mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Adhesive.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
D : Type u''
inst✝⁷ : Category.{v'', u''} D
inst✝⁶ : HasPullbacks D
inst✝⁵ : Adhesive C
inst✝⁴ : HasPullbacks C
inst✝³ : HasPushouts C
H₂ : ∀ {X Y S : D} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝² : Gr.Full
inst✝¹ : Gr.Faithful
inst✝ : PreservesLimitsOfShape WalkingCospan Gl
this✝ : PreservesColimitsOfSize.{?u.95, ?u.94, v, v'', u, u''} Gl
this : PreservesLimitsOfSize.{?u.102, ?u.101, v'', v, u'', u} Gr
W : D
⊢ ∀ {X Y Z : D} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} [Mono f] (H : IsPushout f g h i), H.IsVanKampen

Tactic:
X

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
D : Type u''
inst✝⁷ : Category.{v'', u''} D
inst✝⁶ : HasPullbacks D
inst✝⁵ : Adhesive C
inst✝⁴ : HasPullbacks C
inst✝³ : HasPushouts C
H₂ : ∀ {X Y S : D} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝² : Gr.Full
inst✝¹ : Gr.Faithful
inst✝ : PreservesLimitsOfShape WalkingCospan Gl
this✝ : PreservesColimitsOfSize.{?u.95, ?u.94, v, v'', u, u''} Gl
this : PreservesLimitsOfSize.{?u.102, ?u.101, v'', v, u'', u} Gr
W X : D
⊢ ∀ {Y Z : D} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} [Mono f] (H : IsPushout f g h i), H.IsVanKampen
Module: Mathlib.CategoryTheory.Adhesive.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
D : Type u''
inst✝⁷ : Category.{v'', u''} D
inst✝⁶ : HasPullbacks D
inst✝⁵ : Adhesive C
inst✝⁴ : HasPullbacks C
inst✝³ : HasPushouts C
H₂ : ∀ {X Y S : D} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝² : Gr.Full
inst✝¹ : Gr.Faithful
inst✝ : PreservesLimitsOfShape WalkingCospan Gl
this✝ : PreservesColimitsOfSize.{?u.95, ?u.94, v, v'', u, u''} Gl
this : PreservesLimitsOfSize.{?u.102, ?u.101, v'', v, u'', u} Gr
W X Y Z : D
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
⊢ ∀ [Mono f] (H : IsPushout f g h i), H.IsVanKampen

Tactic:
_

Goal state after:
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u''
inst✝⁸ : Category.{v'', u''} D
inst✝⁷ : HasPullbacks D
inst✝⁶ : Adhesive C
inst✝⁵ : HasPullbacks C
inst✝⁴ : HasPushouts C
H₂ : ∀ {X Y S : D} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
inst✝¹ : PreservesLimitsOfShape WalkingCospan Gl
this✝ : PreservesColimitsOfSize.{?u.95, ?u.94, v, v'', u, u''} Gl
this : PreservesLimitsOfSize.{?u.102, ?u.101, v'', v, u'', u} Gr
W X Y Z : D
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
inst✝ : Mono f
⊢ ∀ (H : IsPushout f g h i), H.IsVanKampen
Module: Mathlib.CategoryTheory.Sites.LocallyInjective

Goal state before:
pp: case mpr
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
F : Cᵒᵖ ⥤ D
X : Cᵒᵖ
x : ToType (F.obj X)
⊢ equalizerSieve x x = ⊤

Tactic:
apply equalizerSieve_self_eq_top

Goal state after:
Module: Mathlib.CategoryTheory.Sites.LocallyInjective

Goal state before:
pp: case mp
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
a✝ : IsLocallyInjective J φ
⊢ ∀ ⦃X : Cᵒᵖ⦄ (x y : ToType (F₁.obj X)),
    equalizerSieve ((ConcreteCategory.hom (φ.app X)) x) ((ConcreteCategory.hom (φ.app X)) y) ∈ J (unop X) →
      equalizerSieve x y ∈ J (unop X)

Tactic:
X

Goal state after:
case mp
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
a✝ : IsLocallyInjective J φ
X : Cᵒᵖ
⊢ ∀ (x y : ToType (F₁.obj X)),
    equalizerSieve ((ConcreteCategory.hom (φ.app X)) x) ((ConcreteCategory.hom (φ.app X)) y) ∈ J (unop X) →
      equalizerSieve x y ∈ J (unop X)
Module: Mathlib.CategoryTheory.Sites.LocallyInjective

Goal state before:
pp: case mp
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
hsep : Presieve.IsSeparated J (F₁ ⋙ forget D)
a✝ : IsLocallyInjective J φ
⊢ ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (φ.app X))

Tactic:
X

Goal state after:
case mp
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
hsep : Presieve.IsSeparated J (F₁ ⋙ forget D)
a✝ : IsLocallyInjective J φ
X : Cᵒᵖ
⊢ Function.Injective ⇑(ConcreteCategory.hom (φ.app X))
Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i ↦ f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
⊢ ∀ (j : Discrete (Fin n)),
    IsPullback
      ((Cofan.mk (pullback c₂.inr i) fun j ↦
              pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯).ι.app
        j)
      ((Discrete.natTrans fun i ↦ α.app { as := i.as.succ }).app j) (pullback.fst c₂.inr i) (c₁.ι.app j)

Tactic:
intro j
    simp only [pair_obj_right, Functor.const_obj_obj, Discrete.functor_obj,
      Cofan.mk_pt, Cofan.mk_ι_app, Discrete.natTrans_app]
    refine IsPullback.of_right ?_ ?_ (IsPullback.of_hasPullback (BinaryCofan.inr c₂) i).flip
    · simp only [Functor.const_obj_obj, pair_obj_right, limit.lift_π,
        PullbackCone.mk_pt, PullbackCone.mk_π_app]
      exact H _
    · simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app, Cofan.inj]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.LocallyBijective

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
h₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.hom.app X))
right✝ : IsLocallySurjective f
this : ∀ (X : Cᵒᵖ), Function.Surjective ⇑(ConcreteCategory.hom (f.hom.app X))
X : Cᵒᵖ
⊢ IsIso (((sheafToPresheaf J (Type w)).map f).app X)

Tactic:
rw [isIso_iff_bijective]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
f : F ⟶ G
h₁ : ∀ (X : Cᵒᵖ), Function.Injective ⇑(ConcreteCategory.hom (f.hom.app X))
right✝ : IsLocallySurjective f
this : ∀ (X : Cᵒᵖ), Function.Surjective ⇑(ConcreteCategory.hom (f.hom.app X))
X : Cᵒᵖ
⊢ Function.Bijective ⇑(ConcreteCategory.hom (((sheafToPresheaf J (Type w)).map f).app X))
Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ IsColimit (Cofan.mk (X i) fun k ↦ if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i))

Tactic:
· refine Cofan.IsColimit.mk _ (fun t ↦ (eqToHom (if_pos rfl).symm) ≫ t.inj i) ?_ ?_
    · intro t j
      simp only [Cofan.mk_pt, cofan_mk_inj]
      split
      · subst ‹j = i›; simp
      · rw [Category.assoc, ← IsIso.eq_inv_comp]
        exact initialIsInitial.hom_ext _ _
    · intro t m hm
      simp [← hm i]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.LocallySurjective

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
S : C
ι : Type u_2
inst✝ : Small.{max w v, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
c : Cofan fun i ↦ uliftYoneda.{w, v, u}.obj (X i)
hc : IsColimit c
e : (Discrete.functor fun i ↦ uliftYoneda.{w, v, u}.obj (X i)) ≅
  Discrete.functor fun i ↦ shrinkYoneda.{max w v, v, u}.obj (X i) :=
  Discrete.natIso fun i ↦ uliftYonedaIsoShrinkYoneda.app (X i.as)
hc' : IsColimit ((Cocone.precompose e.inv).obj c) := (IsColimit.precomposeInvEquiv e c).invFun hc
i : ι
⊢ uliftYonedaIsoShrinkYoneda.hom.app (X i) ≫ shrinkYoneda.{max w v, v, u}.map (f i) =
    c.inj i ≫ Cofan.IsColimit.desc hc' fun i ↦ shrinkYoneda.{max v w, v, u}.map (f i)

Tactic:
← Cofan.IsColimit.fac hc' (fun i ↦ shrinkYoneda.map (f i)) i

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
S : C
ι : Type u_2
inst✝ : Small.{max w v, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
c : Cofan fun i ↦ uliftYoneda.{w, v, u}.obj (X i)
hc : IsColimit c
e : (Discrete.functor fun i ↦ uliftYoneda.{w, v, u}.obj (X i)) ≅
  Discrete.functor fun i ↦ shrinkYoneda.{max w v, v, u}.obj (X i) :=
  Discrete.natIso fun i ↦ uliftYonedaIsoShrinkYoneda.app (X i.as)
hc' : IsColimit ((Cocone.precompose e.inv).obj c) := (IsColimit.precomposeInvEquiv e c).invFun hc
i : ι
⊢ (uliftYonedaIsoShrinkYoneda.hom.app (X i) ≫
      Cofan.inj ((Cocone.precompose e.inv).obj c) i ≫
        Cofan.IsColimit.desc hc' fun i ↦ shrinkYoneda.{max v w, v, u}.map (f i)) =
    c.inj i ≫ Cofan.IsColimit.desc hc' fun i ↦ shrinkYoneda.{max v w, v, u}.map (f i)
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
⊢ projectiveDimension X = ⊥ ↔ HasProjectiveDimensionLT X 0

Tactic:
← projectiveDimension_lt_iff,

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
⊢ projectiveDimension X = ⊥ ↔ projectiveDimension X < ↑0
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case left_id
J : Type v₁
inst✝¹ : SmallCategory J
C : Type u₁
inst✝ : Category.{v₁, u₁} C
F : J ⥤ C
c₁ c₂ : Cone F
⊢ (Bicone.casesOn Bicone.left c₁.pt c₂.pt fun j ↦ F.obj j) ⟶ Bicone.casesOn Bicone.left c₁.pt c₂.pt fun j ↦ F.obj j

Tactic:
exact 𝟙 _

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case right_id
J : Type v₁
inst✝¹ : SmallCategory J
C : Type u₁
inst✝ : Category.{v₁, u₁} C
F : J ⥤ C
c₁ c₂ : Cone F
⊢ (Bicone.casesOn Bicone.right c₁.pt c₂.pt fun j ↦ F.obj j) ⟶ Bicone.casesOn Bicone.right c₁.pt c₂.pt fun j ↦ F.obj j

Tactic:
exact 𝟙 _

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.obj.obj X)
m m' : ↑(A.obj.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
hS : S ∈ J (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ToType (R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.obj.map f.op)) r
m₀ : ↑(M₀.obj (Opposite.op Y))
hm₀ : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀ = (ConcreteCategory.hom (A.obj.map f.op)) m
m₀' : ↑(M₀.obj (Opposite.op Y))
hm₀' : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀' = (ConcreteCategory.hom (A.obj.map f.op)) m'
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀ + r₀ • m₀') =
    (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀) + (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀')

Tactic:
_root_.map_add

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r : ↑(R.obj.obj X)
m m' : ↑(A.obj.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
hS : S ∈ J (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ToType (R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.obj.map f.op)) r
m₀ : ↑(M₀.obj (Opposite.op Y))
hm₀ : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀ = (ConcreteCategory.hom (A.obj.map f.op)) m
m₀' : ↑(M₀.obj (Opposite.op Y))
hm₀' : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀' = (ConcreteCategory.hom (A.obj.map f.op)) m'
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀) + (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀') =
    (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀) + (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀')
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r r' : ↑(R.obj.obj X)
m : ↑(A.obj.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ↑(R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.obj.map f.op)) r
r₀' : ↑(R₀.obj (Opposite.op Y))
hr₀' : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀' = (ConcreteCategory.hom (R.obj.map f.op)) r'
m₀ : ToType (M₀.presheaf.obj (Opposite.op Y))
hm₀ : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀ = (ConcreteCategory.hom (A.obj.map f.op)) m
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀ + r₀' • m₀) =
    (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀) + (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀' • m₀)

Tactic:
_root_.map_add

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X : Cᵒᵖ
r r' : ↑(R.obj.obj X)
m : ↑(A.obj.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ↑(R₀.obj (Opposite.op Y))
hr₀ : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀ = (ConcreteCategory.hom (R.obj.map f.op)) r
r₀' : ↑(R₀.obj (Opposite.op Y))
hr₀' : (ConcreteCategory.hom (α.app (Opposite.op Y))) r₀' = (ConcreteCategory.hom (R.obj.map f.op)) r'
m₀ : ToType (M₀.presheaf.obj (Opposite.op Y))
hm₀ : (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀ = (ConcreteCategory.hom (A.obj.map f.op)) m
⊢ (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀) + (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀' • m₀) =
    (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀ • m₀) + (ConcreteCategory.hom (φ.app (Opposite.op Y))) (r₀' • m₀)
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
Y : C
f : Y ⟶ X
⊢ ⊤.arrows f → (J₁.close S).arrows f

Tactic:
_

Goal state after:
case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
Y : C
f : Y ⟶ X
a✝ : ⊤.arrows f
⊢ (J₁.close S).arrows f
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type ?u.26
⊢ ∀ {𝒥 : SmallCategory J} (x : FinCategory J), PreservesLimitsOfShape J F

Tactic:
_

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type ?u.26
𝒥✝ : SmallCategory J
⊢ ∀ (x : FinCategory J), PreservesLimitsOfShape J F
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type ?u.26
𝒥✝ : SmallCategory J
⊢ ∀ (x : FinCategory J), PreservesLimitsOfShape J F

Tactic:
_

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type ?u.26
𝒥✝ : SmallCategory J
x✝ : FinCategory J
⊢ PreservesLimitsOfShape J F
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
Y : C
f : Y ⟶ X
⊢ J.Covers ⊤ f → ⊤.arrows f

Tactic:
_

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
Y : C
f : Y ⟶ X
a✝ : J.Covers ⊤ f
⊢ ⊤.arrows f
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
J : Precoverage C
K : GrothendieckTopology C
hK : K ∈ {K | ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ K X}
⊢ J.toGrothendieck ≤ K

Tactic:
_

Goal state after:
case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
J : Precoverage C
K : GrothendieckTopology C
hK : K ∈ {K | ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ K X}
i✝ : C
⊢ J.toGrothendieck i✝ ⊆ K i✝
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
J : Precoverage C
K : GrothendieckTopology C
hK : K ∈ {K | ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ K X}
i✝ : C
⊢ J.toGrothendieck i✝ ⊆ K i✝

Tactic:
_

Goal state after:
case refine_1
C : Type u_2
inst✝ : Category.{u_1, u_2} C
J : Precoverage C
K : GrothendieckTopology C
hK : K ∈ {K | ∀ ⦃X : C⦄, ∀ S ∈ J.coverings X, Sieve.generate S ∈ K X}
i✝ : C
a✝ : Sieve i✝
⊢ a✝ ∈ J.toGrothendieck i✝ → a✝ ∈ K i✝
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
⊢ ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows

Tactic:
_

Goal state after:
case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ : C
⊢ ∀ {Y : C} {f : Y ⟶ X✝}, ∀ R ∈ J.coverings X✝, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ : C
⊢ ∀ {Y : C} {f : Y ⟶ X✝}, ∀ R ∈ J.coverings X✝, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows

Tactic:
_

Goal state after:
case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ Y✝ : C
⊢ ∀ {f : Y✝ ⟶ X✝}, ∀ R ∈ J.coverings X✝, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ Y✝ : C
⊢ ∀ {f : Y✝ ⟶ X✝}, ∀ R ∈ J.coverings X✝, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows

Tactic:
_

Goal state after:
case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ Y✝ : C
f✝ : Y✝ ⟶ X✝
⊢ ∀ R ∈ J.coverings X✝, Presieve.IsSheafFor P (Sieve.pullback f✝ (Sieve.generate R)).arrows
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ Y✝ : C
f✝ : Y✝ ⟶ X✝
⊢ ∀ R ∈ J.coverings X✝, Presieve.IsSheafFor P (Sieve.pullback f✝ (Sieve.generate R)).arrows

Tactic:
_

Goal state after:
case mp
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : Presieve.IsSheaf J.toGrothendieck P
X✝ Y✝ : C
f✝ : Y✝ ⟶ X✝
R✝ : Presieve X✝
⊢ R✝ ∈ J.coverings X✝ → Presieve.IsSheafFor P (Sieve.pullback f✝ (Sieve.generate R✝)).arrows
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: C D : Type u₁
inst✝¹⁰ : SmallCategory C
inst✝⁹ : SmallCategory D
E : Type u₂
inst✝⁸ : Category.{u₁, u₂} E
FE : E → E → Type u_1
CE : E → Type u₁
inst✝⁷ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
inst✝⁶ : ConcreteCategory E FE
inst✝⁵ : HasLimits E
inst✝⁴ : HasColimits E
inst✝³ : ReflectsLimits (forget E)
inst✝² : PreservesFilteredColimits (forget E)
inst✝¹ : PreservesLimits (forget E)
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type u₁
⊢ ∀ {𝒥 : SmallCategory J} (x : FinCategory J), PreservesLimitsOfShape J F.op.lan

Tactic:
_

Goal state after:
C D : Type u₁
inst✝¹⁰ : SmallCategory C
inst✝⁹ : SmallCategory D
E : Type u₂
inst✝⁸ : Category.{u₁, u₂} E
FE : E → E → Type u_1
CE : E → Type u₁
inst✝⁷ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
inst✝⁶ : ConcreteCategory E FE
inst✝⁵ : HasLimits E
inst✝⁴ : HasColimits E
inst✝³ : ReflectsLimits (forget E)
inst✝² : PreservesFilteredColimits (forget E)
inst✝¹ : PreservesLimits (forget E)
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type u₁
𝒥✝ : SmallCategory J
⊢ ∀ (x : FinCategory J), PreservesLimitsOfShape J F.op.lan
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: C D : Type u₁
inst✝¹⁰ : SmallCategory C
inst✝⁹ : SmallCategory D
E : Type u₂
inst✝⁸ : Category.{u₁, u₂} E
FE : E → E → Type u_1
CE : E → Type u₁
inst✝⁷ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
inst✝⁶ : ConcreteCategory E FE
inst✝⁵ : HasLimits E
inst✝⁴ : HasColimits E
inst✝³ : ReflectsLimits (forget E)
inst✝² : PreservesFilteredColimits (forget E)
inst✝¹ : PreservesLimits (forget E)
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type u₁
𝒥✝ : SmallCategory J
⊢ ∀ (x : FinCategory J), PreservesLimitsOfShape J F.op.lan

Tactic:
_

Goal state after:
C D : Type u₁
inst✝¹⁰ : SmallCategory C
inst✝⁹ : SmallCategory D
E : Type u₂
inst✝⁸ : Category.{u₁, u₂} E
FE : E → E → Type u_1
CE : E → Type u₁
inst✝⁷ : (X Y : E) → FunLike (FE X Y) (CE X) (CE Y)
inst✝⁶ : ConcreteCategory E FE
inst✝⁵ : HasLimits E
inst✝⁴ : HasColimits E
inst✝³ : ReflectsLimits (forget E)
inst✝² : PreservesFilteredColimits (forget E)
inst✝¹ : PreservesLimits (forget E)
F : C ⥤ D
inst✝ : RepresentablyFlat F
J : Type u₁
𝒥✝ : SmallCategory J
x✝ : FinCategory J
⊢ PreservesLimitsOfShape J F.op.lan
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case mpr
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
⊢ Presieve.IsSheaf J.toGrothendieck P

Tactic:
X

Goal state after:
case mpr
C : Type u_3
inst✝ : Category.{u_2, u_3} C
J : Precoverage C
P : Cᵒᵖ ⥤ Type u_1
H : ∀ {X Y : C} {f : Y ⟶ X}, ∀ R ∈ J.coverings X, Presieve.IsSheafFor P (Sieve.pullback f (Sieve.generate R)).arrows
X : C
⊢ ∀ S ∈ J.toGrothendieck X, Presieve.IsSheafFor P S.arrows