mul [MulZeroClass β] [ContinuousMul β] (f g : C_c(α, β)) : ⇑(f * g) = f * g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [MulZeroClass β] [ContinuousMul β] (f g : α →C_c β)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- CompactlySupportedContinuousMap.mul_apply
theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C_c(α, β)) : (f * g) x = f x * g x 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (x : α) [MulZeroClass β] [ContinuousMul β] (f g : α →C_c β)
Goal: (f * g) x = f x * g x
-- CompactlySupportedContinuousMap.coe_smulc
@[simp]
theorem coe_smulc [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
    {F : Type*} [FunLike F α γ] [ContinuousMapClass F α γ] (f : F) (g : C_c(α, β)) :
    ⇑(f • g) = fun x => f x • g x 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β] {F : Type u_5} [FunLike F α γ] [ContinuousMapClass F α γ] (f : F) (g : α →C_c β)
Goal: ⇑(f • g) = fun x => f x • g x
-- CompactlySupportedContinuousMap.smulc_apply
theorem smulc_apply [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
    {F : Type*} [FunLike F α γ] [ContinuousMapClass F α γ] (f : F) (g : C_c(α, β)) (x : α) :
    (f • g) x = f x • g x 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β] {F : Type u_5} [FunLike F α γ] [ContinuousMapClass F α γ] (f : F) (g : α →C_c β) (x : α)
Goal: (f • g) x = f x • g x
-- CompactlySupportedContinuousMap.coe_add
@[simp]
theorem coe_add [AddZeroClass β] [ContinuousAdd β] (f g : C_c(α, β)) : ⇑(f + g) = f + g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddZeroClass β] [ContinuousAdd β] (f g : α →C_c β)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- CompactlySupportedContinuousMap.add_apply
theorem add_apply [AddZeroClass β] [ContinuousAdd β] (f g : C_c(α, β)) : (f + g) x = f x + g x 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddZeroClass β] [ContinuousAdd β] (f g : α →C_c β)
Goal: (f + g) x = f x + g x
-- CompactlySupportedContinuousMap.coe_smul
@[simp, norm_cast]
theorem coe_smul [Zero β] {R : Type*} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R)
    (f : C_c(α, β)) : ⇑(r • f) = r • ⇑f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {R : Type u_5} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R) (f : α →C_c β)
Goal: ⇑(r • f) = r • ⇑f
-- CompactlySupportedContinuousMap.smul_apply
theorem smul_apply [Zero β] {R : Type*} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R)
    (f : C_c(α, β)) (x : α) : (r • f) x = r • f x 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [Zero β] {R : Type u_5} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R) (f : α →C_c β) (x : α)
Goal: (r • f) x = r • f x
-- CompactlySupportedContinuousMap.coe_sum
@[simp]
theorem coe_sum [AddCommMonoid β] [ContinuousAdd β] {ι : Type*} (s : Finset ι) (f : ι → C_c(α, β)) :
    ⇑(∑ i in s, f i) = ∑ i in s, (f i : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddCommMonoid β] [ContinuousAdd β] {ι : Type u_5} (s : Finset ι) (f : ι → α →C_c β)
Goal: ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, ⇑(f i)
-- CompactlySupportedContinuousMap.sum_apply
theorem sum_apply [AddCommMonoid β] [ContinuousAdd β] {ι : Type*} (s : Finset ι) (f : ι → C_c(α, β))
    (a : α) : (∑ i in s, f i) a = ∑ i in s, f i a 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddCommMonoid β] [ContinuousAdd β] {ι : Type u_5} (s : Finset ι) (f : ι → α →C_c β) (a : α)
Goal: (∑ i ∈ s, f i) a = ∑ i ∈ s, (f i) a
-- CompactlySupportedContinuousMap.coe_neg
@[simp]
theorem coe_neg : ⇑(-f) = -f 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (f : α →C_c β)
Goal: ⇑(-f) = -⇑f
-- CompactlySupportedContinuousMap.neg_apply
theorem neg_apply : (-f) x = -f x 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddGroup β] [TopologicalAddGroup β] (f : α →C_c β)
Goal: (-f) x = -f x
-- CompactlySupportedContinuousMap.coe_sub
@[simp]
theorem coe_sub : ⇑(f - g) = f - g 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (f g : α →C_c β)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- CompactlySupportedContinuousMap.sub_apply
theorem sub_apply : (f - g) x = f x - g x 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] (x : α) [AddGroup β] [TopologicalAddGroup β] (f g : α →C_c β)
Goal: (f - g) x = f x - g x
-- CompactlySupportedContinuousMap.coe_star
@[simp]
theorem coe_star (f : C_c(α, β)) : ⇑(star f) = star (⇑f) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [StarAddMonoid β] [ContinuousStar β] (f : α →C_c β)
Goal: ⇑(star f) = star ⇑f
-- CompactlySupportedContinuousMap.star_apply
theorem star_apply (f : C_c(α, β)) (x : α) : (star f) x = star (f x) 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [StarAddMonoid β] [ContinuousStar β] (f : α →C_c β) (x : α)
Goal: (star f) x = star (f x)
-- CompactlySupportedContinuousMap.coe_comp_to_continuous_fun
@[simp]
theorem coe_comp_to_continuous_fun (f : C_c(γ, δ)) (g : β →co γ) : ((f.comp g) : β → δ) = f ∘ g 
Arguments: {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : γ →C_c δ) (g : CocompactMap β γ)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- CompactlySupportedContinuousMap.comp_id
@[simp]
theorem comp_id (f : C_c(γ, δ)) : f.comp (CocompactMap.id γ) = f 
Arguments: {γ : Type u_4} {δ : Type u_5} [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : γ →C_c δ)
Goal: f.comp (CocompactMap.id γ) = f
-- CompactlySupportedContinuousMap.comp_assoc
@[simp]
theorem comp_assoc (f : C_c(γ, δ)) (g : β →co γ) (h : α →co β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [TopologicalSpace α] {δ : Type u_5} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (f : γ →C_c δ) (g : CocompactMap β γ) (h : CocompactMap α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- CompactlySupportedContinuousMap.zero_comp
@[simp]
theorem zero_comp (g : β →co γ) : (0 : C_c(γ, δ)).comp g = 0 
Arguments: {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] [Zero δ] (g : CocompactMap β γ)
Goal: CompactlySupportedContinuousMap.comp 0 g = 0
-- CompactlySupportedContinuousMapClass.of_compactSpace
lemma of_compactSpace (G : Type*) [FunLike G α β]
    [ContinuousMapClass G α β] [CompactSpace α] : CompactlySupportedContinuousMapClass G α β 
Arguments: {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [Zero β] [TopologicalSpace β] (G : Type u_5) [FunLike G α β] [ContinuousMapClass G α β] [CompactSpace α]
Goal: CompactlySupportedContinuousMapClass G α β
-- CompactlySupportedContinuousMapClass.uniformContinuous
theorem uniformContinuous (f : F) : UniformContinuous (f : β → γ) 
Arguments: {F : Type u_1} {β : Type u_3} {γ : Type u_4} [UniformSpace β] [UniformSpace γ] [Zero γ] [FunLike F β γ] [CompactlySupportedContinuousMapClass F β γ] (f : F)
Goal: UniformContinuous ⇑f
-- ContinuousSMul.continuous_smul
class ContinuousSMul (M X : Type*) [SMul M X] [TopologicalSpace M] [TopologicalSpace X] :
    Prop where
    continuous_smul : Continuous fun p : M × X => p.1 • p.2
Arguments: {M : Type u_1} {X : Type u_2} [SMul M X] [TopologicalSpace M] [TopologicalSpace X] [self : ContinuousSMul M X]
Goal: Continuous fun p => p.1 • p.2
-- ContinuousVAdd.continuous_vadd
class ContinuousVAdd (M X : Type*) [VAdd M X] [TopologicalSpace M] [TopologicalSpace X] :
    Prop where
    continuous_vadd : Continuous fun p : M × X => p.1 +ᵥ p.2
Arguments: {M : Type u_1} {X : Type u_2} [VAdd M X] [TopologicalSpace M] [TopologicalSpace X] [self : ContinuousVAdd M X]
Goal: Continuous fun p => p.1 +ᵥ p.2
-- ContinuousSMul.induced
theorem ContinuousSMul.induced {R : Type*} {α : Type*} {β : Type*} {F : Type*} [FunLike F α β]
    [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β]
    [TopologicalSpace R] [LinearMapClass F R α β] [tβ : TopologicalSpace β] [ContinuousSMul R β]
    (f : F) : @ContinuousSMul R α _ _ (tβ.induced f) 
Arguments: {R : Type u_5} {α : Type u_6} {β : Type u_7} {F : Type u_8} [FunLike F α β] [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [TopologicalSpace R] [LinearMapClass F R α β] [tβ : TopologicalSpace β] [ContinuousSMul R β] (f : F)
Goal: ContinuousSMul R α
-- Filter.Tendsto.vadd
@[to_additive]
theorem Filter.Tendsto.smul {f : α → M} {g : α → X} {l : Filter α} {c : M} {a : X}
    (hf : Tendsto f l (𝓝 c)) (hg : Tendsto g l (𝓝 a)) :
    Tendsto (fun x => f x • g x) l (𝓝 <| c • a) 
Arguments: {M : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace M] [TopologicalSpace X] [VAdd M X] [ContinuousVAdd M X] {f : α → M} {g : α → X} {l : Filter α} {c : M} {a : X} (hf : Filter.Tendsto f l (𝓝 c)) (hg : Filter.Tendsto g l (𝓝 a))
Goal: Filter.Tendsto (fun x => f x +ᵥ g x) l (𝓝 (c +ᵥ a))
-- Filter.Tendsto.smul
@[to_additive]
theorem Filter.Tendsto.smul {f : α → M} {g : α → X} {l : Filter α} {c : M} {a : X}
    (hf : Tendsto f l (𝓝 c)) (hg : Tendsto g l (𝓝 a)) :
    Tendsto (fun x => f x • g x) l (𝓝 <| c • a) 
Arguments: {M : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace M] [TopologicalSpace X] [SMul M X] [ContinuousSMul M X] {f : α → M} {g : α → X} {l : Filter α} {c : M} {a : X} (hf : Filter.Tendsto f l (𝓝 c)) (hg : Filter.Tendsto g l (𝓝 a))
Goal: Filter.Tendsto (fun x => f x • g x) l (𝓝 (c • a))
-- Filter.Tendsto.smul_const
@[to_additive]
theorem Filter.Tendsto.smul_const {f : α → M} {l : Filter α} {c : M} (hf : Tendsto f l (𝓝 c))
    (a : X) : Tendsto (fun x => f x • a) l (𝓝 (c • a)) 
Arguments: {M : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace M] [TopologicalSpace X] [SMul M X] [ContinuousSMul M X] {f : α → M} {l : Filter α} {c : M} (hf : Filter.Tendsto f l (𝓝 c)) (a : X)
Goal: Filter.Tendsto (fun x => f x • a) l (𝓝 (c • a))
-- Filter.Tendsto.vadd_const
@[to_additive]
theorem Filter.Tendsto.smul_const {f : α → M} {l : Filter α} {c : M} (hf : Tendsto f l (𝓝 c))
    (a : X) : Tendsto (fun x => f x • a) l (𝓝 (c • a)) 
Arguments: {M : Type u_1} {X : Type u_2} {α : Type u_4} [TopologicalSpace M] [TopologicalSpace X] [VAdd M X] [ContinuousVAdd M X] {f : α → M} {l : Filter α} {c : M} (hf : Filter.Tendsto f l (𝓝 c)) (a : X)
Goal: Filter.Tendsto (fun x => f x +ᵥ a) l (𝓝 (c +ᵥ a))
-- ContinuousWithinAt.smul
@[to_additive]
theorem ContinuousWithinAt.smul (hf : ContinuousWithinAt f s b) (hg : ContinuousWithinAt g s b) :
    ContinuousWithinAt (fun x => f x • g x) s b 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {f : Y → M} {g : Y → X} {b : Y} {s : Set Y} (hf : ContinuousWithinAt f s b) (hg : ContinuousWithinAt g s b)
Goal: ContinuousWithinAt (fun x => f x • g x) s b
-- ContinuousWithinAt.vadd
@[to_additive]
theorem ContinuousWithinAt.smul (hf : ContinuousWithinAt f s b) (hg : ContinuousWithinAt g s b) :
    ContinuousWithinAt (fun x => f x • g x) s b 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {f : Y → M} {g : Y → X} {b : Y} {s : Set Y} (hf : ContinuousWithinAt f s b) (hg : ContinuousWithinAt g s b)
Goal: ContinuousWithinAt (fun x => f x +ᵥ g x) s b
-- ContinuousAt.smul
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.smul (hf : ContinuousAt f b) (hg : ContinuousAt g b) :
    ContinuousAt (fun x => f x • g x) b 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [SMul M X] [ContinuousSMul M X] {f : Y → M} {g : Y → X} {b : Y} (hf : ContinuousAt f b) (hg : ContinuousAt g b)
Goal: ContinuousAt (fun x => f x • g x) b
-- ContinuousAt.vadd
@[to_additive (attr := fun_prop)]
theorem ContinuousAt.smul (hf : ContinuousAt f b) (hg : ContinuousAt g b) :
    ContinuousAt (fun x => f x • g x) b 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {f : Y → M} {g : Y → X} {b : Y} (hf : ContinuousAt f b) (hg : ContinuousAt g b)
Goal: ContinuousAt (fun x => f x +ᵥ g x) b
-- ContinuousOn.vadd
@[to_additive (attr := fun_prop)]
theorem ContinuousOn.smul (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => f x • g x) s 
Arguments: {M : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace M] [TopologicalSpace X] [TopologicalSpace Y] [VAdd M X] [ContinuousVAdd M X] {f : Y → M} {g : Y → X} {s : Set Y} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (fun x => f x +ᵥ g x) s