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

Goal state before:
pp: C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹¹ : Category.{t, w} D
inst✝¹⁰ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁸ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁷ : ConcreteCategory D FD
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁵ : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
⊢ IsLimit ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F))

Tactic:
refine ⟨fun S => liftToPlusObjLimitObj F X.unop S, ?_, ?_⟩

Goal state after:
case refine_1
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹¹ : Category.{t, w} D
inst✝¹⁰ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁸ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁷ : ConcreteCategory D FD
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁵ : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
⊢ ∀ (s : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X)) (j : K),
    liftToPlusObjLimitObj F (unop X) s ≫ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j =
      s.π.app j

case refine_2
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹¹ : Category.{t, w} D
inst✝¹⁰ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁸ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁷ : ConcreteCategory D FD
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁵ : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
⊢ ∀ (s : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X))
    (m : s.pt ⟶ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt),
    (∀ (j : K), m ≫ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = s.π.app j) →
      m = liftToPlusObjLimitObj F (unop X) s
Module: Mathlib.CategoryTheory.ObjectProperty.Local

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W W' : MorphismProperty C
Z Z' : C
e : Z ≅ Z'
hZ : W.isLocal Z
X Y : C
f : X ⟶ Y
hf : W f
⊢ Function.Bijective fun g ↦ f ≫ g

Tactic:
rw [← Function.Bijective.of_comp_iff _ (Iso.homToEquiv e).bijective]
    convert! (Iso.homToEquiv e).bijective.comp (hZ f hf) using 1
    aesop

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
H : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))
A : C
f : A ⟶ ⊥_ C
this : IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A))
l : (BinaryCofan.mk (𝟙 A) (𝟙 A)).pt ⟶ A
h₁ : (BinaryCofan.mk (𝟙 A) (𝟙 A)).inl ≫ l = f ≫ initial.to A
h₂ : (BinaryCofan.mk (𝟙 A) (𝟙 A)).inr ≫ l = 𝟙 A
⊢ IsIso f

Tactic:
rcases (Category.id_comp _).symm.trans h₂ with rfl

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
H : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))
A : C
f : A ⟶ ⊥_ C
this : IsColimit (BinaryCofan.mk (𝟙 A) (𝟙 A))
h₁ : (BinaryCofan.mk (𝟙 A) (𝟙 A)).inl ≫ 𝟙 A = f ≫ initial.to A
h₂ : (BinaryCofan.mk (𝟙 A) (𝟙 A)).inr ≫ 𝟙 A = 𝟙 A
⊢ IsIso f
Module: Mathlib.CategoryTheory.Limits.VanKampen

Goal state before:
pp: case mp.mp
C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
H✝ : IsVanKampenColimit c
X' Y' : C
c' : BinaryCofan X' Y'
αX : X' ⟶ X
αY : Y' ⟶ Y
f : c'.pt ⟶ c.pt
hαX : αX ≫ c.inl = c'.inl ≫ f
hαY : αY ≫ c.inr = c'.inr ≫ f
H : ∀ (j : Discrete WalkingPair), IsPullback (c'.ι.app j) ((mapPair αX αY).app j) f (c.ι.app j)
⊢ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr

Tactic:
exact ⟨H _, H _⟩

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

Goal state before:
pp: case mpr.mpr
C : Type u
inst✝ : Category.{v, u} C
X Y : C
c : BinaryCofan X Y
H✝ :
  ∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt),
    αX ≫ c.inl = c'.inl ≫ f →
      αY ≫ c.inr = c'.inr ≫ f → (Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c.inl ∧ IsPullback c'.inr αY f c.inr)
X' Y' : C
α : pair X' Y' ⟶ pair X Y
h : NatTrans.Equifibered α
c' : BinaryCofan X' Y'
f : c'.pt ⟶ c.pt
hα : α ≫ c.ι = c'.ι ≫ (const (Discrete WalkingPair)).map f
H : ∀ (j : Discrete WalkingPair), IsPullback (c'.ι.app j) (α.app j) f (c.ι.app j)
⊢ IsPullback c'.inl (α.app { as := WalkingPair.left })
      (((const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) c.inl ∧
    IsPullback c'.inr (α.app { as := WalkingPair.right })
      (((const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) c.inr

Tactic:
exact ⟨H _, H _⟩

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : GrothendieckTopology C
A : Type u_2
inst✝¹ : Category.{v_2, u_2} A
inst✝ : HasWeakSheafify J A
⊢ (presheafToSheaf J A).IsLocalization J.W

Tactic:
by
  rw [W_eq_inverseImage_isomorphisms]
  exact (sheafificationAdjunction J A).isLocalization

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

Goal state before:
pp: case mp
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F G : Sheaf J A
f : F ⟶ G
inst✝¹ : (forget A).ReflectsIsomorphisms
inst✝ : J.HasSheafCompose (forget A)
left✝ : IsLocallyInjective f
right✝ : IsLocallySurjective f
⊢ IsIso f

Tactic:
rw [← isIso_iff_of_reflects_iso f (sheafCompose J (forget A)),
      ← isLocallyBijective_iff_isIso']

Goal state after:
case mp
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F G : Sheaf J A
f : F ⟶ G
inst✝¹ : (forget A).ReflectsIsomorphisms
inst✝ : J.HasSheafCompose (forget A)
left✝ : IsLocallyInjective f
right✝ : IsLocallySurjective f
⊢ IsLocallyInjective ((sheafCompose J (forget A)).map f) ∧ IsLocallySurjective ((sheafCompose J (forget A)).map f)
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R : Sheaf J RingCat
⊢ ∀ (P Q R_1 : SheafOfModules R) (f : P ⟶ Q) (g g' : Q ⟶ R_1), f ≫ (g + g') = f ≫ g + f ≫ g'

Tactic:
tauto

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
m : ↑(A.obj.obj X)
Y : C
f : Y ⟶ Opposite.unop X
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))) (1 • m₀) = (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀

Tactic:
one_smul

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
m : ↑(A.obj.obj X)
Y : C
f : Y ⟶ Opposite.unop X
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))) m₀ = (ConcreteCategory.hom (φ.app (Opposite.op Y))) m₀
Module: Mathlib.CategoryTheory.Limits.Preserves.SigmaConst

Goal state before:
pp: case pos
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
R : C
α : Type u_1
β : Type u_2
f : α → β
inst✝² : HasCoproduct fun x ↦ R
inst✝¹ : HasCoproduct fun x ↦ R
inst✝ : HasCoproduct fun x ↦ R
s : Cofork (Sigma.map' f fun x ↦ 𝟙 R) 0
b : β
hb : b ∈ Set.range f
⊢ (Sigma.ι (fun x ↦ R) b ≫
      Cofork.π (sigmaConstCokernelCofork R f) ≫
        Sigma.desc fun x ↦
          match x with
          | ⟨b, property⟩ => Sigma.ι (fun x ↦ R) b ≫ s.π) =
    Sigma.ι (fun x ↦ R) b ≫ s.π

pp: case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
R : C
α : Type u_1
β : Type u_2
f : α → β
inst✝² : HasCoproduct fun x ↦ R
inst✝¹ : HasCoproduct fun x ↦ R
inst✝ : HasCoproduct fun x ↦ R
s : Cofork (Sigma.map' f fun x ↦ 𝟙 R) 0
b : β
hb : b ∉ Set.range f
⊢ (Sigma.ι (fun x ↦ R) b ≫
      Cofork.π (sigmaConstCokernelCofork R f) ≫
        Sigma.desc fun x ↦
          match x with
          | ⟨b, property⟩ => Sigma.ι (fun x ↦ R) b ≫ s.π) =
    Sigma.ι (fun x ↦ R) b ≫ s.π

Tactic:
· obtain ⟨a, rfl⟩ := hb
        simpa [-CokernelCofork.condition] using Sigma.ι (fun _ ↦ R) a ≫= s.condition.symm

Goal state after:
case neg
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
R : C
α : Type u_1
β : Type u_2
f : α → β
inst✝² : HasCoproduct fun x ↦ R
inst✝¹ : HasCoproduct fun x ↦ R
inst✝ : HasCoproduct fun x ↦ R
s : Cofork (Sigma.map' f fun x ↦ 𝟙 R) 0
b : β
hb : b ∉ Set.range f
⊢ (Sigma.ι (fun x ↦ R) b ≫
      Cofork.π (sigmaConstCokernelCofork R f) ≫
        Sigma.desc fun x ↦
          match x with
          | ⟨b, property⟩ => Sigma.ι (fun x ↦ R) b ≫ s.π) =
    Sigma.ι (fun x ↦ R) b ≫ s.π
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case left_id.left_id
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
h : Bicone.left ⟶ Z✝
⊢ (BiconeHom.left_id ≫ BiconeHom.left_id) ≫ h = BiconeHom.left_id ≫ BiconeHom.left_id ≫ h

Tactic:
cases h

Goal state after:
case left_id.left_id.left_id
J : Type u₁
inst✝ : Category.{v₁, u₁} J
⊢ (BiconeHom.left_id ≫ BiconeHom.left_id) ≫ BiconeHom.left_id =
    BiconeHom.left_id ≫ BiconeHom.left_id ≫ BiconeHom.left_id

case left_id.left_id.left
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ : J
⊢ (BiconeHom.left_id ≫ BiconeHom.left_id) ≫ BiconeHom.left j✝ =
    BiconeHom.left_id ≫ BiconeHom.left_id ≫ BiconeHom.left j✝
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case left_id.left
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
j✝ : J
h : Bicone.diagram j✝ ⟶ Z✝
⊢ (BiconeHom.left_id ≫ BiconeHom.left j✝) ≫ h = BiconeHom.left_id ≫ BiconeHom.left j✝ ≫ h

Tactic:
cases h

Goal state after:
case left_id.left.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ k✝ : J
f✝ : j✝ ⟶ k✝
⊢ (BiconeHom.left_id ≫ BiconeHom.left j✝) ≫ BiconeHom.diagram f✝ =
    BiconeHom.left_id ≫ BiconeHom.left j✝ ≫ BiconeHom.diagram f✝
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case right_id.right_id
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
h : Bicone.right ⟶ Z✝
⊢ (BiconeHom.right_id ≫ BiconeHom.right_id) ≫ h = BiconeHom.right_id ≫ BiconeHom.right_id ≫ h

Tactic:
cases h

Goal state after:
case right_id.right_id.right_id
J : Type u₁
inst✝ : Category.{v₁, u₁} J
⊢ (BiconeHom.right_id ≫ BiconeHom.right_id) ≫ BiconeHom.right_id =
    BiconeHom.right_id ≫ BiconeHom.right_id ≫ BiconeHom.right_id

case right_id.right_id.right
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ : J
⊢ (BiconeHom.right_id ≫ BiconeHom.right_id) ≫ BiconeHom.right j✝ =
    BiconeHom.right_id ≫ BiconeHom.right_id ≫ BiconeHom.right j✝
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case right_id.right
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
j✝ : J
h : Bicone.diagram j✝ ⟶ Z✝
⊢ (BiconeHom.right_id ≫ BiconeHom.right j✝) ≫ h = BiconeHom.right_id ≫ BiconeHom.right j✝ ≫ h

Tactic:
cases h

Goal state after:
case right_id.right.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ k✝ : J
f✝ : j✝ ⟶ k✝
⊢ (BiconeHom.right_id ≫ BiconeHom.right j✝) ≫ BiconeHom.diagram f✝ =
    BiconeHom.right_id ≫ BiconeHom.right j✝ ≫ BiconeHom.diagram f✝
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case left.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
j✝ k✝ : J
f✝ : j✝ ⟶ k✝
h : Bicone.diagram k✝ ⟶ Z✝
⊢ (BiconeHom.left j✝ ≫ BiconeHom.diagram f✝) ≫ h = BiconeHom.left j✝ ≫ BiconeHom.diagram f✝ ≫ h

Tactic:
cases h

Goal state after:
case left.diagram.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ k✝¹ : J
f✝¹ : j✝ ⟶ k✝¹
k✝ : J
f✝ : k✝¹ ⟶ k✝
⊢ (BiconeHom.left j✝ ≫ BiconeHom.diagram f✝¹) ≫ BiconeHom.diagram f✝ =
    BiconeHom.left j✝ ≫ BiconeHom.diagram f✝¹ ≫ BiconeHom.diagram f✝
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case right.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
j✝ k✝ : J
f✝ : j✝ ⟶ k✝
h : Bicone.diagram k✝ ⟶ Z✝
⊢ (BiconeHom.right j✝ ≫ BiconeHom.diagram f✝) ≫ h = BiconeHom.right j✝ ≫ BiconeHom.diagram f✝ ≫ h

Tactic:
cases h

Goal state after:
case right.diagram.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ k✝¹ : J
f✝¹ : j✝ ⟶ k✝¹
k✝ : J
f✝ : k✝¹ ⟶ k✝
⊢ (BiconeHom.right j✝ ≫ BiconeHom.diagram f✝¹) ≫ BiconeHom.diagram f✝ =
    BiconeHom.right j✝ ≫ BiconeHom.diagram f✝¹ ≫ BiconeHom.diagram f✝
Module: Mathlib.CategoryTheory.Limits.Bicones

Goal state before:
pp: case diagram.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
Z✝ : Bicone J
j✝ k✝¹ : J
f✝¹ : j✝ ⟶ k✝¹
k✝ : J
f✝ : k✝¹ ⟶ k✝
h : Bicone.diagram k✝ ⟶ Z✝
⊢ (BiconeHom.diagram f✝¹ ≫ BiconeHom.diagram f✝) ≫ h = BiconeHom.diagram f✝¹ ≫ BiconeHom.diagram f✝ ≫ h

Tactic:
cases h

Goal state after:
case diagram.diagram.diagram
J : Type u₁
inst✝ : Category.{v₁, u₁} J
j✝ k✝² : J
f✝² : j✝ ⟶ k✝²
k✝¹ : J
f✝¹ : k✝² ⟶ k✝¹
k✝ : J
f✝ : k✝¹ ⟶ k✝
⊢ (BiconeHom.diagram f✝² ≫ BiconeHom.diagram f✝¹) ≫ BiconeHom.diagram f✝ =
    BiconeHom.diagram f✝² ≫ BiconeHom.diagram f✝¹ ≫ BiconeHom.diagram f✝
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case refine_1.transitive
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✝² : Sieve i✝
X✝ : C
S✝ R✝ : Sieve X✝
a✝¹ : J.Saturate X✝ S✝
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X✝⦄, S✝.arrows f → J.Saturate Y (Sieve.pullback f R✝)
ih1 : S✝ ∈ K X✝
ih2 : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X✝⦄, S✝.arrows f → Sieve.pullback f R✝ ∈ K Y
⊢ R✝ ∈ K X✝

Tactic:
exact K.transitive ih1 _ ih2

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

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
H : (Cᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)) ⥤ Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
adj : H ⊣ (whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max u₁ v₁ u₂ v₂))).obj F.op
inst✝ : F.IsContinuous J K
G G' : Cᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
f : G ⟶ G'
hf : J.W f
U : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
hU : Presheaf.IsSheaf K U
⊢ Function.Bijective fun g ↦ f ≫ g

Tactic:
apply hf

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
H : (Cᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)) ⥤ Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
adj : H ⊣ (whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max u₁ v₁ u₂ v₂))).obj F.op
inst✝ : F.IsContinuous J K
G G' : Cᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
f : G ⟶ G'
hf : J.W f
U : Dᵒᵖ ⥤ Type (max u₁ v₁ u₂ v₂)
hU : Presheaf.IsSheaf K U
⊢ Presheaf.IsSheaf J (((whiskeringLeft Cᵒᵖ Dᵒᵖ (Type (max u₁ v₁ u₂ v₂))).obj F.op).obj U)
Module: Mathlib.CategoryTheory.Sites.Continuous

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
A : Type u
inst✝¹ : Category.{t, u} A
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
H : (Cᵒᵖ ⥤ A) ⥤ Dᵒᵖ ⥤ A
adj : H ⊣ (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj F.op
inst✝ : F.IsContinuous J K
G G' : Cᵒᵖ ⥤ A
f : G ⟶ G'
hf : J.W f
⊢ K.W (H.map f)

Tactic:
intro U hU
  rw [adj.map_comp_bijective_iff]
  exact hf _ (F.op_comp_isSheaf_of_isSheaf _ _ _ hU)

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : C ⥤ D
hG : CompatiblePreserving K G
ℱ : Sheaf K (Type w)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.obj) T
h : x.Compatible
Y : C
f : Y ⟶ Z
hf : T f
X : C
g : X ⟶ Z
f' : G.obj Y ⟶ G.obj X
hg : T g
eq : G.map f = f' ≫ G.map g
⊢ (FunctorPushforwardStructure.casesOn { preobj := X, premap := g, lift := f', cover := hg, fac := eq }
      fun Z_1 g h h₁ fac ↦ (ConcreteCategory.hom (ℱ.obj.map h.op)) (x g h₁)) =
    x f hf

Tactic:
simpa using hG.compatible ℱ h f' (𝟙 _) hg hf (by simp [eq])

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
G : C ⥤ D
K : GrothendieckTopology D
h✝¹ : ∀ (B : D), ∃ X f, Sieve.generate (Presieve.singleton f) ∈ K B
B : D
X : C
f✝ : G.obj X ⟶ B
h✝ : Sieve.generate (Presieve.singleton f✝) ∈ K B
Y : D
f : Y ⟶ B
Z : D
g : Y ⟶ Z
w✝ : Z ⟶ B
h : Presieve.singleton f✝ w✝
w : g ≫ w✝ = f
⊢ (Sieve.coverByImage G B).arrows f

Tactic:
cases h

Goal state after:
case mk
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝ : Category.{v_2, u_2} D
G : C ⥤ D
K : GrothendieckTopology D
h✝ : ∀ (B : D), ∃ X f, Sieve.generate (Presieve.singleton f) ∈ K B
B : D
X : C
f✝ : G.obj X ⟶ B
h : Sieve.generate (Presieve.singleton f✝) ∈ K B
Y : D
f : Y ⟶ B
g : Y ⟶ G.obj X
w : g ≫ f✝ = f
⊢ (Sieve.coverByImage G B).arrows f
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology

Goal state before:
pp: case of
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
G : C ⥤ D
K : GrothendieckTopology D
inst✝² : G.LocallyCoverDense K
inst✝¹ : G.IsLocallyFull K
inst✝ : G.IsLocallyFaithful K
U✝ : C
S✝ : Sieve U✝
X : C
S : Presieve X
hS : S ∈ (Precoverage.comap G K.toPrecoverage).coverings X
⊢ Sieve.functorPushforward G (Sieve.generate S) ∈ K (G.obj X)

Tactic:
rwa [← Sieve.generate_map_eq_functorPushforward]

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

Goal state before:
pp: case mp
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
e : C ≌ D
inst✝¹ : e.functor.IsCocontinuous J K
inst✝ : e.inverse.IsCocontinuous K J
X : C
S : Sieve X
H : Sieve.functorPushforward e.functor S ∈ K (e.functor.obj X)
⊢ S ∈ J X

Tactic:
refine J.superset_covering ?_ (e.functor.cover_lift J K H)

Goal state after:
case mp
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
K : GrothendieckTopology D
e : C ≌ D
inst✝¹ : e.functor.IsCocontinuous J K
inst✝ : e.inverse.IsCocontinuous K J
X : C
S : Sieve X
H : Sieve.functorPushforward e.functor S ∈ K (e.functor.obj X)
⊢ Sieve.functorPullback e.functor (Sieve.functorPushforward e.functor S) ≤ S
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E F : PreOneHypercover S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ E.Y ((congrIndexOneOfEq ⋯ ⋯) k) = E.Y (e.inv.s₁ (e.hom.s₁ k))

Tactic:
by congr 1; simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E F : PreOneHypercover S
e : E ≅ F
i j : F.I₀
k : F.I₁ i j
⊢ F.Y ((congrIndexOneOfEq ⋯ ⋯) k) = F.Y (e.hom.s₁ (e.inv.s₁ k))

Tactic:
by congr 1; simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{v_1, u_1} A
S✝ : C
E✝¹ E✝ : PreOneHypercover S✝
F✝ : PreOneHypercover S✝
G : PreOneHypercover S✝
S : C
E F : PreOneHypercover S
e : E ≅ F
i j : E.I₀
k : E.I₁ i j
⊢ E.Y (e.inv.s₁ (e.hom.s₁ k)) = E.Y ((congrIndexOneOfEq ?m.87 ?m.88) k)

Tactic:
by congr 1; simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.CoversTop.Basic

Goal state before:
pp: case mp
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
hX : IsTerminal X
I : Type u_1
Y : I → C
⊢ J.CoversTop Y → Sieve.ofObjects Y X ∈ J X

Tactic:
tauto

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Final.Type

Goal state before:
pp: case refine_1
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
P : D ⥤ Type w
inst✝ : F.Final
φ : (Y : D) → P.obj Y → (F ⋙ P).ColimitType
hφ : ∀ (Y : D) (j : StructuredArrow Y F), (F ⋙ P).ιColimitType j.right ∘ ⇑(ConcreteCategory.hom (P.map j.hom)) = φ Y
⊢ Function.Injective (F.colimitTypePrecomp P)

Tactic:
let c : P.CoconeTypes :=
      { pt := (F ⋙ P).ColimitType
        ι Y := φ Y
        ι_naturality {Y₁ Y₂} f := by
          ext
          have X : StructuredArrow Y₂ F := Classical.arbitrary _
          rw [← hφ Y₂ X, ← hφ Y₁ ((StructuredArrow.map f).obj X)]
          simp }

Goal state after:
case refine_1
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
P : D ⥤ Type w
inst✝ : F.Final
φ : (Y : D) → P.obj Y → (F ⋙ P).ColimitType
hφ : ∀ (Y : D) (j : StructuredArrow Y F), (F ⋙ P).ιColimitType j.right ∘ ⇑(ConcreteCategory.hom (P.map j.hom)) = φ Y
c : P.CoconeTypes := { pt := (F ⋙ P).ColimitType, ι := fun Y ↦ φ Y, ι_naturality := ⋯ }
⊢ Function.Injective (F.colimitTypePrecomp P)
Module: Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
D' : Type u₃
inst✝⁵ : Category.{v₃, u₃} D'
D'' : Type u₄
inst✝⁴ : Category.{v₄, u₄} D''
J : GrothendieckTopology C
K : GrothendieckTopology D
F : C ⥤ D
S : Sheaf J RingCat
R : Sheaf K RingCat
inst✝³ : F.IsContinuous J K
φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R
inst✝² : (PresheafOfModules.pushforward φ.hom).IsRightAdjoint
inst✝¹ : HasWeakSheafify K AddCommGrpCat
inst✝ : K.WEqualsLocallyBijective AddCommGrpCat
⊢ ∀ {X : SheafOfModules S} {Y Y' : SheafOfModules R}
    (f : (forget S ⋙ PresheafOfModules.pullback φ.hom ⋙ PresheafOfModules.sheafification (𝟙 R.obj)).obj X ⟶ Y)
    (g : Y ⟶ Y'),
    (((PresheafOfModules.sheafificationAdjunction (𝟙 R.obj)).homEquiv
              ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) Y').trans
          (((PresheafOfModules.pullbackPushforwardAdjunction φ.hom).homEquiv X.val Y'.val).trans
            (fullyFaithfulForget S).homEquiv.symm))
        (f ≫ g) =
      (((PresheafOfModules.sheafificationAdjunction (𝟙 R.obj)).homEquiv
                ((PresheafOfModules.pullback φ.hom).obj ((forget S).obj X)) Y).trans
            (((PresheafOfModules.pullbackPushforwardAdjunction φ.hom).homEquiv X.val Y.val).trans
              (fullyFaithfulForget S).homEquiv.symm))
          f ≫
        (pushforward φ).map g

Tactic:
tauto

Goal state after: