ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- _private.Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear.0.mem_aux
private theorem mem_aux {e : PartialHomeomorph (B × F) (B × F)} :
    (e ∈ ⋃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
      (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
      (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
        {e | e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn
          h2φ.continuousOn)}) ↔
      ∃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U)
        (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U)
        (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
          e.EqOnSource
            (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn h2φ.continuousOn) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] {IB : ModelWithCorners 𝕜 EB HB} {e : PartialHomeomorph (B × F) (B × F)}
Goal: e ∈
    ⋃ φ,
      ⋃ U,
        ⋃ (hU : IsOpen U),
          ⋃ (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) U),
            ⋃ (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) U),
              {e | e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU ⋯ ⋯)} ↔
  ∃ φ U,
    ∃ (hU : IsOpen U) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) U) (h2φ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) U), e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU ⋯ ⋯)
-- mem_smoothFiberwiseLinear_iff
@[simp]
theorem mem_smoothFiberwiseLinear_iff (e : PartialHomeomorph (B × F) (B × F)) :
    e ∈ smoothFiberwiseLinear B F IB ↔
      ∃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U) (hφ :
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => φ x : B → F →L[𝕜] F) U) (h2φ :
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x).symm : B → F →L[𝕜] F) U),
        e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU hφ.continuousOn h2φ.continuousOn) 
Arguments: {𝕜 : Type u_1} (B : Type u_2) (F : Type u_3) [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] (IB : ModelWithCorners 𝕜 EB HB) (e : PartialHomeomorph (B × F) (B × F))
Goal: e ∈ smoothFiberwiseLinear B F IB ↔
  ∃ φ U,
    ∃ (hU : IsOpen U) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) U) (h2φ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) U), e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU ⋯ ⋯)
-- AlgebraicGeometry.LocallyRingedSpace.localRing
structure LocallyRingedSpace extends SheafedSpace CommRingCat.{u} where
    localRing : ∀ x, LocalRing (presheaf.stalk x)
Arguments: (self : AlgebraicGeometry.LocallyRingedSpace) (x : ↑↑self.toPresheafedSpace)
Goal: LocalRing ↑(self.presheaf.stalk x)
-- AlgebraicGeometry.LocallyRingedSpace.Hom.prop
@[ext]
structure Hom (X Y : LocallyRingedSpace.{u}) : Type _ where
    val : X.toSheafedSpace ⟶ Y.toSheafedSpace
    prop : ∀ x, IsLocalRingHom (val.stalkMap x)
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (self : X.Hom Y) (x : ↑↑X.toPresheafedSpace)
Goal: IsLocalRingHom (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap self.val x)
-- AlgebraicGeometry.LocallyRingedSpace.Hom.ext'
@[ext] lemma Hom.ext' (X Y : LocallyRingedSpace.{u}) {f g : X ⟶ Y} (h : f.val = g.val) : f = g 
Arguments: (X Y : AlgebraicGeometry.LocallyRingedSpace) {f g : X ⟶ Y} (h : f.val = g.val)
Goal: f = g
-- AlgebraicGeometry.LocallyRingedSpace.id_val
@[simps]
def id (X : LocallyRingedSpace.{u}) : Hom X X 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace)
Goal: X.id.val = 𝟙 X.toSheafedSpace
-- AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace_map
@[simps]
def forgetToSheafedSpace : LocallyRingedSpace.{u} ⥤ SheafedSpace CommRingCat.{u} 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y)
Goal: AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace.map f = f.val
-- AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace_obj
@[simps]
def forgetToSheafedSpace : LocallyRingedSpace.{u} ⥤ SheafedSpace CommRingCat.{u} 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace)
Goal: AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace.obj X = X.toSheafedSpace
-- AlgebraicGeometry.LocallyRingedSpace.forgetToTop_obj
@[simps!]
def forgetToTop : LocallyRingedSpace.{u} ⥤ TopCat.{u} 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace)
Goal: AlgebraicGeometry.LocallyRingedSpace.forgetToTop.obj X =
  (AlgebraicGeometry.SheafedSpace.forget CommRingCat).obj X.toSheafedSpace
-- AlgebraicGeometry.LocallyRingedSpace.forgetToTop_map
@[simps!]
def forgetToTop : LocallyRingedSpace.{u} ⥤ TopCat.{u} 
Goal: ∀ {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y),
  AlgebraicGeometry.LocallyRingedSpace.forgetToTop.map f = (AlgebraicGeometry.SheafedSpace.forget CommRingCat).map f.val
-- AlgebraicGeometry.LocallyRingedSpace.comp_val
@[simp]
theorem comp_val {X Y Z : LocallyRingedSpace.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) :
    (f ≫ g).val = f.val ≫ g.val 
Arguments: {X Y Z : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (f ≫ g).val = f.val ≫ g.val
-- AlgebraicGeometry.LocallyRingedSpace.id_val'
@[simp] theorem id_val' (X : LocallyRingedSpace.{u}) : Hom.val (𝟙 X) = 𝟙 X.toSheafedSpace 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace)
Goal: (𝟙 X).val = 𝟙 X.toSheafedSpace
-- AlgebraicGeometry.LocallyRingedSpace.comp_val_c
@[simp]
theorem comp_val_c {X Y Z : LocallyRingedSpace.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) :
    (f.1 ≫ g.1).c = g.val.c ≫ (Presheaf.pushforward _ g.val.base).map f.val.c 
Arguments: {X Y Z : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (g : Y ⟶ Z)
Goal: (f.val ≫ g.val).c = g.val.c ≫ (TopCat.Presheaf.pushforward CommRingCat g.val.base).map f.val.c
-- AlgebraicGeometry.LocallyRingedSpace.comp_val_c_app
theorem comp_val_c_app {X Y Z : LocallyRingedSpace.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (U : (Opens Z)ᵒᵖ) :
    (f ≫ g).val.c.app U = g.val.c.app U ≫ f.val.c.app (op <| (Opens.map g.val.base).obj U.unop) 
Arguments: {X Y Z : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (g : Y ⟶ Z) (U : (TopologicalSpace.Opens ↑Z.toTopCat)ᵒᵖ)
Goal: (f ≫ g).val.c.app U =
  g.val.c.app U ≫ f.val.c.app (Opposite.op ((TopologicalSpace.Opens.map g.val.base).obj (Opposite.unop U)))
-- AlgebraicGeometry.LocallyRingedSpace.homOfSheafedSpaceHomOfIsIso_val
@[simps]
def homOfSheafedSpaceHomOfIsIso {X Y : LocallyRingedSpace.{u}}
    (f : X.toSheafedSpace ⟶ Y.toSheafedSpace) [IsIso f] : X ⟶ Y 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X.toSheafedSpace ⟶ Y.toSheafedSpace) [CategoryTheory.IsIso f]
Goal: (AlgebraicGeometry.LocallyRingedSpace.homOfSheafedSpaceHomOfIsIso f).val = f
-- AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_map
@[simps!]
def restrict {U : TopCat} (X : LocallyRingedSpace.{u}) {f : U ⟶ X.toTopCat} (h : OpenEmbedding f) :
    LocallyRingedSpace 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f)
Goal: ∀ {X_1 Y : (TopologicalSpace.Opens ↑U)ᵒᵖ} (f_1 : X_1 ⟶ Y),
  (X.restrict h).presheaf.map f_1 = X.presheaf.map (⋯.functor.map f_1.unop).op
-- AlgebraicGeometry.LocallyRingedSpace.restrict_presheaf_obj
@[simps!]
def restrict {U : TopCat} (X : LocallyRingedSpace.{u}) {f : U ⟶ X.toTopCat} (h : OpenEmbedding f) :
    LocallyRingedSpace 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f)
Goal: ∀ (X_1 : (TopologicalSpace.Opens ↑U)ᵒᵖ),
  (X.restrict h).presheaf.obj X_1 = X.presheaf.obj (Opposite.op (⋯.functor.obj (Opposite.unop X_1)))
-- AlgebraicGeometry.LocallyRingedSpace.restrict_carrier
@[simps!]
def restrict {U : TopCat} (X : LocallyRingedSpace.{u}) {f : U ⟶ X.toTopCat} (h : OpenEmbedding f) :
    LocallyRingedSpace 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f)
Goal: ↑(X.restrict h).toPresheafedSpace = U
-- AlgebraicGeometry.LocallyRingedSpace.Γ_def
theorem Γ_def : Γ = forgetToSheafedSpace.op ⋙ SheafedSpace.Γ 
Goal: AlgebraicGeometry.LocallyRingedSpace.Γ =
  AlgebraicGeometry.LocallyRingedSpace.forgetToSheafedSpace.op ⋙ AlgebraicGeometry.SheafedSpace.Γ
-- AlgebraicGeometry.LocallyRingedSpace.Γ_obj
@[simp]
theorem Γ_obj (X : LocallyRingedSpace.{u}ᵒᵖ) : Γ.obj X = X.unop.presheaf.obj (op ⊤) 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpaceᵒᵖ)
Goal: AlgebraicGeometry.LocallyRingedSpace.Γ.obj X = (Opposite.unop X).presheaf.obj (Opposite.op ⊤)
-- AlgebraicGeometry.LocallyRingedSpace.Γ_obj_op
theorem Γ_obj_op (X : LocallyRingedSpace.{u}) : Γ.obj (op X) = X.presheaf.obj (op ⊤) 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace)
Goal: AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X) = X.presheaf.obj (Opposite.op ⊤)
-- AlgebraicGeometry.LocallyRingedSpace.Γ_map
@[simp]
theorem Γ_map {X Y : LocallyRingedSpace.{u}ᵒᵖ} (f : X ⟶ Y) : Γ.map f = f.unop.1.c.app (op ⊤) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpaceᵒᵖ} (f : X ⟶ Y)
Goal: AlgebraicGeometry.LocallyRingedSpace.Γ.map f = f.unop.val.c.app (Opposite.op ⊤)
-- AlgebraicGeometry.LocallyRingedSpace.Γ_map_op
theorem Γ_map_op {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) : Γ.map f.op = f.1.c.app (op ⊤) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y)
Goal: AlgebraicGeometry.LocallyRingedSpace.Γ.map f.op = f.val.c.app (Opposite.op ⊤)
-- AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen
theorem preimage_basicOpen {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) {U : Opens Y}
    (s : Y.presheaf.obj (op U)) :
    (Opens.map f.1.base).obj (Y.toRingedSpace.basicOpen s) =
      @RingedSpace.basicOpen X.toRingedSpace ((Opens.map f.1.base).obj U) (f.1.c.app _ s) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) {U : TopologicalSpace.Opens ↑Y.toTopCat} (s : ↑(Y.presheaf.obj (Opposite.op U)))
Goal: (TopologicalSpace.Opens.map f.val.base).obj (Y.toRingedSpace.basicOpen s) =
  X.toRingedSpace.basicOpen ((f.val.c.app (Opposite.op U)) s)
-- AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero
theorem basicOpen_zero (X : LocallyRingedSpace.{u}) (U : Opens X.carrier) :
    X.toRingedSpace.basicOpen (0 : X.presheaf.obj <| op U) = ⊥ 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace)
Goal: X.toRingedSpace.basicOpen 0 = ⊥
-- AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_of_isNilpotent
@[simp]
lemma basicOpen_eq_bot_of_isNilpotent (X : LocallyRingedSpace.{u}) (U : Opens X.carrier)
    (f : (X.presheaf.obj <| op U)) (hf : IsNilpotent f) :
    X.toRingedSpace.basicOpen f = ⊥ 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (f : ↑(X.presheaf.obj (Opposite.op U))) (hf : IsNilpotent f)
Goal: X.toRingedSpace.basicOpen f = ⊥
-- AlgebraicGeometry.LocallyRingedSpace.iso_hom_val_base_inv_val_base
@[simp]
lemma iso_hom_val_base_inv_val_base {X Y : LocallyRingedSpace.{u}} (e : X ≅ Y) :
    e.hom.val.base ≫ e.inv.val.base = 𝟙 _ 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y)
Goal: e.hom.val.base ≫ e.inv.val.base = 𝟙 ↑X.toPresheafedSpace
-- AlgebraicGeometry.LocallyRingedSpace.iso_hom_val_base_inv_val_base_apply
@[simp]
lemma iso_hom_val_base_inv_val_base_apply {X Y : LocallyRingedSpace.{u}} (e : X ≅ Y) (x : X) :
    (e.inv.val.base (e.hom.val.base x)) = x 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y) (x : ↑X.toTopCat)
Goal: e.inv.val.base (e.hom.val.base x) = x
-- AlgebraicGeometry.LocallyRingedSpace.iso_inv_val_base_hom_val_base
@[simp]
lemma iso_inv_val_base_hom_val_base {X Y : LocallyRingedSpace.{u}} (e : X ≅ Y) :
    e.inv.val.base ≫ e.hom.val.base = 𝟙 _ 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y)
Goal: e.inv.val.base ≫ e.hom.val.base = 𝟙 ↑Y.toPresheafedSpace
-- AlgebraicGeometry.LocallyRingedSpace.iso_inv_val_base_hom_val_base_apply
@[simp]
lemma iso_inv_val_base_hom_val_base_apply {X Y : LocallyRingedSpace.{u}} (e : X ≅ Y) (y : Y) :
    (e.hom.val.base (e.inv.val.base y)) = y 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y) (y : ↑Y.toTopCat)
Goal: e.hom.val.base (e.inv.val.base y) = y
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_id
@[simp]
lemma stalkMap_id (X : LocallyRingedSpace.{u}) (x : X) :
    (𝟙 X : X ⟶ X).stalkMap x = 𝟙 (X.presheaf.stalk x) 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) (x : ↑X.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap (𝟙 X) x = 𝟙 (X.presheaf.stalk x)
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_comp
lemma stalkMap_comp (x : X) :
    (f ≫ g : X ⟶ Z).stalkMap x = g.stalkMap (f.val.base x) ≫ f.stalkMap x 
Arguments: {X Y Z : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↑X.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap (f ≫ g) x =
  AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap g (f.val.base x) ≫
    AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x
-- AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap
@[reassoc]
lemma stalkSpecializes_stalkMap (x x' : X) (h : x ⤳ x') :
    Y.presheaf.stalkSpecializes (f.val.base.map_specializes h) ≫ f.stalkMap x =
      f.stalkMap x' ≫ X.presheaf.stalkSpecializes h 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x x' : ↑X.toTopCat) (h : x ⤳ x')
Goal: Y.presheaf.stalkSpecializes ⋯ ≫ AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x =
  AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x' ≫ X.presheaf.stalkSpecializes h
-- AlgebraicGeometry.LocallyRingedSpace.stalkSpecializes_stalkMap_apply
lemma stalkSpecializes_stalkMap_apply (x x' : X) (h : x ⤳ x') (y) :
    f.stalkMap x (Y.presheaf.stalkSpecializes (f.val.base.map_specializes h) y) =
      (X.presheaf.stalkSpecializes h (f.stalkMap x' y)) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x x' : ↑X.toTopCat) (h : x ⤳ x') (y : ↑(Y.presheaf.stalk (f.val.base x')))
Goal: (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) ((Y.presheaf.stalkSpecializes ⋯) y) =
  (X.presheaf.stalkSpecializes h) ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x') y)
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr
@[reassoc]
lemma stalkMap_congr (f g : X ⟶ Y) (hfg : f = g) (x x' : X) (hxx' : x = x') :
    f.stalkMap x ≫ X.presheaf.stalkSpecializes (specializes_of_eq hxx'.symm) =
      Y.presheaf.stalkSpecializes (specializes_of_eq <| hfg ▸ hxx' ▸ rfl) ≫ g.stalkMap x' 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f g : X ⟶ Y) (hfg : f = g) (x x' : ↑X.toTopCat) (hxx' : x = x')
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x ≫ X.presheaf.stalkSpecializes ⋯ =
  Y.presheaf.stalkSpecializes ⋯ ≫ AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap g x'
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_hom
@[reassoc]
lemma stalkMap_congr_hom (f g : X ⟶ Y) (hfg : f = g) (x : X) :
    f.stalkMap x = Y.presheaf.stalkSpecializes (specializes_of_eq <| hfg ▸ rfl) ≫
      g.stalkMap x 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f g : X ⟶ Y) (hfg : f = g) (x : ↑X.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x =
  Y.presheaf.stalkSpecializes ⋯ ≫ AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap g x
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_congr_point
@[reassoc]
lemma stalkMap_congr_point {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x x' : X) (hxx' : x = x') :
    f.stalkMap x ≫ X.presheaf.stalkSpecializes (specializes_of_eq hxx'.symm) =
      Y.presheaf.stalkSpecializes (specializes_of_eq <| hxx' ▸ rfl) ≫ f.stalkMap x' 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x x' : ↑X.toTopCat) (hxx' : x = x')
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x ≫ X.presheaf.stalkSpecializes ⋯ =
  Y.presheaf.stalkSpecializes ⋯ ≫ AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x'
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv
@[reassoc (attr := simp)]
lemma stalkMap_hom_inv (e : X ≅ Y) (y : Y) :
    e.hom.stalkMap (e.inv.val.base y) ≫ e.inv.stalkMap y =
      Y.presheaf.stalkSpecializes (specializes_of_eq <| by simp) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y) (y : ↑Y.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom (e.inv.val.base y) ≫
    AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv y =
  Y.presheaf.stalkSpecializes ⋯
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv_apply
@[simp]
lemma stalkMap_hom_inv_apply (e : X ≅ Y) (y : Y) (z) :
    e.inv.stalkMap y (e.hom.stalkMap (e.inv.val.base y) z) =
      Y.presheaf.stalkSpecializes (specializes_of_eq <| by simp) z 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y) (y : ↑Y.toTopCat) (z : ↑(Y.presheaf.stalk (e.hom.val.base (e.inv.val.base y))))
Goal: (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv y)
    ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom (e.inv.val.base y)) z) =
  (Y.presheaf.stalkSpecializes ⋯) z
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom
@[reassoc (attr := simp)]
lemma stalkMap_inv_hom (e : X ≅ Y) (x : X) :
    e.inv.stalkMap (e.hom.val.base x) ≫ e.hom.stalkMap x =
      X.presheaf.stalkSpecializes (specializes_of_eq <| by simp) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y) (x : ↑X.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv (e.hom.val.base x) ≫
    AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom x =
  X.presheaf.stalkSpecializes ⋯
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_inv_hom_apply
@[simp]
lemma stalkMap_inv_hom_apply (e : X ≅ Y) (x : X) (y) :
    e.hom.stalkMap x (e.inv.stalkMap (e.hom.val.base x) y) =
      X.presheaf.stalkSpecializes (specializes_of_eq <| by simp) y 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (e : X ≅ Y) (x : ↑X.toTopCat) (y : ↑(X.presheaf.stalk (e.inv.val.base (e.hom.val.base x))))
Goal: (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom x)
    ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv (e.hom.val.base x)) y) =
  (X.presheaf.stalkSpecializes ⋯) y
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ
@[reassoc]
lemma stalkMap_germ (U : Opens Y)
    (x : (Opens.map f.val.base).obj U) :
    Y.presheaf.germ ⟨f.val.base x.val, x.property⟩ ≫ f.stalkMap x.val =
      f.val.c.app (op U) ≫ X.presheaf.germ x 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (U : TopologicalSpace.Opens ↑Y.toTopCat) (x : ↥((TopologicalSpace.Opens.map f.val.base).obj U))
Goal: Y.presheaf.germ ⟨f.val.base ↑x, ⋯⟩ ≫ AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f ↑x =
  f.val.c.app (Opposite.op U) ≫ X.presheaf.germ x
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ_apply
lemma stalkMap_germ_apply (U : Opens Y) (x : (Opens.map f.val.base).obj U) (y) :
    f.stalkMap x.val (Y.presheaf.germ ⟨f.val.base x.val, x.property⟩ y) =
      X.presheaf.germ x (f.val.c.app (op U) y) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (U : TopologicalSpace.Opens ↑Y.toTopCat) (x : ↥((TopologicalSpace.Opens.map f.val.base).obj U)) (y : ↑(Y.presheaf.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f ↑x) ((Y.presheaf.germ ⟨f.val.base ↑x, ⋯⟩) y) =
  (X.presheaf.germ x) ((f.val.c.app (Opposite.op U)) y)
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ'
@[reassoc (attr := simp)]
lemma stalkMap_germ' (U : Opens Y) (x : X) (hx : f.val.base x ∈ U) :
    Y.presheaf.germ ⟨f.val.base x, hx⟩ ≫ f.stalkMap x =
      f.val.c.app (op U) ≫ X.presheaf.germ (U := (Opens.map f.val.base).obj U) ⟨x, hx⟩ 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (U : TopologicalSpace.Opens ↑Y.toTopCat) (x : ↑X.toTopCat) (hx : f.val.base x ∈ U)
Goal: Y.presheaf.germ ⟨f.val.base x, hx⟩ ≫ AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x =
  f.val.c.app (Opposite.op U) ≫ X.presheaf.germ ⟨x, hx⟩
-- AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ'_apply
@[simp]
lemma stalkMap_germ'_apply
    (U : Opens Y) (x : X) (hx : f.val.base x ∈ U) (y : Y.presheaf.obj (op U)) :
    f.stalkMap x (Y.presheaf.germ (U := U) ⟨f.val.base x, hx⟩ y) =
      X.presheaf.germ (U := (Opens.map f.val.base).obj U) ⟨x, hx⟩ (f.val.c.app (op U) y) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (U : TopologicalSpace.Opens ↑Y.toTopCat) (x : ↑X.toTopCat) (hx : f.val.base x ∈ U) (y : ↑(Y.presheaf.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x) ((Y.presheaf.germ ⟨f.val.base x, hx⟩) y) =
  (X.presheaf.germ ⟨x, hx⟩) ((f.val.c.app (Opposite.op U)) y)
-- AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ
@[reassoc (attr := simp)]
lemma restrictStalkIso_hom_eq_germ :
    (X.restrict h).presheaf.germ ⟨x, hx⟩ ≫ (X.restrictStalkIso h x).hom =
      X.presheaf.germ ⟨f x, show f x ∈ h.isOpenMap.functor.obj V from ⟨x, hx, rfl⟩⟩ 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑U) (x : ↑U) (hx : x ∈ V)
Goal: (X.restrict h).presheaf.germ ⟨x, hx⟩ ≫ (X.restrictStalkIso h x).hom = X.presheaf.germ ⟨f x, ⋯⟩
-- AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_hom_eq_germ_apply
lemma restrictStalkIso_hom_eq_germ_apply (y) :
    (X.restrictStalkIso h x).hom ((X.restrict h).presheaf.germ ⟨x, hx⟩ y) =
      X.presheaf.germ ⟨f x, show f x ∈ h.isOpenMap.functor.obj V from ⟨x, hx, rfl⟩⟩ y 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑U) (x : ↑U) (hx : x ∈ V) (y : ↑((X.restrict h).presheaf.obj (Opposite.op V)))
Goal: (X.restrictStalkIso h x).hom (((X.restrict h).presheaf.germ ⟨x, hx⟩) y) = (X.presheaf.germ ⟨f x, ⋯⟩) y
-- AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ
@[reassoc (attr := simp)]
lemma restrictStalkIso_inv_eq_germ :
    X.presheaf.germ ⟨f x, show f x ∈ h.isOpenMap.functor.obj V from ⟨x, hx, rfl⟩⟩ ≫
      (X.restrictStalkIso h x).inv = (X.restrict h).presheaf.germ ⟨x, hx⟩ 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑U) (x : ↑U) (hx : x ∈ V)
Goal: X.presheaf.germ ⟨f x, ⋯⟩ ≫ (X.restrictStalkIso h x).inv = (X.restrict h).presheaf.germ ⟨x, hx⟩
-- AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_germ_apply
lemma restrictStalkIso_inv_eq_germ_apply (y) :
    (X.restrictStalkIso h x).inv
      (X.presheaf.germ ⟨f x, show f x ∈ h.isOpenMap.functor.obj V from ⟨x, hx, rfl⟩⟩ y) =
        (X.restrict h).presheaf.germ ⟨x, hx⟩ y 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f) (V : TopologicalSpace.Opens ↑U) (x : ↑U) (hx : x ∈ V) (y : ↑(X.presheaf.obj (Opposite.op (⋯.functor.obj V))))
Goal: (X.restrictStalkIso h x).inv ((X.presheaf.germ ⟨f x, ⋯⟩) y) = ((X.restrict h).presheaf.germ ⟨x, hx⟩) y
-- AlgebraicGeometry.LocallyRingedSpace.restrictStalkIso_inv_eq_ofRestrict
lemma restrictStalkIso_inv_eq_ofRestrict :
    (X.restrictStalkIso h x).inv = (X.ofRestrict h).stalkMap x 
Arguments: {U : TopCat} (X : AlgebraicGeometry.LocallyRingedSpace) {f : U ⟶ X.toTopCat} (h : OpenEmbedding ⇑f) (x : ↑U)
Goal: (X.restrictStalkIso h x).inv = AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap (X.ofRestrict h) x
-- uniqueMDiffWithinAt_univ
theorem uniqueMDiffWithinAt_univ : UniqueMDiffWithinAt I univ x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {x : M}
Goal: UniqueMDiffWithinAt I Set.univ x