awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
j :
  PreZeroHypercover.I₀.{u_1, u, u + 1} (C := Scheme) (S := W)
    { I₀ := 𝒰.I₀,
      X := fun (x : 𝒰.I₀) ↦
        (pullback (X := OverClass.asOver (𝒰.X x) S) (Y := OverClass.asOver W S) (Z := OverClass.asOver X S)
            (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
      f := fun (x : 𝒰.I₀) ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }
refine_2 :
  P (X :=
    pullback (X := Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver (𝒰.X j) S)) (Y :=
      Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver W S)) (Z :=
      Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver X S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver (𝒰.f j) S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver f S)))
    (pullback.snd (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver (𝒰.f j) S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver f S)))
⊢ P (X :=
    (pullback (X := OverClass.asOver (𝒰.X j) S) (Y := OverClass.asOver W S) (Z := OverClass.asOver X S)
        (Hom.asOver (𝒰.f j) S) (Hom.asOver f S)).left)
    (Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f j) S) (Hom.asOver f S)))

Tactic:
rw [← Over.forget_map, ← PreservesPullback.iso_hom_snd, P.cancel_left_of_respectsIso]

Premises used: CategoryTheory.Over.forget_map, CategoryTheory.MorphismProperty.cancel_left_of_respectsIso, CategoryTheory.Limits.PreservesPullback.iso_hom_snd
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
j :
  PreZeroHypercover.I₀.{u_1, u, u + 1} (C := Scheme) (S := W)
    { I₀ := 𝒰.I₀,
      X := fun (x : 𝒰.I₀) ↦
        (pullback (X := OverClass.asOver (𝒰.X x) S) (Y := OverClass.asOver W S) (Z := OverClass.asOver X S)
            (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
      f := fun (x : 𝒰.I₀) ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }
⊢ P (X :=
    pullback (X := Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver (𝒰.X j) S)) (Y :=
      Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver W S)) (Z :=
      Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver X S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver (𝒰.f j) S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver f S)))
    (pullback.snd (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver (𝒰.f j) S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver f S)))

Tactic:
P.pullback_snd _ _ (𝒰.map_prop j)

Premises used: CategoryTheory.MorphismProperty.pullback_snd, AlgebraicGeometry.Scheme.Cover.map_prop
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
j :
  PreZeroHypercover.I₀.{u_1, u, u + 1} (C := Scheme) (S := W)
    { I₀ := 𝒰.I₀,
      X := fun (x : 𝒰.I₀) ↦
        (pullback (X := OverClass.asOver (𝒰.X x) S) (Y := OverClass.asOver W S) (Z := OverClass.asOver X S)
            (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)).left,
      f := fun (x : 𝒰.I₀) ↦ Over.Hom.left (pullback.snd (Hom.asOver (𝒰.f x) S) (Hom.asOver f S)) }
⊢ P (X :=
    pullback (X := Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver (𝒰.X j) S)) (Y :=
      Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver W S)) (Z :=
      Functor.obj (C := Over S) (Over.forget S) (OverClass.asOver X S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver (𝒰.f j) S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver f S)))
    (pullback.snd (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver (𝒰.f j) S))
      (CategoryTheory.Functor.map (Over.forget S) (Hom.asOver f S)))

Tactic:
exact P.pullback_snd _ _ (𝒰.map_prop j)

Premises used: CategoryTheory.MorphismProperty.pullback_snd, AlgebraicGeometry.Scheme.Cover.map_prop
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
j : (Cover.pullbackCoverOver' (P := P) S 𝒰 f).I₀
⊢ (Cover.pullbackCoverOver' (P := P) S 𝒰 f).f j ≫ W ↘ S = (Cover.pullbackCoverOver' (P := P) S 𝒰 f).X j ↘ S

Tactic:
Over.w (pullback.snd ((𝒰.f j).asOver S) (f.asOver S))

Premises used: AlgebraicGeometry.Scheme.Hom.asOver, CategoryTheory.PreZeroHypercover.f, CategoryTheory.Limits.pullback.snd, CategoryTheory.over, CategoryTheory.Over.w, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
j : (Cover.pullbackCoverOver' (P := P) S 𝒰 f).I₀
⊢ (Cover.pullbackCoverOver' (P := P) S 𝒰 f).f j ≫ W ↘ S = (Cover.pullbackCoverOver' (P := P) S 𝒰 f).X j ↘ S

Tactic:
exact Over.w (pullback.snd ((𝒰.f j).asOver S) (f.asOver S))

Premises used: AlgebraicGeometry.Scheme.Hom.asOver, CategoryTheory.PreZeroHypercover.f, CategoryTheory.Limits.pullback.snd, CategoryTheory.Over.w, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
x : ↥W
i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀
hy : ∃ (y : ↥((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i)), ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i) y = x
⊢ Membership.mem (γ := Set ↥W)
    (Set.range (ι :=
      ↥(PreZeroHypercover.X
          { I₀ := 𝒰.I₀,
            X := fun (x : 𝒰.I₀) ↦
              (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
                    X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S)
                    (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
            f := fun (x : 𝒰.I₀) ↦
              CommaMorphism.left
                (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) :
            PreZeroHypercover.{u_1, u, u + 1} W }
          i))
      ⇑(PreZeroHypercover.f
          { I₀ := 𝒰.I₀,
            X := fun (x : 𝒰.I₀) ↦
              (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
                    X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S)
                    (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
            f := fun (x : 𝒰.I₀) ↦
              CommaMorphism.left
                (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) :
            PreZeroHypercover.{u_1, u, u + 1} W }
          i))
    x

Tactic:
(mem_range_iff_of_surjective ((𝒰.pullback₁ f).f i) _
        ((PreservesPullback.iso (MorphismProperty.Over.forget Q _ _ ⋙ Over.forget S)
          (f.asOverProp S) ((𝒰.f _).asOverProp S)).inv)
        (PreservesPullback.iso_inv_fst _ _ _) x).mp hy

Premises used: CategoryTheory.PreZeroHypercover.f, Set.Mem, AlgebraicGeometry.Scheme.Hom.asOverProp, CategoryTheory.Functor.comp, CategoryTheory.Limits.PreservesPullback.iso_inv_fst, CategoryTheory.Limits.PreservesPullback.iso, CategoryTheory.Over.forget, AlgebraicGeometry.mem_range_iff_of_surjective, Iff.mp, CategoryTheory.Precoverage.ZeroHypercover.pullback₁, CategoryTheory.MorphismProperty.Over.forget, CategoryTheory.Iso.inv, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
x : ↥W
i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀
hy : ∃ (y : ↥((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i)), ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).f i) y = x
⊢ Membership.mem (γ := Set ↥W)
    (Set.range (ι :=
      ↥(PreZeroHypercover.X
          { I₀ := 𝒰.I₀,
            X := fun (x : 𝒰.I₀) ↦
              (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
                    X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S)
                    (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
            f := fun (x : 𝒰.I₀) ↦
              CommaMorphism.left
                (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) :
            PreZeroHypercover.{u_1, u, u + 1} W }
          i))
      ⇑(PreZeroHypercover.f
          { I₀ := 𝒰.I₀,
            X := fun (x : 𝒰.I₀) ↦
              (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
                    X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S)
                    (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
            f := fun (x : 𝒰.I₀) ↦
              CommaMorphism.left
                (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
                  (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) :
            PreZeroHypercover.{u_1, u, u + 1} W }
          i))
    x

Tactic:
exact (mem_range_iff_of_surjective ((𝒰.pullback₁ f).f i) _
        ((PreservesPullback.iso (MorphismProperty.Over.forget Q _ _ ⋙ Over.forget S)
          (f.asOverProp S) ((𝒰.f _).asOverProp S)).inv)
        (PreservesPullback.iso_inv_fst _ _ _) x).mp hy

Premises used: CategoryTheory.PreZeroHypercover.f, AlgebraicGeometry.Scheme.Hom.asOverProp, CategoryTheory.Functor.comp, CategoryTheory.Limits.PreservesPullback.iso_inv_fst, CategoryTheory.Limits.PreservesPullback.iso, CategoryTheory.Over.forget, AlgebraicGeometry.mem_range_iff_of_surjective, Iff.mp, CategoryTheory.Precoverage.ZeroHypercover.pullback₁, CategoryTheory.MorphismProperty.Over.forget, CategoryTheory.Iso.inv, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j :
  PreZeroHypercover.I₀.{u_1, u, u + 1} (C := Scheme) (S := W)
    { I₀ := 𝒰.I₀,
      X := fun (x : 𝒰.I₀) ↦
        (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
            (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
              X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
      f := fun (x : 𝒰.I₀) ↦
        CommaMorphism.left
          (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
            (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) }
refine_2 :
  P (X :=
    (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
        (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X j).asOverProp (P := Q) S ⋯) (Z :=
          X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f j) S))).left)
    (Y := W)
    (CategoryTheory.Functor.map (Over.forget S)
        (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
          (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f j) S))) :
      Functor.obj (Over.forget S)
          (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
            (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X j).asOverProp (P := Q) S ⋯) (Z :=
              X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f j) S))) ⟶
        Functor.obj (Over.forget S) (W.asOver S))
⊢ P
    (PreZeroHypercover.f
      { I₀ := 𝒰.I₀,
        X := fun (x : 𝒰.I₀) ↦
          (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
              (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
                X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
        f := fun (x : 𝒰.I₀) ↦
          CommaMorphism.left
            (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
              (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) :
        PreZeroHypercover.{u_1, u, u + 1} W }
      j)

Tactic:
simp only [← CategoryTheory.Over.forget_map]

Premises used: CategoryTheory.Over.forget_map
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j :
  PreZeroHypercover.I₀.{u_1, u, u + 1} (C := Scheme) (S := W)
    { I₀ := 𝒰.I₀,
      X := fun (x : 𝒰.I₀) ↦
        (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
            (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
              X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
      f := fun (x : 𝒰.I₀) ↦
        CommaMorphism.left
          (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
            (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) }
⊢ P (X :=
    pullback (X :=
      Functor.obj (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (W.asOverProp (P := Q) S hW))
      (Y :=
      Functor.obj (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        ((𝒰.X j).asOverProp (P := Q) S ⋯))
      (Z :=
      Functor.obj (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (X.asOverProp (P := Q) S hX))
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) f S))
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) (𝒰.f j) S)))
    (pullback.fst
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) f S))
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) (𝒰.f j) S)))

Tactic:
P.pullback_fst _ _ (𝒰.map_prop j)

Premises used: CategoryTheory.MorphismProperty.pullback_fst, AlgebraicGeometry.Scheme.Cover.map_prop
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j :
  PreZeroHypercover.I₀.{u_1, u, u + 1} (C := Scheme) (S := W)
    { I₀ := 𝒰.I₀,
      X := fun (x : 𝒰.I₀) ↦
        (MorphismProperty.Comma.toComma (P := Q) (Q := ⊤) (W := ⊤)
            (pullback (X := W.asOverProp (P := Q) S hW) (Y := (𝒰.X x).asOverProp (P := Q) S ⋯) (Z :=
              X.asOverProp (P := Q) S hX) (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))).left,
      f := fun (x : 𝒰.I₀) ↦
        CommaMorphism.left
          (MorphismProperty.Comma.Hom.toCommaMorphism (P := Q) (Q := ⊤) (W := ⊤)
            (pullback.fst (Hom.asOverProp (P := Q) f S) (Hom.asOverProp (P := Q) (𝒰.f x) S))) }
⊢ P (X :=
    pullback (X :=
      Functor.obj (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (W.asOverProp (P := Q) S hW))
      (Y :=
      Functor.obj (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        ((𝒰.X j).asOverProp (P := Q) S ⋯))
      (Z :=
      Functor.obj (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (X.asOverProp (P := Q) S hX))
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) f S))
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) (𝒰.f j) S)))
    (pullback.fst
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) f S))
      (CategoryTheory.Functor.map (MorphismProperty.Comma.forget (𝟭 Scheme) (Functor.fromPUnit S) Q ⊤ ⊤ ⋙ Over.forget S)
        (Hom.asOverProp (P := Q) (𝒰.f j) S)))

Tactic:
exact P.pullback_fst _ _ (𝒰.map_prop j)

Premises used: CategoryTheory.MorphismProperty.pullback_fst, AlgebraicGeometry.Scheme.Cover.map_prop
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j : (Cover.pullbackCoverOverProp (P := P) S 𝒰 f (Q := Q) hX hW hQ).I₀
⊢ (Cover.pullbackCoverOverProp (P := P) S 𝒰 f (Q := Q) hX hW hQ).f j ≫ W ↘ S =
    (Cover.pullbackCoverOverProp (P := P) S 𝒰 f (Q := Q) hX hW hQ).X j ↘ S

Tactic:
(pullback.fst (f.asOverProp S) ((𝒰.f j).asOverProp S)).w

Premises used: CategoryTheory.NatTrans.vcomp, CategoryTheory.PreZeroHypercover.f, Set.Mem, AlgebraicGeometry.Scheme.Hom.asOverProp, setOf, ContinuousMap.comp, CategoryTheory.CommaMorphism.w, CategoryTheory.over, Set.preimage, AlgebraicGeometry.PresheafedSpace.comp, CategoryTheory.Limits.pullback.fst, AlgebraicGeometry.LocallyRingedSpace.comp, CategoryTheory.MorphismProperty.Comma.Hom.toCommaMorphism, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover
-- Module: Mathlib.AlgebraicGeometry.Cover.Over
Goal:
P : MorphismProperty Scheme
S : Scheme
inst : P.IsStableUnderBaseChange
inst_1 : IsJointlySurjectivePreserving P
X W : Scheme
𝒰 : Cover.{u_1, u} (precoverage P) X
f : W ⟶ X
inst_2 : W.Over S
inst_3 : X.Over S
inst_4 : Cover.Over S (P := P) 𝒰
inst_5 : Hom.IsOver f S
Q : MorphismProperty Scheme
inst_6 : Q.HasOfPostcompProperty Q
inst_7 : Q.IsStableUnderBaseChange
inst_8 : Q.IsStableUnderComposition
hX : Q (X ↘ S)
hW : Q (W ↘ S)
hQ : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j : (Cover.pullbackCoverOverProp (P := P) S 𝒰 f (Q := Q) hX hW hQ).I₀
⊢ (Cover.pullbackCoverOverProp (P := P) S 𝒰 f (Q := Q) hX hW hQ).f j ≫ W ↘ S =
    (Cover.pullbackCoverOverProp (P := P) S 𝒰 f (Q := Q) hX hW hQ).X j ↘ S

Tactic:
exact (pullback.fst (f.asOverProp S) ((𝒰.f j).asOverProp S)).w

Premises used: CategoryTheory.PreZeroHypercover.f, AlgebraicGeometry.Scheme.Hom.asOverProp, CategoryTheory.CommaMorphism.w, CategoryTheory.Limits.pullback.fst, CategoryTheory.MorphismProperty.Comma.Hom.toCommaMorphism, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover