awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
⊢ obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶
    (StructuredArrow.proj (op (G.obj U)) G.op ⋙ Y.obj (P := Presheaf.IsSheaf J)).obj g
have refine_1 : ∀
  (I₁ I₂ :
    GrothendieckTopology.Cover.Arrow
      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
        Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
        (G.imageSieve g.hom.unop) sorry))
  (r : I₁.Relation I₂),
  Eq (α := obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op r.Z))
    ((fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
                  Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2)))
                    ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
                  (G.imageSieve g.hom.unop) sorry)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop sorry).op)
        I₁ ≫
      Y.obj (P := Presheaf.IsSheaf J).map r.g₁.op)
    ((fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
                  Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2)))
                    ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
                  (G.imageSieve g.hom.unop) sorry)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop sorry).op)
        I₂ ≫
      Y.obj (P := Presheaf.IsSheaf J).map r.g₂.op)
Tactic: refine Y.2.amalgamate ⟨_, IsDenseSubsite.imageSieve_mem J K G g.hom.unop⟩
          (fun I ↦ f ≫ Y.1.map (l _ _ _ _ _ I.hf).op) fun I₁ I₂ r ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
⊢ Eq (α := obj (obj (const (StructuredArrow (op (G.obj U)) G.op)) X) g ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op r.Z))
    ((fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
                  Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2)))
                    ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
                  (G.imageSieve g.hom.unop) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op)
        I₁ ≫
      Y.obj (P := Presheaf.IsSheaf J).map r.g₁.op)
    ((fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
                  Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2)))
                    ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
                  (G.imageSieve g.hom.unop) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y (unop g.2) U I.f g.hom.unop ⋯).op)
        I₂ ≫
      Y.obj (P := Presheaf.IsSheaf J).map r.g₂.op)
have apply : G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop sorry) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop sorry)
Tactic: apply (Y.2 X _ (IsDenseSubsite.equalizer_mem J K G (r.g₁ ≫ l _ _ _ _ _ I₁.hf)
          (r.g₂ ≫ l _ _ _ _ _ I₂.hf) ?_)).isSeparatedFor.ext fun V iUV (hiUV : _ = _) ↦ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop sorry) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop sorry)
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop sorry)
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw₂ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map (l I₂.Y (unop g.2) U I₂.f g.hom.unop sorry)
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₂ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map (l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw₃ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₂ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map (l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₃ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw₄ : G.map (r.g₁ ≫ I₁.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₂ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map (l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₃ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
rw₄ : G.map (r.g₁ ≫ I₁.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw₅ : G.map (r.g₂ ≫ I₂.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
g : StructuredArrow (op (G.obj U)) G.op
I₁ I₂ :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} (C := C) (unop g.2)) ↦
      Membership.mem (γ := Set (Sieve.{v_1, u_1} (C := C) (unop g.2))) ((J : (a : C) → Set (Sieve a)) (unop g.2)) S)
      (G.imageSieve g.hom.unop) ⋯)
r : I₁.Relation I₂
rw : G.map r.g₁ ≫ G.map (l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₁ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₂ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map (l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
rw₃ : G.map r.g₁ ≫ G.map I₁.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
rw₄ : G.map (r.g₁ ≫ I₁.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
rw₅ : G.map (r.g₂ ≫ I₂.f) ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
⊢ G.map (r.g₁ ≫ l I₁.Y (unop g.2) U I₁.f g.hom.unop ⋯) = G.map (r.g₂ ≫ l I₂.Y (unop g.2) U I₂.f g.hom.unop ⋯)
have rw₆ : G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop = G.map r.g₂ ≫ G.map I₂.f ≫ g.hom.unop
Tactic: rw [map_comp, hl, map_comp, hl, ← map_comp_assoc, r.w, map_comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) =
    StructuredArrow.hom (StructuredArrow.mk (op (G.map i ≫ g₁)))
⊢ 𝟙 X ≫
      ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
          (G.imageSieve (G.map i ≫ g₁)) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
                  (G.imageSieve (G.map i ≫ g₁)) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯).op)
        ⋯ =
    ⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁)
                  (G.imageSieve g₁) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map (op i)
have refine_2 : Eq (α := X ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op W₂))
  (sorry.amalgamate
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
      (G.imageSieve (G.map i ≫ g₁)) sorry)
    (fun
        (I :
          GrothendieckTopology.Cover.Arrow
            (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
              (G.imageSieve (G.map i ≫ g₁)) sorry)) ↦
      f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) sorry).op)
    sorry)
  (sorry.amalgamate
      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) sorry)
      (fun
          (I :
            GrothendieckTopology.Cover.Arrow
              (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁)
                sorry)) ↦
        f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ sorry).op)
      sorry ≫
    Y.obj (P := Presheaf.IsSheaf J).map (op i))
Tactic: rw [Category.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
G : Functor.{v_1, v_2, u_1, u_2} C D
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
A : Type w
inst_2 : Category.{w', w} A
inst_3 : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A
inst_4 : IsDenseSubsite J K G
Y : Sheaf.{v_1, w', u_1, w} J A
U : C
X : A
f : obj (obj yoneda ((obj (𝟭 (Functor.{v_1, w', u_1, w} Cᵒᵖ A)) (Y.obj (P := Presheaf.IsSheaf J))).obj (op U))) (op X)
l : (X Y Z : C) → (f : X ⟶ Y) → (g : G.obj Y ⟶ G.obj Z) → (hf : (G.imageSieve g).arrows f) → X ⟶ Z
hl :
  ∀ (X Y Z : C) (f : X ⟶ Y) (g : G.obj Y ⟶ G.obj Z) (hf : (G.imageSieve g).arrows f),
    G.map (l X Y Z f g hf) = G.map f ≫ g
W₁ : C
g₁ : G.obj W₁ ⟶ G.obj U
W₂ : C
i : W₂ ⟶ W₁
hi :
  StructuredArrow.hom (StructuredArrow.mk (op g₁)) ≫ G.op.map (op i) =
    StructuredArrow.hom (StructuredArrow.mk (op (G.map i ≫ g₁)))
I :
  GrothendieckTopology.Cover.Arrow
    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
      (G.imageSieve (G.map i ≫ g₁)) ⋯)
⊢ Eq (α := X ⟶ Y.obj (P := Presheaf.IsSheaf J).obj (op I.Y))
    (⋯.amalgamate
        (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
          (G.imageSieve (G.map i ≫ g₁)) ⋯)
        (fun
            (I :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₂) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₂)
                  (G.imageSieve (G.map i ≫ g₁)) ⋯)) ↦
          f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) ⋯).op)
        ⋯ ≫
      Y.obj (P := Presheaf.IsSheaf J).map I.f.op)
    ((⋯.amalgamate
          (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) ⋯)
          (fun
              (I :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁)
                    (G.imageSieve g₁) ⋯)) ↦
            f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ ⋯).op)
          ⋯ ≫
        Y.obj (P := Presheaf.IsSheaf J).map (op i)) ≫
      Y.obj (P := Presheaf.IsSheaf J).map I.f.op)
have refine_2 : f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₂ U I.f (G.map i ≫ g₁) sorry).op =
  sorry.amalgamate
      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁) sorry)
      (fun
          (I :
            GrothendieckTopology.Cover.Arrow
              (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} W₁) ↦ S ∈ (J : (a : C) → Set (Sieve a)) W₁) (G.imageSieve g₁)
                sorry)) ↦
        f ≫ Y.obj (P := Presheaf.IsSheaf J).map (l I.Y W₁ U I.f g₁ sorry).op)
      sorry ≫
    Y.obj (P := Presheaf.IsSheaf J).map (op i ≫ I.f.op)
Tactic: simp only [Presheaf.IsSheaf.amalgamate_map, Category.assoc, ← Functor.map_comp]