l3lab/ntp-mathlib huggingface dataset:
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
⊢ ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).map fV.op (gluedSection ℱ hS hx) ≫
      limit.π (Ran.diagram G.op ℱ.val (op V)) W =
    x fV hV ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W

Next tactic:
simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
⊢ limit.lift (StructuredArrow.map fV.op ⋙ Ran.diagram G.op ℱ.val (op U))
        (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone ℱ hS hx)) ≫
      limit.π (Ran.diagram G.op ℱ.val (op V)) W =
    x fV hV ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W

Next tactic:
erw [limit.lift_π]
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  erw [limit.lift_π]
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
⊢ (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone ℱ hS hx)).π.app W =
    x fV hV ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W

Next tactic:
symm
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  erw [limit.lift_π]
  symm
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
⊢ x fV hV ≫ limit.π (Ran.diagram G.op ℱ.val (op V)) W =
    (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone ℱ hS hx)).π.app W

Next tactic:
convert helper ℱ hS hx _ (x fV hV) _ _ using 1
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  erw [limit.lift_π]
  symm
  convert helper ℱ hS hx _ (x fV hV) _ _ using 1
  

Proof state:
case convert_3
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
⊢ ∀ {V' : C} {fV_1 : G.obj V' ⟶ V} (hV_1 : S.arrows (fV_1 ≫ fV)),
    x fV hV ≫ ((ran G.op).obj ℱ.val).map fV_1.op = x (fV_1 ≫ fV) hV_1

Next tactic:
intro V' fV' hV'
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  erw [limit.lift_π]
  symm
  convert helper ℱ hS hx _ (x fV hV) _ _ using 1
  intro V' fV' hV'
  

Proof state:
case convert_3
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
V' : C
fV' : G.obj V' ⟶ V
hV' : S.arrows (fV' ≫ fV)
⊢ x fV hV ≫ ((ran G.op).obj ℱ.val).map fV'.op = x (fV' ≫ fV) hV'

Next tactic:
convert hx fV' (𝟙 _) hV hV' (by rw [Category.id_comp])
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  erw [limit.lift_π]
  symm
  convert helper ℱ hS hx _ (x fV hV) _ _ using 1
  intro V' fV' hV'
  convert hx fV' (𝟙 _) hV hV' (by 

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
V' : C
fV' : G.obj V' ⟶ V
hV' : S.arrows (fV' ≫ fV)
⊢ fV' ≫ fV = 𝟙 (G.obj V') ≫ fV' ≫ fV

Next tactic:
rw [Category.id_comp]
/-- Verify that the `glued_section` is an amalgamation of `x`. -/
theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by
  intro V fV hV
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op V) G.op) => ?_)
  simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]
  erw [limit.lift_π]
  symm
  convert helper ℱ hS hx _ (x fV hV) _ _ using 1
  intro V' fV' hV'
  convert hx fV' (𝟙 _) hV hV' (by rw [Category.id_comp])
  

Proof state:
case h.e'_3.h
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
V : D
fV : V ⟶ U
hV : S.arrows fV
W : StructuredArrow (op V) G.op
V' : C
fV' : G.obj V' ⟶ V
hV' : S.arrows (fV' ≫ fV)
e_1✝ :
  (X ⟶ ((ran G.op).obj ℱ.val).obj (op (G.obj V'))) = ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op (G.obj V'))
⊢ x (fV' ≫ fV) hV' = ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).map (𝟙 (G.obj V')).op (x (fV' ≫ fV) hV')

Next tactic:
simp only [op_id, FunctorToTypes.map_id_apply]
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
⊢ y = gluedSection ℱ hS hx

Next tactic:
unfold gluedSection limit.lift
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
⊢ y = IsLimit.lift (limit.isLimit (Ran.diagram G.op ℱ.val (op U))) (gluedLimitCone ℱ hS hx)

Next tactic:
refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
⊢ y ≫ limit.π (Ran.diagram G.op ℱ.val (op U)) W =
    IsLimit.lift (limit.isLimit (Ran.diagram G.op ℱ.val (op U))) (gluedLimitCone ℱ hS hx) ≫
      limit.π (Ran.diagram G.op ℱ.val (op U)) W

Next tactic:
erw [limit.lift_π]
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  erw [limit.lift_π]
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
⊢ y ≫ limit.π (Ran.diagram G.op ℱ.val (op U)) W = (gluedLimitCone ℱ hS hx).π.app W

Next tactic:
convert helper ℱ hS hx (𝟙 _) y W _
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  erw [limit.lift_π]
  convert helper ℱ hS hx (𝟙 _) y W _
  · 

Proof state:
case h.e'_3.h.h.e'_8
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
e_1✝ : ((op X).unop ⟶ (Ran.diagram G.op ℱ.val (op U)).obj W) = (X ⟶ (Ran.diagram G.op ℱ.val (op U)).obj W)
⊢ W = (StructuredArrow.map (𝟙 U).op).obj W

Next tactic:
simp only [op_id, StructuredArrow.map_id]
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  erw [limit.lift_π]
  convert helper ℱ hS hx (𝟙 _) y W _
  · simp only [op_id, StructuredArrow.map_id]
  · 

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
⊢ ∀ {V' : C} {fV : G.obj V' ⟶ U} (hV : S.arrows (fV ≫ 𝟙 U)), y ≫ ((ran G.op).obj ℱ.val).map fV.op = x (fV ≫ 𝟙 U) hV

Next tactic:
intro V' fV' hV'
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  erw [limit.lift_π]
  convert helper ℱ hS hx (𝟙 _) y W _
  · simp only [op_id, StructuredArrow.map_id]
  · intro V' fV' hV'
    

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
V' : C
fV' : G.obj V' ⟶ U
hV' : S.arrows (fV' ≫ 𝟙 U)
⊢ y ≫ ((ran G.op).obj ℱ.val).map fV'.op = x (fV' ≫ 𝟙 U) hV'

Next tactic:
convert hy fV' (by simpa only [Category.comp_id] using hV')
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  erw [limit.lift_π]
  convert helper ℱ hS hx (𝟙 _) y W _
  · simp only [op_id, StructuredArrow.map_id]
  · intro V' fV' hV'
    convert hy fV' (by 

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
V' : C
fV' : G.obj V' ⟶ U
hV' : S.arrows (fV' ≫ 𝟙 U)
⊢ S.arrows fV'

Next tactic:
simpa only [Category.comp_id] using hV'
/-- Verify that the amalgamation is indeed unique. -/
theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by
  unfold gluedSection limit.lift
  -- porting note: next line was `ext W`
  -- Now `ext` can't see that `ran` is defined as a limit.
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  refine limit.hom_ext (λ (W : StructuredArrow (op U) G.op) => ?_)
  erw [limit.lift_π]
  convert helper ℱ hS hx (𝟙 _) y W _
  · simp only [op_id, StructuredArrow.map_id]
  · intro V' fV' hV'
    convert hy fV' (by simpa only [Category.comp_id] using hV')
    

Proof state:
case h.e'_3.h.h.e'_2
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)
hy : IsAmalgamation x y
W : StructuredArrow (op U) G.op
V' : C
fV' : G.obj V' ⟶ U
hV' : S.arrows (fV' ≫ 𝟙 U)
e_1✝ :
  (X ⟶ ((ran G.op).obj ℱ.val).obj (op (G.obj V'))) = ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op (G.obj V'))
⊢ fV' ≫ 𝟙 U = fV'

Next tactic:
erw [Category.comp_id]
/-- If `G` is cocontinuous, then `Ran G.op` pushes sheaves to sheaves.

This result is basically https://stacks.math.columbia.edu/tag/00XK,
but without the condition that `C` or `D` has pullbacks.
-/
theorem ran_isSheaf_of_isCocontinuous (ℱ : Sheaf J A) :
    Presheaf.IsSheaf K ((ran G.op).obj ℱ.val) := by
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
⊢ Presheaf.IsSheaf K ((ran G.op).obj ℱ.val)

Next tactic:
intro X U S hS x hx
/-- If `G` is cocontinuous, then `Ran G.op` pushes sheaves to sheaves.

This result is basically https://stacks.math.columbia.edu/tag/00XK,
but without the condition that `C` or `D` has pullbacks.
-/
theorem ran_isSheaf_of_isCocontinuous (ℱ : Sheaf J A) :
    Presheaf.IsSheaf K ((ran G.op).obj ℱ.val) := by
  intro X U S hS x hx
  

Proof state:
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
⊢ ∃! t, IsAmalgamation x t

Next tactic:
constructor
/-- If `G` is cocontinuous, then `Ran G.op` pushes sheaves to sheaves.

This result is basically https://stacks.math.columbia.edu/tag/00XK,
but without the condition that `C` or `D` has pullbacks.
-/
theorem ran_isSheaf_of_isCocontinuous (ℱ : Sheaf J A) :
    Presheaf.IsSheaf K ((ran G.op).obj ℱ.val) := by
  intro X U S hS x hx
  constructor; 

Proof state:
case h
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
⊢ (fun t => IsAmalgamation x t) ?w ∧
    ∀ (y : ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)), (fun t => IsAmalgamation x t) y → y = ?w
case w
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
⊢ ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)

Next tactic:
swap
/-- If `G` is cocontinuous, then `Ran G.op` pushes sheaves to sheaves.

This result is basically https://stacks.math.columbia.edu/tag/00XK,
but without the condition that `C` or `D` has pullbacks.
-/
theorem ran_isSheaf_of_isCocontinuous (ℱ : Sheaf J A) :
    Presheaf.IsSheaf K ((ran G.op).obj ℱ.val) := by
  intro X U S hS x hx
  constructor; swap
  · 

Proof state:
case w
C D : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v, u} D
G : C ⥤ D
A : Type w
inst✝² : Category.{max u v, w} A
inst✝¹ : HasLimits A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝ : Functor.IsCocontinuous G J K
ℱ : Sheaf J A
X : A
U : D
S : Sieve U
hS : S ∈ GrothendieckTopology.sieves K U
x : FamilyOfElements ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)) S.arrows
hx : Compatible x
⊢ ((ran G.op).obj ℱ.val ⋙ coyoneda.obj (op X)).obj (op U)

Next tactic:
apply RanIsSheafOfIsCocontinuous.gluedSection ℱ hS hx