ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Profinite.NobelingProof.GoodProducts.chain'_cons_of_lt
theorem chain'_cons_of_lt (l : MaxProducts C ho)
    (q : Products I) (hq : q < l.val.Tail) :
    List.Chain' (fun x x_1 ↦ x > x_1) (term I ho :: q.val) 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (C : Set (I → Bool)) {o : Ordinal.{u}} (hsC : Profinite.NobelingProof.contained C (Order.succ o)) (ho : o < Ordinal.type fun x x_1 => x < x_1) (l : ↑(Profinite.NobelingProof.GoodProducts.MaxProducts C ho)) (q : Profinite.NobelingProof.Products I) (hq : q < (↑l).Tail)
Goal: List.Chain' (fun x x_1 => x > x_1) (Profinite.NobelingProof.term I ho :: ↑q)
-- Profinite.NobelingProof.GoodProducts.good_lt_maxProducts
theorem good_lt_maxProducts (q : GoodProducts (π C (ord I · < o)))
    (l : MaxProducts C ho) : List.Lex (·<·) q.val.val l.val.val 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (C : Set (I → Bool)) {o : Ordinal.{u}} (hsC : Profinite.NobelingProof.contained C (Order.succ o)) (ho : o < Ordinal.type fun x x_1 => x < x_1) (q : ↑(Profinite.NobelingProof.GoodProducts (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o))) (l : ↑(Profinite.NobelingProof.GoodProducts.MaxProducts C ho))
Goal: List.Lex (fun x x_1 => x < x_1) ↑↑q ↑↑l
-- Profinite.NobelingProof.GoodProducts.maxTail_isGood
theorem maxTail_isGood (l : MaxProducts C ho)
    (h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
    l.val.Tail.isGood (C' C ho) 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (C : Set (I → Bool)) {o : Ordinal.{u}} (hC : IsClosed C) (hsC : Profinite.NobelingProof.contained C (Order.succ o)) (ho : o < Ordinal.type fun x x_1 => x < x_1) (l : ↑(Profinite.NobelingProof.GoodProducts.MaxProducts C ho)) (h₁ :
  ⊤ ≤
    Submodule.span ℤ
      (Set.range
        (Profinite.NobelingProof.GoodProducts.eval
          (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o))))
Goal: Profinite.NobelingProof.Products.isGood (Profinite.NobelingProof.C' C ho) (↑l).Tail
-- Profinite.NobelingProof.GoodProducts.maxToGood_injective
theorem maxToGood_injective
    (h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
    (MaxToGood C hC hsC ho h₁).Injective 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (C : Set (I → Bool)) {o : Ordinal.{u}} (hC : IsClosed C) (hsC : Profinite.NobelingProof.contained C (Order.succ o)) (ho : o < Ordinal.type fun x x_1 => x < x_1) (h₁ :
  ⊤ ≤
    Submodule.span ℤ
      (Set.range
        (Profinite.NobelingProof.GoodProducts.eval
          (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o))))
Goal: Function.Injective (Profinite.NobelingProof.GoodProducts.MaxToGood C hC hsC ho h₁)
-- Profinite.NobelingProof.GoodProducts.linearIndependent_comp_of_eval
theorem linearIndependent_comp_of_eval
    (h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
    LinearIndependent ℤ (eval (C' C ho)) →
    LinearIndependent ℤ (ModuleCat.ofHom (Linear_CC' C hsC ho) ∘ SumEval C ho ∘ Sum.inr) 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (C : Set (I → Bool)) {o : Ordinal.{u}} (hC : IsClosed C) (hsC : Profinite.NobelingProof.contained C (Order.succ o)) (ho : o < Ordinal.type fun x x_1 => x < x_1) (h₁ :
  ⊤ ≤
    Submodule.span ℤ
      (Set.range
        (Profinite.NobelingProof.GoodProducts.eval
          (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o))))
Goal: LinearIndependent ℤ (Profinite.NobelingProof.GoodProducts.eval (Profinite.NobelingProof.C' C ho)) →
  LinearIndependent (ι := ↑(Profinite.NobelingProof.GoodProducts.MaxProducts C ho)) ℤ
    (⇑(ModuleCat.ofHom (Profinite.NobelingProof.Linear_CC' C hsC ho)) ∘
      Profinite.NobelingProof.GoodProducts.SumEval C ho ∘ Sum.inr)
-- Profinite.NobelingProof.GoodProducts.P0
theorem GoodProducts.P0 : P I 0 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1]
Goal: Profinite.NobelingProof.P I 0
-- Profinite.NobelingProof.GoodProducts.Plimit
theorem GoodProducts.Plimit (o : Ordinal) (ho : Ordinal.IsLimit o) :
    (∀ (o' : Ordinal), o' < o → P I o') → P I o 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (o : Ordinal.{u}) (ho : o.IsLimit)
Goal: (∀ o' < o, Profinite.NobelingProof.P I o') → Profinite.NobelingProof.P I o
-- Profinite.NobelingProof.GoodProducts.linearIndependentAux
theorem GoodProducts.linearIndependentAux (μ : Ordinal) : P I μ 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (μ : Ordinal.{u})
Goal: Profinite.NobelingProof.P I μ
-- Profinite.NobelingProof.GoodProducts.linearIndependent
theorem GoodProducts.linearIndependent (hC : IsClosed C) :
    LinearIndependent ℤ (GoodProducts.eval C) 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] (C : Set (I → Bool)) (hC : IsClosed C)
Goal: LinearIndependent ℤ (Profinite.NobelingProof.GoodProducts.eval C)
-- Profinite.NobelingProof.Nobeling_aux
theorem Nobeling_aux : Module.Free ℤ (LocallyConstant S ℤ) 
Arguments: {I : Type u} [LinearOrder I] [IsWellOrder I fun x x_1 => x < x_1] {S : Profinite} {ι : ↑S.toTop → I → Bool} (hι : ClosedEmbedding ι)
Goal: Module.Free ℤ (LocallyConstant ↑S.toTop ℤ)
-- Profinite.Nobeling.embedding
theorem Nobeling.embedding : ClosedEmbedding (Nobeling.ι S) 
Arguments: (S : Profinite)
Goal: ClosedEmbedding (Profinite.Nobeling.ι S)
-- TopCat.Presheaf.SheafConditionEqualizerProducts.res_π
@[simp, elementwise]
theorem res_π (i : ι) : res F U ≫ limit.π _ ⟨i⟩ = F.map (Opens.leSupr U i).op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (i : ι)
Goal: TopCat.Presheaf.SheafConditionEqualizerProducts.res F U ≫
    CategoryTheory.Limits.limit.π (CategoryTheory.Discrete.functor fun i => F.obj (Opposite.op (U i))) { as := i } =
  F.map (TopologicalSpace.Opens.leSupr U i).op
-- TopCat.Presheaf.SheafConditionEqualizerProducts.w
@[elementwise]
theorem w : res F U ≫ leftRes F U = res F U ≫ rightRes F U 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: TopCat.Presheaf.SheafConditionEqualizerProducts.res F U ≫ TopCat.Presheaf.SheafConditionEqualizerProducts.leftRes F U =
  TopCat.Presheaf.SheafConditionEqualizerProducts.res F U ≫ TopCat.Presheaf.SheafConditionEqualizerProducts.rightRes F U
-- TopCat.Presheaf.SheafConditionEqualizerProducts.fork_pt
@[simp]
theorem fork_pt : (fork F U).pt = F.obj (op (iSup U)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionEqualizerProducts.fork F U).pt = F.obj (Opposite.op (iSup U))
-- TopCat.Presheaf.SheafConditionEqualizerProducts.fork_ι
@[simp]
theorem fork_ι : (fork F U).ι = res F U 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionEqualizerProducts.fork F U).ι = TopCat.Presheaf.SheafConditionEqualizerProducts.res F U
-- TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_zero
@[simp]
theorem fork_π_app_walkingParallelPair_zero : (fork F U).π.app WalkingParallelPair.zero = res F U 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionEqualizerProducts.fork F U).π.app CategoryTheory.Limits.WalkingParallelPair.zero =
  TopCat.Presheaf.SheafConditionEqualizerProducts.res F U
-- TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_one
@[simp (high)]
theorem fork_π_app_walkingParallelPair_one :
    (fork F U).π.app WalkingParallelPair.one = res F U ≫ leftRes F U 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionEqualizerProducts.fork F U).π.app CategoryTheory.Limits.WalkingParallelPair.one =
  TopCat.Presheaf.SheafConditionEqualizerProducts.res F U ≫ TopCat.Presheaf.SheafConditionEqualizerProducts.leftRes F U
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_pt
@[simps]
def coneEquivFunctorObj (c : Cone ((diagram U).op ⋙ F)) :
    Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F))
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj F U c).pt = c.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj_π_app
@[simps]
def coneEquivFunctorObj (c : Cone ((diagram U).op ⋙ F)) :
    Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F)) (Z : CategoryTheory.Limits.WalkingParallelPair)
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctorObj F U c).π.app Z =
  CategoryTheory.Limits.WalkingParallelPair.casesOn Z
    (CategoryTheory.Limits.Pi.lift fun i => c.π.app (Opposite.op (CategoryTheory.Pairwise.single i)))
    (CategoryTheory.Limits.Pi.lift fun b => c.π.app (Opposite.op (CategoryTheory.Pairwise.pair b.1 b.2)))
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_map_hom
@[simps!]
def coneEquivFunctor :
    Limits.Cone ((diagram U).op ⋙ F) ⥤
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) {c c' : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F)} (f : c ⟶ c')
Goal: ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor F U).map f).hom = f.hom
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_π_app
@[simps!]
def coneEquivFunctor :
    Limits.Cone ((diagram U).op ⋙ F) ⥤
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F)) (Z : CategoryTheory.Limits.WalkingParallelPair)
Goal: ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor F U).obj c).π.app Z =
  CategoryTheory.Limits.WalkingParallelPair.rec
    (CategoryTheory.Limits.Pi.lift fun i => c.π.app (Opposite.op (CategoryTheory.Pairwise.single i)))
    (CategoryTheory.Limits.Pi.lift fun b => c.π.app (Opposite.op (CategoryTheory.Pairwise.pair b.1 b.2))) Z
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor_obj_pt
@[simps!]
def coneEquivFunctor :
    Limits.Cone ((diagram U).op ⋙ F) ⥤
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F))
Goal: ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor F U).obj c).pt = c.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_π_app
@[simps]
def coneEquivInverseObj (c : Limits.Cone (SheafConditionEqualizerProducts.diagram F U)) :
    Limits.Cone ((diagram U).op ⋙ F) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U)) (x : (CategoryTheory.Pairwise ι)ᵒᵖ)
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj F U c).π.app x =
  Opposite.rec'
    (fun x =>
      CategoryTheory.Pairwise.casesOn x
        (fun i =>
          c.π.app CategoryTheory.Limits.WalkingParallelPair.zero ≫
            CategoryTheory.Limits.Pi.π (fun i => F.obj (Opposite.op (U i))) i)
        fun i j =>
        c.π.app CategoryTheory.Limits.WalkingParallelPair.one ≫
          CategoryTheory.Limits.Pi.π (fun p => F.obj (Opposite.op (U p.1 ⊓ U p.2))) (i, j))
    x
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj_pt
@[simps]
def coneEquivInverseObj (c : Limits.Cone (SheafConditionEqualizerProducts.diagram F U)) :
    Limits.Cone ((diagram U).op ⋙ F) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U))
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverseObj F U c).pt = c.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_pt
@[simps!]
def coneEquivInverse :
    Limits.Cone (SheafConditionEqualizerProducts.diagram F U) ⥤
      Limits.Cone ((diagram U).op ⋙ F) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U))
Goal: ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse F U).obj c).pt = c.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_map_hom
@[simps!]
def coneEquivInverse :
    Limits.Cone (SheafConditionEqualizerProducts.diagram F U) ⥤
      Limits.Cone ((diagram U).op ⋙ F) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) {c c' : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U)} (f : c ⟶ c')
Goal: ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse F U).map f).hom = f.hom
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse_obj_π_app
@[simps!]
def coneEquivInverse :
    Limits.Cone (SheafConditionEqualizerProducts.diagram F U) ⥤
      Limits.Cone ((diagram U).op ⋙ F) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U)) (x : (CategoryTheory.Pairwise ι)ᵒᵖ)
Goal: ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse F U).obj c).π.app x =
  CategoryTheory.Pairwise.rec
    (fun a => CategoryTheory.Limits.Fork.ι c ≫ CategoryTheory.Limits.Pi.π (fun i => F.obj (Opposite.op (U i))) a)
    (fun a a_1 =>
      CategoryTheory.Limits.Fork.ι c ≫
        TopCat.Presheaf.SheafConditionEqualizerProducts.leftRes F U ≫
          CategoryTheory.Limits.Pi.π (fun p => F.obj (Opposite.op (U p.1 ⊓ U p.2))) (a, a_1))
    (Opposite.unop x)
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_hom_hom
@[simps]
def coneEquivUnitIsoApp (c : Cone ((diagram U).op ⋙ F)) :
    (𝟭 (Cone ((diagram U).op ⋙ F))).obj c ≅
      (coneEquivFunctor F U ⋙ coneEquivInverse F U).obj c 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F))
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp F U c).hom.hom =
  𝟙 ((𝟭 (CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F))).obj c).pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp_inv_hom
@[simps]
def coneEquivUnitIsoApp (c : Cone ((diagram U).op ⋙ F)) :
    (𝟭 (Cone ((diagram U).op ⋙ F))).obj c ≅
      (coneEquivFunctor F U ⋙ coneEquivInverse F U).obj c 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X) (c : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F))
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIsoApp F U c).inv.hom =
  𝟙
    ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor F U ⋙
            TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse F U).obj
        c).pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_inv_app_hom
@[simps!]
def coneEquivUnitIso :
    𝟭 (Limits.Cone ((diagram U).op ⋙ F)) ≅ coneEquivFunctor F U ⋙ coneEquivInverse F U 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: ∀ (X_1 : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F)),
  ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso F U).inv.app X_1).hom = 𝟙 X_1.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso_hom_app_hom
@[simps!]
def coneEquivUnitIso :
    𝟭 (Limits.Cone ((diagram U).op ⋙ F)) ≅ coneEquivFunctor F U ⋙ coneEquivInverse F U 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: ∀ (X_1 : CategoryTheory.Limits.Cone ((CategoryTheory.Pairwise.diagram U).op ⋙ F)),
  ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso F U).hom.app X_1).hom = 𝟙 X_1.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_inv_app_hom
@[simps!]
def coneEquivCounitIso :
    coneEquivInverse F U ⋙ coneEquivFunctor F U ≅
      𝟭 (Limits.Cone (SheafConditionEqualizerProducts.diagram F U)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: ∀ (X_1 : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U)),
  ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso F U).inv.app X_1).hom = 𝟙 X_1.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso_hom_app_hom
@[simps!]
def coneEquivCounitIso :
    coneEquivInverse F U ⋙ coneEquivFunctor F U ≅
      𝟭 (Limits.Cone (SheafConditionEqualizerProducts.diagram F U)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: ∀ (X_1 : CategoryTheory.Limits.Cone (TopCat.Presheaf.SheafConditionEqualizerProducts.diagram F U)),
  ((TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso F U).hom.app X_1).hom = 𝟙 X_1.pt
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_functor
@[simps]
def coneEquiv :
    Limits.Cone ((diagram U).op ⋙ F) ≌
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv F U).functor =
  TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivFunctor F U
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_unitIso
@[simps]
def coneEquiv :
    Limits.Cone ((diagram U).op ⋙ F) ≌
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv F U).unitIso =
  TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso F U
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_counitIso
@[simps]
def coneEquiv :
    Limits.Cone ((diagram U).op ⋙ F) ≌
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv F U).counitIso =
  TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivCounitIso F U
-- TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv_inverse
@[simps]
def coneEquiv :
    Limits.Cone ((diagram U).op ⋙ F) ≌
      Limits.Cone (SheafConditionEqualizerProducts.diagram F U) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X) {ι : Type v'} (U : ι → TopologicalSpace.Opens ↑X)
Goal: (TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv F U).inverse =
  TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivInverse F U
-- TopCat.Presheaf.isSheaf_iff_isSheafEqualizerProducts
theorem isSheaf_iff_isSheafEqualizerProducts (F : Presheaf C X) :
    F.IsSheaf ↔ F.IsSheafEqualizerProducts 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasProducts C] {X : TopCat} (F : TopCat.Presheaf C X)
Goal: F.IsSheaf ↔ F.IsSheafEqualizerProducts
-- Filter.tendsto_cocompact_mul_left₀
theorem Filter.tendsto_cocompact_mul_left₀ [ContinuousMul K] {a : K} (ha : a ≠ 0) :
    Filter.Tendsto (fun x : K => a * x) (Filter.cocompact K) (Filter.cocompact K) 
Arguments: {K : Type u_1} [DivisionRing K] [TopologicalSpace K] [ContinuousMul K] {a : K} (ha : a ≠ 0)
Goal: Filter.Tendsto (fun x => a * x) (Filter.cocompact K) (Filter.cocompact K)
-- Filter.tendsto_cocompact_mul_right₀
theorem Filter.tendsto_cocompact_mul_right₀ [ContinuousMul K] {a : K} (ha : a ≠ 0) :
    Filter.Tendsto (fun x : K => x * a) (Filter.cocompact K) (Filter.cocompact K) 
Arguments: {K : Type u_1} [DivisionRing K] [TopologicalSpace K] [ContinuousMul K] {a : K} (ha : a ≠ 0)
Goal: Filter.Tendsto (fun x => x * a) (Filter.cocompact K) (Filter.cocompact K)
-- Subfield.le_topologicalClosure
theorem Subfield.le_topologicalClosure (s : Subfield α) : s ≤ s.topologicalClosure 
Arguments: {α : Type u_2} [Field α] [TopologicalSpace α] [TopologicalDivisionRing α] (s : Subfield α)
Goal: s ≤ s.topologicalClosure
-- Subfield.isClosed_topologicalClosure
theorem Subfield.isClosed_topologicalClosure (s : Subfield α) :
    IsClosed (s.topologicalClosure : Set α) 
Arguments: {α : Type u_2} [Field α] [TopologicalSpace α] [TopologicalDivisionRing α] (s : Subfield α)
Goal: IsClosed ↑s.topologicalClosure
-- Subfield.topologicalClosure_minimal
theorem Subfield.topologicalClosure_minimal (s : Subfield α) {t : Subfield α} (h : s ≤ t)
    (ht : IsClosed (t : Set α)) : s.topologicalClosure ≤ t 
Arguments: {α : Type u_2} [Field α] [TopologicalSpace α] [TopologicalDivisionRing α] (s : Subfield α) {t : Subfield α} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- affineHomeomorph_symm_apply
@[simps]
def affineHomeomorph (a b : 𝕜) (h : a ≠ 0) : 𝕜 ≃ₜ 𝕜 
Arguments: {𝕜 : Type u_2} [Field 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] (a b : 𝕜) (h : a ≠ 0) (y : 𝕜)
Goal: (affineHomeomorph a b h).symm y = (y - b) / a