mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.zero =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.zero

pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one

Tactic:
·

Goal state after:
case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.zero =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.zero
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one

Tactic:
·

Goal state after:
case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ 𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app WalkingParallelPair.one =
    ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app WalkingParallelPair.one
Module: Mathlib.Topology.Sheaves.Sheafify

Goal state before:
pp: case h.w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : ToType (F.obj (op U))
⊢ (y : ↥↑⟨U, m⟩) → F.stalk ↑y

pp: case h.h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : ToType (F.obj (op U))
⊢ ∃ (_ : (Sheafify.isLocallyGerm F).pred ?h.w), ?h.w ⟨x, ⋯⟩ = (ConcreteCategory.hom (F.germ U x m)) s

Tactic:
·

Goal state after:
case h.w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : ToType (F.obj (op U))
⊢ (y : ↥↑⟨U, m⟩) → F.stalk ↑y
Module: Mathlib.Topology.Sheaves.Sheafify

Goal state before:
pp: case h.h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : ToType (F.obj (op U))
⊢ ∃ (_ : (Sheafify.isLocallyGerm F).pred fun y ↦ (ConcreteCategory.hom (F.germ ↑⟨U, m⟩ ↑y ⋯)) s),
    (fun y ↦ (ConcreteCategory.hom (F.germ ↑⟨U, m⟩ ↑y ⋯)) s) ⟨x, ⋯⟩ = (ConcreteCategory.hom (F.germ U x m)) s

Tactic:
·

Goal state after:
case h.h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U : Opens ↑X
m : x ∈ U
s : ToType (F.obj (op U))
⊢ ∃ (_ : (Sheafify.isLocallyGerm F).pred fun y ↦ (ConcreteCategory.hom (F.germ ↑⟨U, m⟩ ↑y ⋯)) s),
    (fun y ↦ (ConcreteCategory.hom (F.germ ↑⟨U, m⟩ ↑y ⋯)) s) ⟨x, ⋯⟩ = (ConcreteCategory.hom (F.germ U x m)) s
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case op.single
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝ : ι
⊢ 𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
      (Functor.mapCone F (cocone U).op).π.app (op (single a✝)) =
    ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (single a✝))

pp: case op.pair
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝¹ a✝ : ι
⊢ 𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
      (Functor.mapCone F (cocone U).op).π.app (op (Pairwise.pair a✝¹ a✝)) =
    ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (Pairwise.pair a✝¹ a✝))

Tactic:
·

Goal state after:
case op.single
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝ : ι
⊢ 𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
      (Functor.mapCone F (cocone U).op).π.app (op (single a✝)) =
    ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (single a✝))
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case op.pair
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝¹ a✝ : ι
⊢ 𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
      (Functor.mapCone F (cocone U).op).π.app (op (Pairwise.pair a✝¹ a✝)) =
    ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (Pairwise.pair a✝¹ a✝))

Tactic:
·

Goal state after:
case op.pair
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝¹ a✝ : ι
⊢ 𝟙 ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).pt ≫
      (Functor.mapCone F (cocone U).op).π.app (op (Pairwise.pair a✝¹ a✝)) =
    ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (Pairwise.pair a✝¹ a✝))
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case op.single
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝ : ι
⊢ 𝟙 (Functor.mapCone F (cocone U).op).pt ≫
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (single a✝)) =
    (Functor.mapCone F (cocone U).op).π.app (op (single a✝))

pp: case op.pair
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝¹ a✝ : ι
⊢ 𝟙 (Functor.mapCone F (cocone U).op).pt ≫
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (Pairwise.pair a✝¹ a✝)) =
    (Functor.mapCone F (cocone U).op).π.app (op (Pairwise.pair a✝¹ a✝))

Tactic:
·

Goal state after:
case op.single
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝ : ι
⊢ 𝟙 (Functor.mapCone F (cocone U).op).pt ≫
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (single a✝)) =
    (Functor.mapCone F (cocone U).op).π.app (op (single a✝))
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case op.pair
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝¹ a✝ : ι
⊢ 𝟙 (Functor.mapCone F (cocone U).op).pt ≫
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (Pairwise.pair a✝¹ a✝)) =
    (Functor.mapCone F (cocone U).op).π.app (op (Pairwise.pair a✝¹ a✝))

Tactic:
·

Goal state after:
case op.pair
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
P : IsLimit (SheafConditionEqualizerProducts.fork F U)
a✝¹ a✝ : ι
⊢ 𝟙 (Functor.mapCone F (cocone U).op).pt ≫
      ((coneEquiv F U).symm.functor.obj (SheafConditionEqualizerProducts.fork F U)).π.app (op (Pairwise.pair a✝¹ a✝)) =
    (Functor.mapCone F (cocone U).op).π.app (op (Pairwise.pair a✝¹ a✝))
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).pt ≫
      (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.zero =
    ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.zero

pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).pt ≫
      (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.one =
    ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.one

Tactic:
·

Goal state after:
case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).pt ≫
      (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.zero =
    ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.zero
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).pt ≫
      (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.one =
    ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.one

Tactic:
·

Goal state after:
case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).pt ≫
      (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.one =
    ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.one
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 (SheafConditionEqualizerProducts.fork F U).pt ≫
      ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.zero =
    (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.zero

pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 (SheafConditionEqualizerProducts.fork F U).pt ≫
      ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.one =
    (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.one

Tactic:
·

Goal state after:
case zero
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 (SheafConditionEqualizerProducts.fork F U).pt ≫
      ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.zero =
    (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.zero
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 (SheafConditionEqualizerProducts.fork F U).pt ≫
      ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.one =
    (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.one

Tactic:
·

Goal state after:
case one
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
Q : IsLimit (Functor.mapCone F (cocone U).op)
⊢ 𝟙 (SheafConditionEqualizerProducts.fork F U).pt ≫
      ((coneEquiv F U).functor.obj (Functor.mapCone F (cocone U).op)).π.app WalkingParallelPair.one =
    (SheafConditionEqualizerProducts.fork F U).π.app WalkingParallelPair.one
Module: Mathlib.Topology.Sheaves.Sheafify

Goal state before:
pp: case h.refine_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ ↑U
gU : F.obj (op U')
wU : ∀ (x_1 : ↥U'), (fun x_2 ↦ fU (iU x_2)) x_1 = (ConcreteCategory.hom (F.germ U' ↑x_1 ⋯)) gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ ↑V
gV : F.obj (op V')
e : (ConcreteCategory.hom (F.germ U' x ⋯)) gU = (ConcreteCategory.hom (F.germ V' x ⋯)) gV
wV : ∀ (x_1 : ↥V'), (fun x_2 ↦ fV (iV x_2)) x_1 = (ConcreteCategory.hom (F.germ V' ↑x_1 ⋯)) gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : (ConcreteCategory.hom (F.map iU'.op)) gU = (ConcreteCategory.hom (F.map iV'.op)) gV
⊢ ⟨W ⊓ (U' ⊓ V'), ⋯⟩ ⟶ V

pp: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ ↑U
gU : F.obj (op U')
wU : ∀ (x_1 : ↥U'), (fun x_2 ↦ fU (iU x_2)) x_1 = (ConcreteCategory.hom (F.germ U' ↑x_1 ⋯)) gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ ↑V
gV : F.obj (op V')
e : (ConcreteCategory.hom (F.germ U' x ⋯)) gU = (ConcreteCategory.hom (F.germ V' x ⋯)) gV
wV : ∀ (x_1 : ↥V'), (fun x_2 ↦ fV (iV x_2)) x_1 = (ConcreteCategory.hom (F.germ V' ↑x_1 ⋯)) gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : (ConcreteCategory.hom (F.map iU'.op)) gU = (ConcreteCategory.hom (F.map iV'.op)) gV
⊢ ∀ (w : ↥↑⟨W ⊓ (U' ⊓ V'), ⋯⟩), fU ((id (W.infLERight (U' ⊓ V') ≫ U'.infLELeft V' ≫ iU)) w) = fV (?h.refine_2 w)

Tactic:
·

Goal state after:
case h.refine_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ ↑U
gU : F.obj (op U')
wU : ∀ (x_1 : ↥U'), (fun x_2 ↦ fU (iU x_2)) x_1 = (ConcreteCategory.hom (F.germ U' ↑x_1 ⋯)) gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ ↑V
gV : F.obj (op V')
e : (ConcreteCategory.hom (F.germ U' x ⋯)) gU = (ConcreteCategory.hom (F.germ V' x ⋯)) gV
wV : ∀ (x_1 : ↥V'), (fun x_2 ↦ fV (iV x_2)) x_1 = (ConcreteCategory.hom (F.germ V' ↑x_1 ⋯)) gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : (ConcreteCategory.hom (F.map iU'.op)) gU = (ConcreteCategory.hom (F.map iV'.op)) gV
⊢ ⟨W ⊓ (U' ⊓ V'), ⋯⟩ ⟶ V
Module: Mathlib.Topology.Sheaves.Sheafify

Goal state before:
pp: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ ↑U
gU : F.obj (op U')
wU : ∀ (x_1 : ↥U'), (fun x_2 ↦ fU (iU x_2)) x_1 = (ConcreteCategory.hom (F.germ U' ↑x_1 ⋯)) gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ ↑V
gV : F.obj (op V')
e : (ConcreteCategory.hom (F.germ U' x ⋯)) gU = (ConcreteCategory.hom (F.germ V' x ⋯)) gV
wV : ∀ (x_1 : ↥V'), (fun x_2 ↦ fV (iV x_2)) x_1 = (ConcreteCategory.hom (F.germ V' ↑x_1 ⋯)) gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : (ConcreteCategory.hom (F.map iU'.op)) gU = (ConcreteCategory.hom (F.map iV'.op)) gV
⊢ ∀ (w : ↥↑⟨W ⊓ (U' ⊓ V'), ⋯⟩),
    fU ((id (W.infLERight (U' ⊓ V') ≫ U'.infLELeft V' ≫ iU)) w) =
      fV ((id (W.infLERight (U' ⊓ V') ≫ U'.infLERight V' ≫ iV)) w)

Tactic:
·

Goal state after:
case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ ↑U
gU : F.obj (op U')
wU : ∀ (x_1 : ↥U'), (fun x_2 ↦ fU (iU x_2)) x_1 = (ConcreteCategory.hom (F.germ U' ↑x_1 ⋯)) gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ ↑V
gV : F.obj (op V')
e : (ConcreteCategory.hom (F.germ U' x ⋯)) gU = (ConcreteCategory.hom (F.germ V' x ⋯)) gV
wV : ∀ (x_1 : ↥V'), (fun x_2 ↦ fV (iV x_2)) x_1 = (ConcreteCategory.hom (F.germ V' ↑x_1 ⋯)) gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : (ConcreteCategory.hom (F.map iU'.op)) gU = (ConcreteCategory.hom (F.map iV'.op)) gV
⊢ ∀ (w : ↥↑⟨W ⊓ (U' ⊓ V'), ⋯⟩),
    fU ((id (W.infLERight (U' ⊓ V') ≫ U'.infLELeft V' ≫ iU)) w) =
      fV ((id (W.infLERight (U' ⊓ V') ≫ U'.infLERight V' ≫ iV)) w)
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: case pos
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

pp: case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∉ unop U
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
·

Goal state after:
case pos
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: case pos
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
h2 : p₀ ∈ unop V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

pp: case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
h2 : p₀ ∉ unop V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
·

Goal state after:
case pos
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
h2 : p₀ ∈ unop V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
h2 : p₀ ∉ unop V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
·

Goal state after:
case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∈ unop U
h2 : p₀ ∉ unop V
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)
Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∉ unop U
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)

Tactic:
·

Goal state after:
case neg
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : C
inst✝¹ : HasTerminal C
inst✝ : Epi (terminalIsTerminal.from A)
U V : (Opens ↑X)ᵒᵖ
r : U ⟶ V
h1 : p₀ ∉ unop U
⊢ Epi ((skyscraperSheaf p₀ A).obj.map r)
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: case mp
X : Type u_1
inst✝ : TopologicalSpace X
⊢ HasSmallInductiveDimensionLT X 1 → IsTopologicalBasis {s | IsClopen s}

pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
⊢ IsTopologicalBasis {s | IsClopen s} → HasSmallInductiveDimensionLT X 1

Tactic:
·

Goal state after:
case mp
X : Type u_1
inst✝ : TopologicalSpace X
⊢ HasSmallInductiveDimensionLT X 1 → IsTopologicalBasis {s | IsClopen s}
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
⊢ IsTopologicalBasis {s | IsClopen s} → HasSmallInductiveDimensionLT X 1

Tactic:
·

Goal state after:
case mpr
X : Type u_1
inst✝ : TopologicalSpace X
⊢ IsTopologicalBasis {s | IsClopen s} → HasSmallInductiveDimensionLT X 1
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ {s | IsOpen[inst✝¹] s ∧ IsCompact s}
⊢ IsCompact s

pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ {s | IsClosed[inst✝¹] s ∧ IsCompact sᶜ}
⊢ IsCompact s

Tactic:
·

Goal state after:
case inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ {s | IsOpen[inst✝¹] s ∧ IsCompact s}
⊢ IsCompact s
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ {s | IsClosed[inst✝¹] s ∧ IsCompact sᶜ}
⊢ IsCompact s

Tactic:
·

Goal state after:
case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ {s | IsClosed[inst✝¹] s ∧ IsCompact sᶜ}
⊢ IsCompact s
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : QuasiSeparatedSpace X
s : Set (Set X)
hs : s ⊆ constructibleTopologySubbasis X
hf : s.Finite
t : Set X
ht : t ∈ s
⊢ IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t

pp: case refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : QuasiSeparatedSpace X
s : Set (Set X)
hs : s ⊆ constructibleTopologySubbasis X
hf : s.Finite
t : Set X
ht : t ∈ s
⊢ IsCompact t

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : QuasiSeparatedSpace X
s : Set (Set X)
hs : s ⊆ constructibleTopologySubbasis X
hf : s.Finite
t : Set X
ht : t ∈ s
⊢ IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : QuasiSeparatedSpace X
s : Set (Set X)
hs : s ⊆ constructibleTopologySubbasis X
hf : s.Finite
t : Set X
ht : t ∈ s
⊢ IsCompact t

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : QuasiSeparatedSpace X
s : Set (Set X)
hs : s ⊆ constructibleTopologySubbasis X
hf : s.Finite
t : Set X
ht : t ∈ s
⊢ IsCompact t