awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
inst_3 : HasPullbacks.{v, u} C
inst_4 : F.Full
inst_5 : F.Faithful
S : C
ι : (S : C) → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun (i : ι S) ↦ F.obj (U S i)) (f S) ∈ (J : (a : C) → Set (Sieve a)) S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
this : F.IsCoverDense J
T : C
Z : C₀
q : Z ⟶ W₀
r : T ⟶ F.obj Z
Y : C
k : ι (pullback (f S i) (f S j))
s : F.obj Z ⟶ F.obj (U (pullback (f S i) (f S j)) k)
fac : s ≫ f (pullback (f S i) (f S j)) k = F.map q ≫ pullback.lift (F.map p₁) (F.map p₂) hp
fac₁ : s ≫ f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j) = F.map q ≫ F.map p₁
fac₂ : s ≫ f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j) = F.map q ≫ F.map p₂
⊢ (Sieve.functorPushforward F
        ({ I₀ := ι S, X := U S, f := f S, I₁ := fun (i j : ι S) ↦ ι (pullback (f S i) (f S j)),
              Y := fun (i j : ι S) ↦ U (pullback (f S i) (f S j)),
              p₁ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
                F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
              p₂ := fun (i j : ι S) (k : ι (pullback (f S i) (f S j))) ↦
                F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)),
              w := ⋯ : PreOneHypercoverDenseData.{w, v₀, v, u₀, u} F S }.sieve₁₀
          p₁ p₂)).arrows
    (r ≫ F.map q)

Tactic:
exact ⟨Z, q, r, ⟨k, F.preimage s, F.map_injective (by simp [fac₁]),
          F.map_injective (by simp [fac₂])⟩, rfl⟩
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
inst_3 : HasPullbacks.{v, u} C
inst_4 : F.Full
inst_5 : F.Faithful
S : C
ι : (S : C) → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun (i : ι S) ↦ F.obj (U S i)) (f S) ∈ (J : (a : C) → Set (Sieve a)) S
⊢ Nonempty (OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J S)

Tactic:
exact ⟨{
      I₀ := ι S
      X := U S
      f := f S
      I₁ i j := ι (pullback (f _ i) (f _ j))
      Y i j := U (pullback (f _ i) (f _ j))
      p₁ i j k := F.preimage (f _ k ≫ pullback.fst _ _)
      p₂ i j k := F.preimage (f _ k ≫ pullback.snd _ _)
      w i j k := by simp [pullback.condition]
      mem₀ := hf S
      mem₁₀ i j W₀ p₁ p₂ hp := by
        have := IsDenseSubsite.isCoverDense J₀ J F
        rw [← functorPushforward_mem_iff J₀ J F]
        refine J.superset_covering ?_
          (IsCoverDense.functorPullback_pushforward_covering
            ⟨_, J.pullback_stable (pullback.lift _ _ hp) (hf (pullback (f _ i) (f _ j)))⟩)
        rintro T _ ⟨Z, q, r, ⟨_, s, _, ⟨k⟩, fac⟩, rfl⟩
        have fac₁ := fac =≫ pullback.fst _ _
        have fac₂ := fac =≫ pullback.snd _ _
        simp only [Category.assoc, pullback.lift_fst, pullback.lift_snd] at fac₁ fac₂
        exact ⟨Z, q, r, ⟨k, F.preimage s, F.map_injective (by simp [fac₁]),
          F.map_injective (by simp [fac₂])⟩, rfl⟩ }⟩
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
i₁ i₂ : data.I₀
W : C
p₁ : W ⟶ F.obj (data.X i₁)
p₂ : W ⟶ F.obj (data.X i₂)
w : p₁ ≫ data.f i₁ = p₂ ≫ data.f i₂
⊢ F.IsCoverDense J

Tactic:
have := IsDenseSubsite.isCoverDense J₀ J F
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
i₁ i₂ : data.I₀
W : C
p₁ : W ⟶ F.obj (data.X i₁)
p₂ : W ⟶ F.obj (data.X i₂)
w : p₁ ≫ data.f i₁ = p₂ ≫ data.f i₂
this : F.IsCoverDense J
S : Sieve.{v, u} W :=
  Sieve.bind (Sieve.coverByImage F W).arrows fun (Y : C) (f : Y ⟶ W) (hf : (Sieve.coverByImage F W).arrows f) ↦
    Sieve.pullback (Nonempty.some hf).lift
      (Sieve.functorPushforward F
        (F.imageSieve ((Nonempty.some hf).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some hf).map ≫ p₂)))
Z : C
g : Z ⟶ W
hg : S.arrows g
str : Presieve.FunctorPushforwardStructure F
  (F.imageSieve ((Nonempty.some ⋯).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some ⋯).map ≫ p₂)).arrows
  ((Presieve.bind.bindStruct (S := (Sieve.coverByImage F W).arrows) (R :=
          fun (Y : C) (f : Y ⟶ W) (H : (Sieve.coverByImage F W).arrows f) ↦
          Sieve.arrows
            ((fun (Y : C) (f : Y ⟶ W) (hf : (Sieve.coverByImage F W).arrows f) ↦
                Sieve.pullback (Nonempty.some hf).lift
                  (Sieve.functorPushforward F
                    (F.imageSieve ((Nonempty.some hf).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some hf).map ≫ p₂))))
              Y f H))
          hg).g
      (S := (Sieve.coverByImage F W).arrows) (R := fun (Y : C) (f : Y ⟶ W) (H : (Sieve.coverByImage F W).arrows f) ↦
      Sieve.arrows
        ((fun (Y : C) (f : Y ⟶ W) (hf : (Sieve.coverByImage F W).arrows f) ↦
            Sieve.pullback (Nonempty.some hf).lift
              (Sieve.functorPushforward F
                (F.imageSieve ((Nonempty.some hf).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some hf).map ≫ p₂))))
          Y f H)) ≫
    (Nonempty.some ⋯).lift) :=
  Presieve.getFunctorPushforwardStructure (S :=
    (F.imageSieve ((Nonempty.some ⋯).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some ⋯).map ≫ p₂)).arrows) ⋯
⊢ Sieve.{v, u} Z

Tactic:
exact Sieve.pullback str.lift
      (Sieve.functorPushforward F (data.sieve₁₀ str.cover.1.choose str.cover.2.choose))

Premises used: CategoryTheory.Presieve.FunctorPushforwardStructure.cover, And.right, And.left, CategoryTheory.Sieve.functorPushforward, CategoryTheory.Functor.PreOneHypercoverDenseData.sieve₁₀, Exists.choose, CategoryTheory.Sieve.pullback, CategoryTheory.Functor.OneHypercoverDenseData.toPreOneHypercoverDenseData, CategoryTheory.Presieve.FunctorPushforwardStructure.lift
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
X_1 : C
data_1 : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X_1
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ Z₀ : C₀
g : Y₀ ⟶ X₀
h : data.SieveStruct f g
p : Z₀ ⟶ Y₀
rw : F.map p ≫ h.q ≫ data.f h.i₀ = F.map (p ≫ g) ≫ f
rw₁ : F.map p ≫ F.map g ≫ f = F.map (p ≫ g) ≫ f
rw₂ rw₃ : F.map p ≫ F.map g ≫ f = F.map p ≫ F.map g ≫ f
⊢ (F.map p ≫ h.q) ≫ data.f h.i₀ = F.map (p ≫ g) ≫ f

Tactic:
rw [assoc, h.fac, map_comp_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Functor.OneHypercoverDenseData.SieveStruct.fac, CategoryTheory.Functor.map_comp_assoc
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
X_1 : C
data_1 : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X_1
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
Y₀ Z₀ : C₀
g : Y₀ ⟶ X₀
h : data.SieveStruct f g
p : Z₀ ⟶ Y₀
⊢ Nonempty (data.SieveStruct f (p ≫ g))

Tactic:
exact ⟨{ i₀ := h.i₀, q := F.map p ≫ h.q, fac := by rw [assoc, h.fac, map_comp_assoc]}⟩
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
⊢ F.IsCoverDense J

Tactic:
have := IsDenseSubsite.isCoverDense J₀ J F
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this : F.IsCoverDense J
⊢ F.IsLocallyFull J

Tactic:
have := IsDenseSubsite.isLocallyFull J₀ J F
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
rw :
  Membership.mem (γ := Set (Sieve.{v, u} (F.obj X₀))) ((J : (a : C) → Set (Sieve a)) (F.obj X₀))
    (Sieve.functorPushforward F (data.sieve f))
⊢ data.sieve f ∈ (J₀ : (a : C₀) → Set (Sieve a)) X₀

Tactic:
rw [← functorPushforward_mem_iff J₀ J F]

Premises used: CategoryTheory.Functor.functorPushforward_mem_iff
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
⊢ Membership.mem (γ := Set (Sieve.{v, u} (F.obj X₀))) ((J : (a : C) → Set (Sieve a)) (F.obj X₀))
    (Sieve.functorPushforward F (data.sieve f))

Tactic:
J.superset_covering ?_
    (J.bind_covering (J.pullback_stable f (data.toOneHypercover.mem₀)) (R := R)
    (fun W p hp ↦ J.bind_covering (F.is_cover_of_isCoverDense J W) ?_))

Premises used: CategoryTheory.GrothendieckTopology.superset_covering, CategoryTheory.GrothendieckTopology.OneHypercover.mem₀, CategoryTheory.Functor.is_cover_of_isCoverDense, CategoryTheory.GrothendieckTopology.pullback_stable, CategoryTheory.GrothendieckTopology.bind_covering, CategoryTheory.Functor.OneHypercoverDenseData.toOneHypercover
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
refine_1 :
  Sieve.bind (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows R ≤
    Sieve.functorPushforward F (data.sieve f)
refine_2 :
  ∀ (W : C) (p : W ⟶ F.obj X₀)
    (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p)
    ⦃Y : C⦄ ⦃f : Y ⟶ W⦄ (H : (Sieve.coverByImage F W).arrows f),
    Sieve.pullback (Nonempty.some H).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some H).map ≫ p))) ∈
      (J : (a : C) → Set (Sieve a)) Y
⊢ Membership.mem (γ := Set (Sieve.{v, u} (F.obj X₀))) ((J : (a : C) → Set (Sieve a)) (F.obj X₀))
    (Sieve.functorPushforward F (data.sieve f))

Tactic:
refine J.superset_covering ?_
    (J.bind_covering (J.pullback_stable f (data.toOneHypercover.mem₀)) (R := R)
    (fun W p hp ↦ J.bind_covering (F.is_cover_of_isCoverDense J W) ?_))

Premises used: CategoryTheory.GrothendieckTopology.superset_covering, CategoryTheory.GrothendieckTopology.OneHypercover.mem₀, CategoryTheory.Functor.is_cover_of_isCoverDense, CategoryTheory.GrothendieckTopology.pullback_stable, CategoryTheory.GrothendieckTopology.bind_covering, CategoryTheory.Functor.OneHypercoverDenseData.toOneHypercover
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W' W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p
Y₀ : C
a : W' ⟶ Y₀
b : Y₀ ⟶ W
hb : (Sieve.coverByImage F W).arrows b
U : C₀
c : U ⟶ (Nonempty.some hb).obj
d : W' ⟶ F.obj U
fac : a ≫ (Nonempty.some hb).lift = d ≫ F.map c
x₁ : U ⟶ X₀
w₁ : F.map x₁ = F.map c ≫ (Nonempty.some hb).map ≫ p
⊢ Sieve.ofArrows.h hp ≫ data.toOneHypercover.f (Sieve.ofArrows.i hp) = p ≫ f

Tactic:
have hp' := Sieve.ofArrows.fac hp
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W' W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p
Y₀ : C
a : W' ⟶ Y₀
b : Y₀ ⟶ W
hb : (Sieve.coverByImage F W).arrows b
U : C₀
c : U ⟶ (Nonempty.some hb).obj
d : W' ⟶ F.obj U
fac : a ≫ (Nonempty.some hb).lift = d ≫ F.map c
x₁ : U ⟶ X₀
w₁ : F.map x₁ = F.map c ≫ (Nonempty.some hb).map ≫ p
hp' : Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) = p ≫ f
⊢ (Sieve.functorPushforward F (data.sieve f)).arrows ((a ≫ b) ≫ p)

Tactic:
⟨U, x₁, d, ⟨Sieve.ofArrows.i hp,
      F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp, ?_⟩, ?_⟩

Premises used: Nonempty.some, CategoryTheory.Sieve.ofArrows.h, CategoryTheory.Presieve.CoverByImageStructure.map, Nonempty.intro, CategoryTheory.Functor.map, And.intro, Exists.intro, CategoryTheory.CategoryStruct.comp, CategoryTheory.Sieve.ofArrows.i, CategoryTheory.Presieve.functorPushforward, CategoryTheory.Functor.OneHypercoverDenseData.SieveStruct.mk
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W' W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p
Y₀ : C
a : W' ⟶ Y₀
b : Y₀ ⟶ W
hb : (Sieve.coverByImage F W).arrows b
U : C₀
c : U ⟶ (Nonempty.some hb).obj
d : W' ⟶ F.obj U
fac : a ≫ (Nonempty.some hb).lift = d ≫ F.map c
x₁ : U ⟶ X₀
w₁ : F.map x₁ = F.map c ≫ (Nonempty.some hb).map ≫ p
hp' : Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) = p ≫ f
refine_1 : (F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp) ≫ data.f (Sieve.ofArrows.i hp) = F.map x₁ ≫ f
refine_1₁ : (a ≫ b) ≫ p = d ≫ F.map x₁
⊢ (Sieve.functorPushforward F (data.sieve f)).arrows ((a ≫ b) ≫ p)

Tactic:
refine ⟨U, x₁, d, ⟨Sieve.ofArrows.i hp,
      F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp, ?_⟩, ?_⟩
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W' W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p
Y₀ : C
a : W' ⟶ Y₀
b : Y₀ ⟶ W
hb : (Sieve.coverByImage F W).arrows b
U : C₀
c : U ⟶ (Nonempty.some hb).obj
d : W' ⟶ F.obj U
fac : a ≫ (Nonempty.some hb).lift = d ≫ F.map c
x₁ : U ⟶ X₀
w₁ : F.map x₁ = F.map c ≫ (Nonempty.some hb).map ≫ p
hp' : Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) = p ≫ f
refine_1 :
  (F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp) ≫ data.f (Sieve.ofArrows.i hp) =
    (F.map c ≫ (Nonempty.some hb).map ≫ p) ≫ f
refine_1₁ :
  F.map c ≫ ((Nonempty.some hb).map ≫ Sieve.ofArrows.h hp) ≫ data.f (Sieve.ofArrows.i hp) =
    (F.map c ≫ (Nonempty.some hb).map ≫ p) ≫ f
refine_1₂ :
  F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) =
    (F.map c ≫ (Nonempty.some hb).map ≫ p) ≫ f
refine_1₃ :
  F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) =
    F.map c ≫ ((Nonempty.some hb).map ≫ p) ≫ f
refine_1₄ :
  F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) =
    F.map c ≫ (Nonempty.some hb).map ≫ p ≫ f
refine_1₅ refine_1₆ : F.map c ≫ (Nonempty.some hb).map ≫ p ≫ f = F.map c ≫ (Nonempty.some hb).map ≫ p ≫ f
⊢ (F.map c ≫ (Nonempty.some hb).map ≫ Sieve.ofArrows.h hp) ≫ data.f (Sieve.ofArrows.i hp) = F.map x₁ ≫ f

Tactic:
rw [w₁, assoc, assoc, assoc, assoc, hp']

Premises used: CategoryTheory.Category.assoc
-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
inst_2 : IsDenseSubsite J₀ J F
X : C
data : OneHypercoverDenseData.{w, v₀, v, u₀, u} F J₀ J X
X₀ : C₀
f : F.obj X₀ ⟶ X
this_1 : F.IsCoverDense J
this : F.IsLocallyFull J
R : ⦃W : C⦄ →
  ⦃p : W ⟶ F.obj X₀⦄ →
    (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p →
      Sieve.{v, u} W :=
  fun (W : C) (p : W ⟶ F.obj X₀)
      (hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p) ↦
    Sieve.bind (Sieve.coverByImage F W).arrows fun (U : C) (π : U ⟶ W) (hπ : (Sieve.coverByImage F W).arrows π) ↦
      Sieve.pullback (Nonempty.some hπ).lift (Sieve.functorPushforward F (F.imageSieve ((Nonempty.some hπ).map ≫ p)))
W' W : C
p : W ⟶ F.obj X₀
hp : (Sieve.pullback f (PreZeroHypercover.sieve₀.{w, v, u} data.toOneHypercover.toPreZeroHypercover)).arrows p
Y₀ : C
a : W' ⟶ Y₀
b : Y₀ ⟶ W
hb : (Sieve.coverByImage F W).arrows b
U : C₀
c : U ⟶ (Nonempty.some hb).obj
d : W' ⟶ F.obj U
fac : a ≫ (Nonempty.some hb).lift = d ≫ F.map c
x₁ : U ⟶ X₀
w₁ : F.map x₁ = F.map c ≫ (Nonempty.some hb).map ≫ p
hp' : Sieve.ofArrows.h hp ≫ data.f (Sieve.ofArrows.i hp) = p ≫ f
refine_1 : (a ≫ b) ≫ p = d ≫ F.map c ≫ (Nonempty.some hb).map ≫ p
refine_1₁ : a ≫ b ≫ p = d ≫ F.map c ≫ (Nonempty.some hb).map ≫ p
refine_1₂ : a ≫ b ≫ p = a ≫ (Nonempty.some hb).lift ≫ (Nonempty.some hb).map ≫ p
refine_1₃ refine_1₄ : a ≫ b ≫ p = a ≫ b ≫ p
⊢ (a ≫ b) ≫ p = d ≫ F.map x₁

Tactic:
rw [w₁, assoc, ← reassoc_of% fac, hb.some.fac_assoc]

Premises used: CategoryTheory.Category.assoc, Nonempty.some, CategoryTheory.Presieve.CoverByImageStructure.fac_assoc