mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => Exists.choose hT ≫ e,
          naturality := ⋯ } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
⊢ f ≫ Hf.desc (aux e ⋯) = e

Tactic:
by
      have := Hf.fac (aux e h) ⟨Over.mk f, 𝟙 _, by simp⟩
      dsimp [aux] at this; rw [this]; clear this
      nth_rewrite 2 [← Category.id_comp e]
      apply h
      generalize_proofs hh
      rw [hh.choose_spec, Category.id_comp]

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => Exists.choose hT ≫ e,
          naturality := ⋯ } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
A : Over X
g : A.left ⟶ Y
hA : g ≫ f = A.hom
⊢ (Sieve.generateSingleton f).arrows.cocone.ι.app { obj := A, property := ⋯ } ≫ m =
    (aux e ⋯).ι.app { obj := A, property := ⋯ }

Tactic:
dsimp

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f : Y ⟶ X
Hf : IsColimit (Sieve.generateSingleton f).arrows.cocone
aux : {W : C} →
  (e : Y ⟶ W) →
    (∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) →
      Cocone (Sieve.generateSingleton f).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => Exists.choose hT ≫ e,
          naturality := ⋯ } }
W : C
e : Y ⟶ W
h : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e
m : X ⟶ W
hm : f ≫ m = e
A : Over X
g : A.left ⟶ Y
hA : g ≫ f = A.hom
⊢ A.hom ≫ m = (aux e ⋯).ι.app { obj := A, property := ⋯ }
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ (restriction data G₀ ((data X).f (Sieve.ofArrows.i hd) ≫ g) ≫
        IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ Sieve.ofArrows.h hd)) ≫
      G₀.obj.map { Y := V₀, f := a, hf := ⋯ }.f.op =
    (presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ IsDenseSubsite.mapPreimage J F G₀ (Sieve.ofArrows.h hc)) ≫
      G₀.obj.map { Y := V₀, f := a, hf := ⋯ }.f.op

Tactic:
dsimp

Goal state after:
C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ (restriction data G₀ ((data X).f (Sieve.ofArrows.i hd) ≫ g) ≫
        IsDenseSubsite.mapPreimage J F G₀ (F.map c ≫ Sieve.ofArrows.h hd)) ≫
      G₀.obj.map a.op =
    (presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ IsDenseSubsite.mapPreimage J F G₀ (Sieve.ofArrows.h hc)) ≫
      G₀.obj.map a.op
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamilyStruct X π
D : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom
F : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram
S : Cocone (Sieve.generateFamily X π).arrows.diagram
T : Over B
a : α
g : T.left ⟶ X a
hT : g ≫ π a = T.hom
⊢ (Sieve.generateFamily X π).arrows.cocone.ι.app { obj := T, property := ⋯ } ≫
      H.desc (fun a ↦ S.ι.app { obj := Over.mk (π a), property := ⋯ }) ⋯ =
    S.ι.app { obj := T, property := ⋯ }

Tactic:
dsimp

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamilyStruct X π
D : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom
F : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram
S : Cocone (Sieve.generateFamily X π).arrows.diagram
T : Over B
a : α
g : T.left ⟶ X a
hT : g ≫ π a = T.hom
⊢ T.hom ≫ H.desc (fun a ↦ S.ι.app { obj := Over.mk (π a), property := ⋯ }) ⋯ = S.ι.app { obj := T, property := ⋯ }
Module: Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact

Goal state before:
pp: P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ QuasiCompactCover
    (Precoverage.ZeroHypercover.pullback₂ (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i)
        (Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover)).toPreZeroHypercover

Tactic:
dsimp

Goal state after:
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ QuasiCompactCover
    (PreZeroHypercover.pullback₂ (V.affineOneHypercover.p₁ k ≫ V.affineCover.f i)
      (PreZeroHypercover.pullback₂ f 𝒱.cover.toPreZeroHypercover))
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamilyStruct X π
D : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom
F : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram
S : Cocone (Sieve.generateFamily X π).arrows.diagram
m : (Sieve.generateFamily X π).arrows.cocone.pt ⟶ S.pt
hm :
  ∀ (j : (Sieve.generateFamily X π).arrows.category), (Sieve.generateFamily X π).arrows.cocone.ι.app j ≫ m = S.ι.app j
⊢ m = H.desc (fun a ↦ S.ι.app { obj := Over.mk (π a), property := ⋯ }) ⋯

Tactic:
dsimp

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : EffectiveEpiFamilyStruct X π
D : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom
F : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram
S : Cocone (Sieve.generateFamily X π).arrows.diagram
m : (Sieve.generateFamily X π).arrows.cocone.pt ⟶ S.pt
hm :
  ∀ (j : (Sieve.generateFamily X π).arrows.category), (Sieve.generateFamily X π).arrows.cocone.ι.app j ≫ m = S.ι.app j
⊢ m = H.desc (fun a ↦ S.ι.app { obj := Over.mk (π a), property := ⋯ }) ⋯
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X : C
⊢ presheafMap data G₀ (𝟙 X) = 𝟙 (presheafObj data G₀ X)

Tactic:
by
  ext i
  rw [presheafMap_π, comp_id, id_comp,
    restriction_eq_of_fac data G₀ ((data X).f i) (𝟙 _) (by simp),
    IsDenseSubsite.mapPreimage_id, comp_id]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact

Goal state before:
pp: P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
left✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
right✝ : Finite 𝒰.I₀
𝒰' : Cover (precoverage P) (Spec R) := 𝒰.forgetQc.sigma
⊢ IsAffine (𝒰.forgetQc.sigma.X default)

Tactic:
dsimp

Goal state after:
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
left✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
right✝ : Finite 𝒰.I₀
𝒰' : Cover (precoverage P) (Spec R) := 𝒰.forgetQc.sigma
⊢ IsAffine (∐ 𝒰.X)
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
⊢ ((Sieve.generateFamily X π).arrows.diagram.map { hom := q } ≫
      match { obj := B, property := ⋯ } with
      | { obj := obj, property := hT } => ⋯.choose ≫ e (Exists.choose hT)) =
    (match { obj := A, property := ⋯ } with
      | { obj := obj, property := hT } => ⋯.choose ≫ e (Exists.choose hT)) ≫
      ((Functor.const (Sieve.generateFamily X π).arrows.category).obj W).map { hom := q }

Tactic:
dsimp

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
⊢ Over.Hom.left q ≫ ⋯.choose ≫ e ⋯.choose = (⋯.choose ≫ e ⋯.choose) ≫ 𝟙 W
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
aux : {W : C} →
  (e : (a : α) → X a ⟶ W) →
    (∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂) →
      Cocone (Sieve.generateFamily X π).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => ⋯.choose ≫ e (Exists.choose hT),
          naturality := ⋯ } }
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
m : B ⟶ W
hm : ∀ (a : α), π a ≫ m = e a
T : Over B
a : α
g : T.left ⟶ X a
ha : g ≫ π a = T.hom
⊢ (Sieve.generateFamily X π).arrows.cocone.ι.app { obj := T, property := ⋯ } ≫ m =
    (aux e ⋯).ι.app { obj := T, property := ⋯ }

Tactic:
dsimp

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
aux : {W : C} →
  (e : (a : α) → X a ⟶ W) →
    (∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂) →
      Cocone (Sieve.generateFamily X π).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => ⋯.choose ≫ e (Exists.choose hT),
          naturality := ⋯ } }
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
m : B ⟶ W
hm : ∀ (a : α), π a ≫ m = e a
T : Over B
a : α
g : T.left ⟶ X a
ha : g ≫ π a = T.hom
⊢ T.hom ≫ m = (aux e ⋯).ι.app { obj := T, property := ⋯ }
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X₀ W₀ : C₀
a : F.obj W₀ ⟶ F.obj X₀
i : (data (F.obj X₀)).I₀
p : F.obj W₀ ⟶ F.obj ((data (F.obj X₀)).X i)
fac : p ≫ (data (F.obj X₀)).f i = a
T₀ : C₀
b : T₀ ⟶ W₀
c : T₀ ⟶ X₀
hc : F.map c = F.map b ≫ a
⊢ (hom data G₀ X₀ ≫ IsDenseSubsite.mapPreimage J F G₀ a) ≫ G₀.obj.map { Y := T₀, f := b, hf := ⋯ }.f.op =
    (presheafObjπ data G₀ (F.obj X₀) i ≫ IsDenseSubsite.mapPreimage J F G₀ p) ≫
      G₀.obj.map { Y := T₀, f := b, hf := ⋯ }.f.op

Tactic:
dsimp

Goal state after:
C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X₀ W₀ : C₀
a : F.obj W₀ ⟶ F.obj X₀
i : (data (F.obj X₀)).I₀
p : F.obj W₀ ⟶ F.obj ((data (F.obj X₀)).X i)
fac : p ≫ (data (F.obj X₀)).f i = a
T₀ : C₀
b : T₀ ⟶ W₀
c : T₀ ⟶ X₀
hc : F.map c = F.map b ≫ a
⊢ (hom data G₀ X₀ ≫ IsDenseSubsite.mapPreimage J F G₀ a) ≫ G₀.obj.map b.op =
    (presheafObjπ data G₀ (F.obj X₀) i ≫ IsDenseSubsite.mapPreimage J F G₀ p) ≫ G₀.obj.map b.op
Module: Mathlib.AlgebraicGeometry.Sites.ConstantSheaf

Goal state before:
pp: S : Scheme
T : Type v
inst✝¹ : TopologicalSpace T
inst✝ : TotallyDisconnectedSpace T
U : Scheme
f : C(ConnectedComponents ↥U, T)
⊢ ⇑((fun f ↦ { toFun := ⋯.connectedComponentsLift, continuous_toFun := ⋯ })
        ((fun f ↦ f.comp { toFun := ConnectedComponents.mk, continuous_toFun := ⋯ }) f)) =
    ⇑f

Tactic:
dsimp

Goal state after:
S : Scheme
T : Type v
inst✝¹ : TopologicalSpace T
inst✝ : TotallyDisconnectedSpace T
U : Scheme
f : C(ConnectedComponents ↥U, T)
⊢ ⋯.connectedComponentsLift = ⇑f
Module: Mathlib.CategoryTheory.Sites.SubcanonicalOver

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X : C
E Z✝ : Over X
ι : Type (max u_1 v_1)
T : ι → Over X
g : (i : ι) → T i ⟶ Z✝
hR : Sieve.ofArrows T g ∈ (J.over X) Z✝
t : Presieve.FamilyOfElements (CategoryTheory.yoneda.obj E) (Sieve.ofArrows T g).arrows
ht : t.Compatible
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : gi ≫ (fun i ↦ Over.Hom.left (g i)) i = gj ≫ (fun i ↦ Over.Hom.left (g i)) j
⊢ gj ≫ (T j).hom = (Over.mk (gi ≫ (T i).hom)).hom

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X : C
E Z✝ : Over X
ι : Type (max u_1 v_1)
T : ι → Over X
g : (i : ι) → T i ⟶ Z✝
hR : Sieve.ofArrows T g ∈ (J.over X) Z✝
t : Presieve.FamilyOfElements (CategoryTheory.yoneda.obj E) (Sieve.ofArrows T g).arrows
ht : t.Compatible
i j : ι
Z : C
gi : Z ⟶ (T i).left
gj : Z ⟶ (T j).left
hgij : gi ≫ (fun i ↦ Over.Hom.left (g i)) i = gj ≫ (fun i ↦ Over.Hom.left (g i)) j
⊢ gj ≫ (T j).hom = gi ≫ (T i).hom
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : colim.PreservesMonomorphisms
X₁ X₂ : J ⥤ C
φ : X₁ ⟶ X₂
inst✝ : Mono φ
c₁ : Cocone X₁
hc₁ : IsColimit c₁
c₂ : Cocone X₂
hc₂ : IsColimit c₂
f : c₁.pt ⟶ c₂.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = φ.app j ≫ c₂.ι.app j
j : J
⊢ c₁.ι.app j ≫ (hc₁.coconePointUniqueUpToIso (colimit.isColimit X₁)).hom ≫ (Arrow.mk (colim.map φ)).hom =
    c₁.ι.app j ≫ (Arrow.mk f).hom ≫ (hc₂.coconePointUniqueUpToIso (colimit.isColimit X₂)).hom

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : colim.PreservesMonomorphisms
X₁ X₂ : J ⥤ C
φ : X₁ ⟶ X₂
inst✝ : Mono φ
c₁ : Cocone X₁
hc₁ : IsColimit c₁
c₂ : Cocone X₂
hc₂ : IsColimit c₂
f : c₁.pt ⟶ c₂.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = φ.app j ≫ c₂.ι.app j
j : J
⊢ c₁.ι.app j ≫ (hc₁.coconePointUniqueUpToIso (colimit.isColimit X₁)).hom ≫ colimMap φ =
    c₁.ι.app j ≫ f ≫ (hc₂.coconePointUniqueUpToIso (colimit.isColimit X₂)).hom
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
J : Type u'
inst✝⁴ : Category.{v', u'} J
X : J ⥤ C
inst✝³ : ∀ (j j' : J) (φ : j ⟶ j'), Mono (X.map φ)
c : Cocone X
hc : IsColimit c
inst✝² : IsFiltered J
j₀ : J
inst✝¹ : HasColimitsOfShape (Under j₀) C
inst✝ : colim.PreservesMonomorphisms
x✝¹ x✝ : Under j₀
g : x✝¹ ⟶ x✝
⊢ ((Functor.const (Under j₀)).obj (X.obj j₀)).map g ≫ X.map x✝.hom = X.map x✝¹.hom ≫ (Under.forget j₀ ⋙ X).map g

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
J : Type u'
inst✝⁴ : Category.{v', u'} J
X : J ⥤ C
inst✝³ : ∀ (j j' : J) (φ : j ⟶ j'), Mono (X.map φ)
c : Cocone X
hc : IsColimit c
inst✝² : IsFiltered J
j₀ : J
inst✝¹ : HasColimitsOfShape (Under j₀) C
inst✝ : colim.PreservesMonomorphisms
x✝¹ x✝ : Under j₀
g : x✝¹ ⟶ x✝
⊢ 𝟙 (X.obj j₀) ≫ X.map x✝.hom = X.map x✝¹.hom ≫ X.map (Under.Hom.right g)
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim

Goal state before:
pp: case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : HasExactColimitsOfShape J C
inst✝ : HasZeroMorphisms C
S : ShortComplex (J ⥤ C)
hS : S.Exact
c₁ : Cocone S.X₁
hc₁ : IsColimit c₁
c₂ : Cocone S.X₂
hc₂ : IsColimit c₂
c₃ : Cocone S.X₃
hc₃ : IsColimit c₃
f : c₁.pt ⟶ c₂.pt
g : c₂.pt ⟶ c₃.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = S.f.app j ≫ c₂.ι.app j
hg : ∀ (j : J), c₂.ι.app j ≫ g = S.g.app j ≫ c₃.ι.app j
j : J
⊢ c₁.ι.app j ≫ (hc₁.coconePointUniqueUpToIso (colimit.isColimit S.X₁)).hom ≫ (S.map colim).f =
    c₁.ι.app j ≫ (mapShortComplex S hc₁ c₂ c₃ f g hf hg).f ≫ (hc₂.coconePointUniqueUpToIso (colimit.isColimit S.X₂)).hom

Tactic:
dsimp

Goal state after:
case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : HasExactColimitsOfShape J C
inst✝ : HasZeroMorphisms C
S : ShortComplex (J ⥤ C)
hS : S.Exact
c₁ : Cocone S.X₁
hc₁ : IsColimit c₁
c₂ : Cocone S.X₂
hc₂ : IsColimit c₂
c₃ : Cocone S.X₃
hc₃ : IsColimit c₃
f : c₁.pt ⟶ c₂.pt
g : c₂.pt ⟶ c₃.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = S.f.app j ≫ c₂.ι.app j
hg : ∀ (j : J), c₂.ι.app j ≫ g = S.g.app j ≫ c₃.ι.app j
j : J
⊢ c₁.ι.app j ≫ (hc₁.coconePointUniqueUpToIso (colimit.isColimit S.X₁)).hom ≫ colimMap S.f =
    c₁.ι.app j ≫ f ≫ (hc₂.coconePointUniqueUpToIso (colimit.isColimit S.X₂)).hom