awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasProducts.{v', v, u} C
X : TopCat
F : Presheaf.{v', v, u} C X
ι : Type v'
U : ι → Opens (↑X : Type v')
Q : IsLimit (Functor.mapCone F (Cocone.op (cocone U)))
⊢ Eq.{v + 1} (α :=
    Cone.pt (F :=
        parallelPair (X := SheafConditionEqualizerProducts.piOpens F U) (SheafConditionEqualizerProducts.leftRes F U)
          (SheafConditionEqualizerProducts.rightRes F U))
        (SheafConditionEqualizerProducts.fork F U) ⟶
      (parallelPair (X := SheafConditionEqualizerProducts.piOpens F U) (SheafConditionEqualizerProducts.leftRes F U)
            (SheafConditionEqualizerProducts.rightRes F U)).obj
        WalkingParallelPair.one)
    (𝟙
        (Cone.pt (F :=
          parallelPair (X := SheafConditionEqualizerProducts.piOpens F U) (SheafConditionEqualizerProducts.leftRes F U)
            (SheafConditionEqualizerProducts.rightRes F U))
          (SheafConditionEqualizerProducts.fork F U)) ≫
      NatTrans.app
        (Cone.π (Functor.obj (Equivalence.functor (coneEquiv F U)) (Functor.mapCone F (Cocone.op (cocone U)))))
        WalkingParallelPair.one)
    (NatTrans.app (Cone.π (SheafConditionEqualizerProducts.fork F U)) WalkingParallelPair.one)
have one : Eq (α := F.obj (op (iSup U)) ⟶ SheafConditionEqualizerProducts.piInters F U)
  (𝟙 (F.obj (op (iSup U))) ≫ Pi.lift fun (b : ι × ι) ↦ F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair b.1 b.2))) :
    F.obj (op (iSup U)) ⟶ SheafConditionEqualizerProducts.piInters F U)
  (SheafConditionEqualizerProducts.res F U ≫ SheafConditionEqualizerProducts.leftRes F U :
    F.obj (op (iSup U)) ⟶ SheafConditionEqualizerProducts.piInters F U)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasProducts.{v', v, u} C
X : TopCat
F : Presheaf.{v', v, u} C X
ι : Type v'
U : ι → Opens (↑X : Type v')
Q : IsLimit (Functor.mapCone F (Cocone.op (cocone U)))
⊢ Eq (α := F.obj (op (iSup U)) ⟶ SheafConditionEqualizerProducts.piInters F U)
    (𝟙 (F.obj (op (iSup U))) ≫ Pi.lift fun (b : ι × ι) ↦ F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair b.1 b.2))) :
      F.obj (op (iSup U)) ⟶ SheafConditionEqualizerProducts.piInters F U)
    (SheafConditionEqualizerProducts.res F U ≫ SheafConditionEqualizerProducts.leftRes F U :
      F.obj (op (iSup U)) ⟶ SheafConditionEqualizerProducts.piInters F U)
have one : ∀ (j : Discrete (ι × ι)),
  CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
      (𝟙 (F.obj (op (iSup U))) ≫ Pi.lift fun (b : ι × ι) ↦ F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair b.1 b.2))))
      (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j) =
    CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
      (SheafConditionEqualizerProducts.res F U ≫ SheafConditionEqualizerProducts.leftRes F U)
      (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j)
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasProducts.{v', v, u} C
X : TopCat
F : Presheaf.{v', v, u} C X
ι : Type v'
U : ι → Opens (↑X : Type v')
Q : IsLimit (Functor.mapCone F (Cocone.op (cocone U)))
j : Discrete (ι × ι)
⊢ CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
      (𝟙 (F.obj (op (iSup U))) ≫ Pi.lift fun (b : ι × ι) ↦ F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair b.1 b.2))))
      (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j) =
    CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
      (SheafConditionEqualizerProducts.res F U ≫ SheafConditionEqualizerProducts.leftRes F U)
      (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j)
have one : Eq.{v + 1} (α := F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
  (CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
      (𝟙 (F.obj (op (iSup U))) ≫ Pi.lift fun (b : ι × ι) ↦ F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair b.1 b.2))))
      (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j) :
    F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
  (CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
      (CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piOpens F U)
        (Pi.lift fun (i : ι) ↦ F.map (leSupr U i).op)
        (Pi.lift fun (p : ι × ι) ↦
          Pi.π (fun (i : ι) ↦ F.obj (op (U i))) p.1 ≫ F.map (Quiver.Hom.op ((U p.1).infLELeft (U p.2)))))
      (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j) :
    F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
Tactic: dsimp [coneEquivInverse, SheafConditionEqualizerProducts.res,
                SheafConditionEqualizerProducts.leftRes]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasProducts.{v', v, u} C
X : TopCat
F : Presheaf.{v', v, u} C X
ι : Type v'
U : ι → Opens (↑X : Type v')
Q : IsLimit (Functor.mapCone F (Cocone.op (cocone U)))
j : Discrete (ι × ι)
⊢ Eq.{v + 1} (α := F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
    (CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
        (𝟙 (F.obj (op (iSup U))) ≫
          Pi.lift fun (b : ι × ι) ↦ F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair b.1 b.2))))
        (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j) :
      F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
    (CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piInters F U)
        (CategoryStruct.comp (Y := SheafConditionEqualizerProducts.piOpens F U)
          (Pi.lift fun (i : ι) ↦ F.map (leSupr U i).op)
          (Pi.lift fun (p : ι × ι) ↦
            Pi.π (fun (i : ι) ↦ F.obj (op (U i))) p.1 ≫ F.map (Quiver.Hom.op ((U p.1).infLELeft (U p.2)))))
        (limit.π (Discrete.functor fun (p : ι × ι) ↦ F.obj (op (U p.1 ⊓ U p.2))) j) :
      F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
have one : Eq (α := F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
  (F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair j.as.1 j.as.2))) :
    F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
  (F.map (Quiver.Hom.op (leSupr U j.as.1)) ≫ F.map (Quiver.Hom.op ((U j.as.1).infLELeft (U j.as.2))) :
    Cone.pt.{v', v, v', u} (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ F.obj (op (U i)))
        (Fan.mk (F.obj (op (iSup U))) fun (i : ι) ↦ F.map (leSupr U i).op) ⟶
      F.obj (op (U j.as.1 ⊓ U j.as.2)))
Tactic: simp only [limit.lift_π, limit.lift_π_assoc, Category.id_comp, Fan.mk_π_app,
                Category.assoc]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasProducts.{v', v, u} C
X : TopCat
F : Presheaf.{v', v, u} C X
ι : Type v'
U : ι → Opens (↑X : Type v')
Q : IsLimit (Functor.mapCone F (Cocone.op (cocone U)))
j : Discrete (ι × ι)
⊢ Eq (α := F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
    (F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair j.as.1 j.as.2))) :
      F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
    (F.map (Quiver.Hom.op (leSupr U j.as.1)) ≫ F.map (Quiver.Hom.op ((U j.as.1).infLELeft (U j.as.2))) :
      Cone.pt.{v', v, v', u} (J := Discrete ι) (F := Discrete.functor fun (i : ι) ↦ F.obj (op (U i)))
          (Fan.mk (F.obj (op (iSup U))) fun (i : ι) ↦ F.map (leSupr U i).op) ⟶
        F.obj (op (U j.as.1 ⊓ U j.as.2)))
have one : Eq (α := F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
  (F.map (Quiver.Hom.op (coconeιApp U (Pairwise.pair j.as.1 j.as.2))) :
    F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
  (F.map (Quiver.Hom.op (leSupr U j.as.1) ≫ Quiver.Hom.op ((U j.as.1).infLELeft (U j.as.2))) :
    F.obj (op (iSup U)) ⟶ F.obj (op (U j.as.1 ⊓ U j.as.2)))
Tactic: rw [← F.map_comp]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
hY : Y = iSup U
j :
  (ObjectProperty.FullSubcategory (C := Over Y) fun (f : Over Y) ↦
      (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map (eqToHom ⋯) ≫ (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op).π.app j =
    (Cone.whisker (generateEquivalenceOpensLe U hY).op.functor (Functor.mapCone F (opensLeCoverCocone U).op)).π.app j
have dsimp : F.map (eqToHom sorry) ≫
    F.map
      (unop j).obj (P := fun (f : Over Y) ↦
            ∃ (Y_1 : Opens (↑X : Type w)) (h : f.left ⟶ Y_1) (g : Y_1 ⟶ Y),
              presieveOfCoveringAux U Y g ∧ h ≫ g = f.hom).hom.op =
  F.map (Quiver.Hom.op ((opensLeCoverCocone U).ι.app (Functor.obj (generateEquivalenceOpensLe_functor' U) (unop j))))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
hY : Y = iSup U
j :
  (ObjectProperty.FullSubcategory (C := Over Y) fun (f : Over Y) ↦
      (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map (eqToHom ⋯) ≫
      F.map
        (unop j).obj (P := fun (f : Over Y) ↦
              ∃ (Y_1 : Opens (↑X : Type w)) (h : f.left ⟶ Y_1) (g : Y_1 ⟶ Y),
                presieveOfCoveringAux U Y g ∧ h ≫ g = f.hom).hom.op =
    F.map (Quiver.Hom.op ((opensLeCoverCocone U).ι.app (Functor.obj (generateEquivalenceOpensLe_functor' U) (unop j))))
have rw : F.map
    (eqToHom sorry ≫
      (unop j).obj (P := fun (f : Over Y) ↦
            ∃ (Y_1 : Opens (↑X : Type w)) (h : f.left ⟶ Y_1) (g : Y_1 ⟶ Y),
              presieveOfCoveringAux U Y g ∧ h ≫ g = f.hom).hom.op) =
  F.map (Quiver.Hom.op ((opensLeCoverCocone U).ι.app (Functor.obj (generateEquivalenceOpensLe_functor' U) (unop j))))
Tactic: rw [← F.map_comp]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
hY : Y = iSup U
j :
  (ObjectProperty.FullSubcategory (C := Over Y) fun (f : Over Y) ↦
      (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map (eqToHom ⋯) ≫
      (Cone.whisker (generateEquivalenceOpensLe U hY).op.functor (Functor.mapCone F (opensLeCoverCocone U).op)).π.app
        j =
    (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op).π.app j
have dsimp : F.map (eqToHom sorry) ≫
    F.map
      (Quiver.Hom.op ((opensLeCoverCocone U).ι.app (Functor.obj (generateEquivalenceOpensLe_functor' U) (unop j)))) =
  F.map
    (unop j).obj (P := fun (f : Over Y) ↦
          ∃ (Y_1 : Opens (↑X : Type w)) (h : f.left ⟶ Y_1) (g : Y_1 ⟶ Y),
            presieveOfCoveringAux U Y g ∧ h ≫ g = f.hom).hom.op
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
hY : Y = iSup U
j :
  (ObjectProperty.FullSubcategory (C := Over Y) fun (f : Over Y) ↦
      (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map (eqToHom ⋯) ≫
      F.map
        (Quiver.Hom.op ((opensLeCoverCocone U).ι.app (Functor.obj (generateEquivalenceOpensLe_functor' U) (unop j)))) =
    F.map
      (unop j).obj (P := fun (f : Over Y) ↦
            ∃ (Y_1 : Opens (↑X : Type w)) (h : f.left ⟶ Y_1) (g : Y_1 ⟶ Y),
              presieveOfCoveringAux U Y g ∧ h ≫ g = f.hom).hom.op
have rw : F.map
    (eqToHom sorry ≫
      Quiver.Hom.op ((opensLeCoverCocone U).ι.app (Functor.obj (generateEquivalenceOpensLe_functor' U) (unop j)))) =
  F.map
    (unop j).obj (P := fun (f : Over Y) ↦
          ∃ (Y_1 : Opens (↑X : Type w)) (h : f.left ⟶ Y_1) (g : Y_1 ⟶ Y),
            presieveOfCoveringAux U Y g ∧ h ≫ g = f.hom).hom.op
Tactic: rw [← F.map_comp]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
hY : Y = iSup U
⊢ CategoryStruct.comp (Y := Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op)
      { hom := F.map (eqToHom ⋯), w := ⋯ } { hom := F.map (eqToHom ⋯), w := ⋯ } =
    𝟙 (Cone.whisker (generateEquivalenceOpensLe U hY).op.functor (Functor.mapCone F (opensLeCoverCocone U).op))
have w : ConeMorphism.hom
    (CategoryStruct.comp (Y := Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op)
      { hom := F.map (eqToHom sorry), w := sorry } { hom := F.map (eqToHom sorry), w := sorry }) =
  ConeMorphism.hom
    (𝟙 (Cone.whisker (generateEquivalenceOpensLe U hY).op.functor (Functor.mapCone F (opensLeCoverCocone U).op)))
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
hY : Y = iSup U
⊢ CategoryStruct.comp (Y :=
      Cone.whisker (generateEquivalenceOpensLe U hY).op.functor (Functor.mapCone F (opensLeCoverCocone U).op))
      { hom := F.map (eqToHom ⋯), w := ⋯ } { hom := F.map (eqToHom ⋯), w := ⋯ } =
    𝟙 (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op)
have w : ConeMorphism.hom
    (CategoryStruct.comp (Y :=
      Cone.whisker (generateEquivalenceOpensLe U hY).op.functor (Functor.mapCone F (opensLeCoverCocone U).op))
      { hom := F.map (eqToHom sorry), w := sorry } { hom := F.map (eqToHom sorry), w := sorry }) =
  ConeMorphism.hom (𝟙 (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op))
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
R : Presieve Y
hR : Sieve.generate R ∈ (Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) Y
⊢ IsLimit (Functor.mapCone F (opensLeCoverCocone (coveringOfPresieve Y R)).op) ≃
    IsLimit (Functor.mapCone F (Sieve.generate R).arrows.cocone.op)
have h : IsLimit (Functor.mapCone F (Sieve.generate R).arrows.cocone.op) =
  IsLimit (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux (coveringOfPresieve Y R) Y)).arrows.cocone.op)
Tactic: convert!
    isLimitOpensLeEquivGenerate₁ F (coveringOfPresieve Y R)
      (coveringOfPresieve.iSup_eq_of_mem_grothendieck Y R hR).symm using 1
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
Y : Opens (↑X : Type w)
R : Presieve Y
hR : Sieve.generate R ∈ (Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) Y
⊢ IsLimit (Functor.mapCone F (Sieve.generate R).arrows.cocone.op) =
    IsLimit (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux (coveringOfPresieve Y R) Y)).arrows.cocone.op)
have h : IsLimit (Functor.mapCone F (Sieve.generate R).arrows.cocone.op) =
  IsLimit (Functor.mapCone F (Sieve.generate R).arrows.cocone.op)
Tactic: rw [covering_presieve_eq_self R]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
h : F.IsSheaf
⊢ Nonempty (IsLimit (Functor.mapCone F (opensLeCoverCocone U).op))
have rw : Nonempty
  (IsLimit (J := (Presieve.category (X := iSup U) (Sieve.arrows (Sieve.generate (presieveOfCoveringAux U (iSup U)))))ᵒᵖ)
    (Functor.mapCone F
      (Cocone.op (Presieve.cocone (Sieve.arrows (Sieve.generate (presieveOfCoveringAux U (iSup U))))))))
Tactic: rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
ι : Type u_2
U : ι → Opens (↑X : Type w)
h : F.IsSheaf
⊢ Nonempty
    (IsLimit (J :=
      (Presieve.category (X := iSup U) (Sieve.arrows (Sieve.generate (presieveOfCoveringAux U (iSup U)))))ᵒᵖ)
      (Functor.mapCone F
        (Cocone.op (Presieve.cocone (Sieve.arrows (Sieve.generate (presieveOfCoveringAux U (iSup U))))))))
have a : Membership.mem (γ := Set (Sieve (iSup U)))
  ((Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) (iSup U))
  (Sieve.generate (presieveOfCoveringAux U (iSup U)))
Tactic: apply (Presheaf.isSheaf_iff_isLimit _ _).mp h
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
⊢ F.IsSheaf ↔ F.IsSheafOpensLeCover
have refine : ∀ (h : F.IsSheafOpensLeCover) (Y : Opens (↑X : Type w)),
  ∀ S ∈ (Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) Y,
    Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))
Tactic: refine ⟨fun h _ ↦ h.isSheafOpensLeCover,
    fun h ↦ (Presheaf.isSheaf_iff_isLimit _ _).mpr fun Y S ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
h : F.IsSheafOpensLeCover
Y : Opens (↑X : Type w)
S : Sieve Y
⊢ S ∈ (Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) Y →
    Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))
have rw : Sieve.generate S.arrows ∈ (Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) Y →
  Nonempty (IsLimit (Functor.mapCone F (Sieve.generate S.arrows).arrows.cocone.op))
Tactic: rw [← Sieve.generate_sieve S]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
F : Presheaf.{w, v_1, u_1} C X
h : F.IsSheafOpensLeCover
Y : Opens (↑X : Type w)
S : Sieve Y
hS : Sieve.generate S.arrows ∈ (Opens.grothendieckTopology (↑X : Type w) : (a : Opens (↑X : Type w)) → Set (Sieve a)) Y
⊢ Nonempty (IsLimit (Functor.mapCone F (Sieve.generate S.arrows).arrows.cocone.op))
have rw : Nonempty
  (IsLimit (J :=
    (ObjectProperty.FullSubcategory (C := Opens (↑X : Type w)) fun (V : Opens (↑X : Type w)) ↦
        ∃ (i : (V : Opens (↑X : Type w)) × { f : V ⟶ Y // S.arrows f }), V ≤ coveringOfPresieve Y S.arrows i)ᵒᵖ)
    (Functor.mapCone F (Cocone.op (opensLeCoverCocone (coveringOfPresieve Y S.arrows)))))
Tactic: rw [← (isLimitOpensLeEquivGenerate₂ F S.1 hS).nonempty_congr]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
ι : Type u_2
U : ι → Opens (↑X : Type w)
V : OpensLeCover U
A B : StructuredArrow V (pairwiseToOpensLeCover U)
⊢ ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))), List.IsChain Zag (A :: l) ∧ (A :: l).getLast ⋯ = B
have single : ∀ (i : ι)
  (a :
    Functor.obj (Functor.fromPUnit V) { as := PUnit.unit : Discrete PUnit.{1} } ⟶
      (pairwiseToOpensLeCover U).obj (single i)),
  ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))),
    List.IsChain Zag
        ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := single i, hom := a :
            Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
          l) ∧
      ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := single i, hom := a :
                Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
              l).getLast
          sorry =
        B
Tactic: rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩
Cleared hypotheses: A
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
ι : Type u_2
U : ι → Opens (↑X : Type w)
V : OpensLeCover U
A B : StructuredArrow V (pairwiseToOpensLeCover U)
single :
  ∀ (i : ι)
    (a :
      Functor.obj (Functor.fromPUnit V) { as := PUnit.unit : Discrete PUnit.{1} } ⟶
        (pairwiseToOpensLeCover U).obj (Pairwise.single i)),
    ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))),
      List.IsChain Zag
          ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := Pairwise.single i, hom := a :
              Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
            l) ∧
        ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := Pairwise.single i, hom := a :
                  Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
                l).getLast
            ⋯ =
          B
⊢ ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))), List.IsChain Zag (A :: l) ∧ (A :: l).getLast ⋯ = B
have pair : ∀ (i j : ι)
  (a :
    Functor.obj (Functor.fromPUnit V) { as := PUnit.unit : Discrete PUnit.{1} } ⟶
      (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)),
  ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))),
    List.IsChain Zag
        ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := Pairwise.pair i j, hom := a :
            Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
          l) ∧
      ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := Pairwise.pair i j, hom := a :
                Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
              l).getLast
          sorry =
        B
Tactic: rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩
Cleared hypotheses: A
-- Lean 4 proof step from Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : TopCat
ι : Type u_2
U : ι → Opens (↑X : Type w)
V : OpensLeCover U
B : StructuredArrow V (pairwiseToOpensLeCover U)
i : ι
a :
  Functor.obj (Functor.fromPUnit V) { as := PUnit.unit : Discrete PUnit.{1} } ⟶
    (pairwiseToOpensLeCover U).obj (single i)
⊢ ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))),
    List.IsChain Zag
        ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := single i, hom := a :
            Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
          l) ∧
      ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := single i, hom := a :
                Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
              l).getLast
          ⋯ =
        B
have single : ∀ (i' : ι)
  (b :
    Functor.obj (Functor.fromPUnit V) { as := PUnit.unit : Discrete PUnit.{1} } ⟶
      (pairwiseToOpensLeCover U).obj (single i')),
  ∃ (l : List (StructuredArrow V (pairwiseToOpensLeCover U))),
    List.IsChain Zag
        ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := single i, hom := a :
            Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
          l) ∧
      ({ left := { as := PUnit.unit : Discrete PUnit.{1} }, right := single i, hom := a :
                Comma (Functor.fromPUnit V) (pairwiseToOpensLeCover U) } ::
              l).getLast
          sorry =
        { left := { as := PUnit.unit }, right := single i', hom := b }
Tactic: rcases B with ⟨⟨⟨⟩⟩, ⟨i'⟩ | ⟨i', j'⟩, b⟩
Cleared hypotheses: B