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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{t, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type s
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
W : (J.Cover X)ᵒᵖ
F : K ⥤ Cᵒᵖ ⥤ D
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj W)
⊢ ∀ (b : (unop W).shape.R),
    liftToDiagramLimitObjAux F E ((unop W).shape.fst b) ≫ ((unop W).index (limit F)).fst b =
      liftToDiagramLimitObjAux F E ((unop W).shape.snd b) ≫ ((unop W).index (limit F)).snd b

Tactic:
intro i

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{t, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
K : Type s
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
W : (J.Cover X)ᵒᵖ
F : K ⥤ Cᵒᵖ ⥤ D
E : Cone (F ⋙ J.diagramFunctor D X ⋙ (evaluation (J.Cover X)ᵒᵖ D).obj W)
i : (unop W).shape.R
⊢ liftToDiagramLimitObjAux F E ((unop W).shape.fst i) ≫ ((unop W).index (limit F)).fst i =
    liftToDiagramLimitObjAux F E ((unop W).shape.snd i) ≫ ((unop W).index (limit F)).snd i
Module: Mathlib.CategoryTheory.Localization.Bousfield

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : G ⊣ F
inst✝¹ : G.Full
inst✝ : G.Faithful
P₁ P₂ : C
f : P₁ ⟶ P₂
⊢ isColocal (fun x ↦ x ∈ Set.range G.obj) f ↔ (MorphismProperty.isomorphisms D).inverseImage F f

Tactic:
isColocal_iff_isIso_map adj

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : G ⊣ F
inst✝¹ : G.Full
inst✝ : G.Faithful
P₁ P₂ : C
f : P₁ ⟶ P₂
⊢ IsIso (F.map f) ↔ (MorphismProperty.isomorphisms D).inverseImage F f
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
hX : IsZero X
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), 0 ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0

Tactic:
intro i

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
hX : IsZero X
this : HasExt C := HasExt.standard C
i : ℕ
⊢ 0 ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X : C
n m : ℕ
h : n ≤ m
inst✝ : HasProjectiveDimensionLT X n
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), m ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0

Tactic:
intro i

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X : C
n m : ℕ
h : n ≤ m
inst✝ : HasProjectiveDimensionLT X n
this : HasExt C := HasExt.standard C
i : ℕ
⊢ m ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X : C
n : ℕ
inst✝ : Projective X
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), 1 ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0

Tactic:
intro i

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X : C
n : ℕ
inst✝ : Projective X
this : HasExt C := HasExt.standard C
i : ℕ
⊢ 1 ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X Y : C
h : Retract X Y
n : ℕ
inst✝ : HasProjectiveDimensionLT Y n
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0

Tactic:
intro i

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X Y : C
h : Retract X Y
n : ℕ
inst✝ : HasProjectiveDimensionLT Y n
this : HasExt C := HasExt.standard C
i : ℕ
⊢ n ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : S.ShortExact
n : ℕ
h₁ : HasProjectiveDimensionLT S.X₁ n
h₃ : HasProjectiveDimensionLT S.X₃ n
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext S.X₂ Y i), e = 0

Tactic:
intro i

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : S.ShortExact
n : ℕ
h₁ : HasProjectiveDimensionLT S.X₁ n
h₃ : HasProjectiveDimensionLT S.X₃ n
this : HasExt C := HasExt.standard C
i : ℕ
⊢ n ≤ i → ∀ ⦃Y : C⦄ (e : Ext S.X₂ Y i), e = 0
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : S.ShortExact
n : ℕ
h₂ : HasProjectiveDimensionLT S.X₂ n
h₃ : HasProjectiveDimensionLT S.X₃ (n + 1)
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext S.X₁ Y i), e = 0

Tactic:
intro i

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : S.ShortExact
n : ℕ
h₂ : HasProjectiveDimensionLT S.X₂ n
h₃ : HasProjectiveDimensionLT S.X₃ (n + 1)
this : HasExt C := HasExt.standard C
i : ℕ
⊢ n ≤ i → ∀ ⦃Y : C⦄ (e : Ext S.X₁ Y i), e = 0
Module: Mathlib.CategoryTheory.Abelian.Projective.Dimension

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y : C
h : Retract X Y
n : WithBot ℕ∞
hn : ∀ (i : ℕ), n < ↑i → HasProjectiveDimensionLT Y i
⊢ ∀ (i : ℕ), n < ↑i → HasProjectiveDimensionLT X i

Tactic:
intro i

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y : C
h : Retract X Y
n : WithBot ℕ∞
hn : ∀ (i : ℕ), n < ↑i → HasProjectiveDimensionLT Y i
i : ℕ
⊢ n < ↑i → HasProjectiveDimensionLT X i
Module: Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension

Goal state before:
pp: case coe.coe
R : Type u
inst✝³ : Ring R
inst✝² : Small.{v, u} R
R' : Type u'
inst✝¹ : Ring R'
inst✝ : Small.{v', u'} R'
e : R ≃+* R'
M : ModuleCat R
N : ModuleCat R'
e' : ↑M ≃ₛₗ[↑e] ↑N
n : ℕ
⊢ projectiveDimension M ≤ ↑↑n ↔ projectiveDimension N ≤ ↑↑n

Tactic:
norm_cast

Goal state after:
case coe.coe
R : Type u
inst✝³ : Ring R
inst✝² : Small.{v, u} R
R' : Type u'
inst✝¹ : Ring R'
inst✝ : Small.{v', u'} R'
e : R ≃+* R'
M : ModuleCat R
N : ModuleCat R'
e' : ↑M ≃ₛₗ[↑e] ↑N
n : ℕ
⊢ projectiveDimension M ≤ ↑n ↔ projectiveDimension N ≤ ↑n
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)
⊢ ∀ (Y : C) (f : Y ⟶ Opposite.unop X),
    (Presheaf.imageSieve α r).arrows f →
      (ConcreteCategory.hom (A.obj.map f.op)) (smul α φ r 0) = (ConcreteCategory.hom (A.obj.map f.op)) 0

Tactic:
rintro Y f ⟨r₀, hr₀⟩

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)
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
⊢ (ConcreteCategory.hom (A.obj.map f.op)) (smul α φ r 0) = (ConcreteCategory.hom (A.obj.map f.op)) 0
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
ι : Type u_1
f : ι → E.I₀
⊢ ∀ (i : (E.restrictIndex f).I₀), E.presieve₀ ((E.restrictIndex f).f i)

Tactic:
intro i

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
ι : Type u_1
f : ι → E.I₀
i : (E.restrictIndex f).I₀
⊢ E.presieve₀ ((E.restrictIndex f).f i)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E F : PreZeroHypercover S
e : E ≅ F
i : E.I₀
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯

Tactic:
obtain ⟨hs, hh⟩ := Hom.ext'_iff.mp e.hom_inv_id

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E F : PreZeroHypercover S
e : E ≅ F
i : E.I₀
hs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀
hh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯
⊢ e.hom.h₀ i ≫ e.inv.h₀ (e.hom.s₀ i) = eqToHom ⋯
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
f : E.Hom F
⊢ ∀ (i : E.I₀), F.sieve₀.arrows (E.f i)

Tactic:
intro i

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
f : E.Hom F
i : E.I₀
⊢ F.sieve₀.arrows (E.f i)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
K : Precoverage C
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasIsos
X : C
E F : PreZeroHypercover X
e : E ≅ F
hE : E.presieve₀ ∈ K.coverings X
⊢ ∀ (i : F.I₀), Presieve.ofArrows (fun i ↦ F.X i.fst) (fun i ↦ F.f i.fst) (F.f i)

Tactic:
intro i

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
K : Precoverage C
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasIsos
X : C
E F : PreZeroHypercover X
e : E ≅ F
hE : E.presieve₀ ∈ K.coverings X
i : F.I₀
⊢ Presieve.ofArrows (fun i ↦ F.X i.fst) (fun i ↦ F.f i.fst) (F.f i)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
K : Precoverage C
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasIsos
X : C
E F : PreZeroHypercover X
e : E ≅ F
hE : E.presieve₀ ∈ K.coverings X
this : F.presieve₀ = Presieve.ofArrows (fun i ↦ F.X i.fst) fun i ↦ e.inv.h₀ i.fst ≫ E.f (e.inv.s₀ i.fst)
⊢ ∀ (i : F.I₀), (Presieve.ofArrows (fun k ↦ F.X i) fun k ↦ e.inv.h₀ i) ∈ K.coverings (E.X (e.inv.s₀ i))

Tactic:
intro i

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
K : Precoverage C
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasIsos
X : C
E F : PreZeroHypercover X
e : E ≅ F
hE : E.presieve₀ ∈ K.coverings X
this : F.presieve₀ = Presieve.ofArrows (fun i ↦ F.X i.fst) fun i ↦ e.inv.h₀ i.fst ≫ E.f (e.inv.s₀ i.fst)
i : F.I₀
⊢ (Presieve.ofArrows (fun k ↦ F.X i) fun k ↦ e.inv.h₀ i) ∈ K.coverings (E.X (e.inv.s₀ i))
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.RespectsIso
S : C
R T : Presieve S
hR : R ∈ J.coverings S
YR : ⦃Z : C⦄ → (g : Z ⟶ S) → R g → C
eR : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : R g) → YR g a ≅ Z
hTeg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : R g), T ((eR g a).hom ≫ g)
YT : ⦃Z : C⦄ → (g : Z ⟶ S) → T g → C
eT : ⦃Z : C⦄ → (g : Z ⟶ S) → (a : T g) → YT g a ≅ Z
hReg : ∀ ⦃Z : C⦄ (g : Z ⟶ S) (a : T g), R ((eT g a).hom ≫ g)
E : PreZeroHypercover S :=
  { I₀ := ↑R.uncurry ⊕ ↑T.uncurry, X := fun i ↦ Sum.elim (fun j ↦ (↑j).fst) (fun j ↦ YT (↑j).snd ⋯) i,
    f := fun i ↦
      match i with
      | Sum.inl i => (↑i).snd
      | Sum.inr i => (eT (↑i).snd ⋯).hom ≫ (↑i).snd }
F : PreZeroHypercover S :=
  { I₀ := ↑R.uncurry ⊕ ↑T.uncurry, X := fun i ↦ Sum.elim (fun j ↦ YR (↑j).snd ⋯) (fun j ↦ (↑j).fst) i,
    f := fun i ↦
      match i with
      | Sum.inl i => (eR (↑i).snd ⋯).hom ≫ (↑i).snd
      | Sum.inr i => (↑i).snd }
i✝ : E.I₀
i : ↑T.uncurry
⊢ E.X (Sum.inr i) ≅ F.X ((Equiv.refl E.I₀) (Sum.inr i))

Tactic:
dsimp [E, F]; apply eT

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

Goal state before:
pp: case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.IsStableUnderBaseChange
S : C
E F : PreZeroHypercover S
e : E ≅ F
h : E.presieve₀ ∈ J.coverings S
⊢ ∀ (i : F.I₀), IsPullback (F.f i) (e.inv.h₀ i) (𝟙 S) (E.f (e.inv.s₀ i))

Tactic:
intro i

Goal state after:
case refine_3
C : Type u
inst✝¹ : Category.{v, u} C
J : Precoverage C
inst✝ : J.IsStableUnderBaseChange
S : C
E F : PreZeroHypercover S
e : E ≅ F
h : E.presieve₀ ∈ J.coverings S
i : F.I₀
⊢ IsPullback (F.f i) (e.inv.h₀ i) (𝟙 S) (E.f (e.inv.s₀ i))
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case refine_1.pullback
C : Type u_2
inst✝⁴ : Category.{u_1, u_2} C
J : Precoverage C
inst✝³ : J.IsStableUnderComposition
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.HasIsos
X✝ : C
S✝ : Sieve X✝
X : C
S : Sieve X
hS : J.Saturate X S
Y : C
f : Y ⟶ X
R : Presieve X
hR : R ∈ J.coverings X
hle : R ≤ S.arrows
this : R.HasPullbacks f
⊢ Presieve.pullbackArrows f R ≤ (Sieve.pullback f S).arrows

Tactic:
← Sieve.generate_le_iff,

Goal state after:
case refine_1.pullback
C : Type u_2
inst✝⁴ : Category.{u_1, u_2} C
J : Precoverage C
inst✝³ : J.IsStableUnderComposition
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.HasIsos
X✝ : C
S✝ : Sieve X✝
X : C
S : Sieve X
hS : J.Saturate X S
Y : C
f : Y ⟶ X
R : Presieve X
hR : R ∈ J.coverings X
hle : R ≤ S.arrows
this : R.HasPullbacks f
⊢ Sieve.generate (Presieve.pullbackArrows f R) ≤ Sieve.pullback f S
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: case refine_1.transitive
C : Type u_2
inst✝⁴ : Category.{u_1, u_2} C
J : Precoverage C
inst✝³ : J.IsStableUnderComposition
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.HasIsos
X✝ : C
S✝ : Sieve X✝
X : C
S T : Sieve X
hS : J.Saturate X S
hT : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → J.Saturate Y (Sieve.pullback f T)
E : J.ZeroHypercover X
hle : Presieve.ofArrows E.X E.f ≤ S.arrows
F : (i : E.I₀) → J.ZeroHypercover (E.X i)
hle' : ∀ (i : E.I₀), (F i).presieve₀ ≤ (Sieve.pullback (E.f i) T).arrows
⊢ ∀ (i : (E.bind F).I₀), T.arrows ((E.bind F).f i)

Tactic:
intro i

Goal state after:
case refine_1.transitive
C : Type u_2
inst✝⁴ : Category.{u_1, u_2} C
J : Precoverage C
inst✝³ : J.IsStableUnderComposition
inst✝² : J.IsStableUnderBaseChange
inst✝¹ : J.HasPullbacks
inst✝ : J.HasIsos
X✝ : C
S✝ : Sieve X✝
X : C
S T : Sieve X
hS : J.Saturate X S
hT : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → J.Saturate Y (Sieve.pullback f T)
E : J.ZeroHypercover X
hle : Presieve.ofArrows E.X E.f ≤ S.arrows
F : (i : E.I₀) → J.ZeroHypercover (E.X i)
hle' : ∀ (i : E.I₀), (F i).presieve₀ ≤ (Sieve.pullback (E.f i) T).arrows
i : (E.bind F).I₀
⊢ T.arrows ((E.bind F).f i)
Module: Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck

Goal state before:
pp: C : Type u_4
inst✝³ : Category.{u_3, u_4} C
D : Type u_1
inst✝² : Category.{v_1, u_1} D
F : C ⥤ D
P : Dᵒᵖ ⥤ Type u_2
X : C
ι : Type (max u_4 u_3)
Y : ι → C
f : (i : ι) → Y i ⟶ X
inst✝¹ : (ofArrows Y f).HasPairwisePullbacks
inst✝ : F.PreservesPairwisePullbacks (ofArrows Y f)
this : (ofArrows (fun i ↦ F.obj (Y i)) fun i ↦ F.map (f i)).HasPairwisePullbacks
x : (i : ι) → P.obj (Opposite.op (F.obj (Y i)))
⊢ ∀ (a b : ι),
    (ConcreteCategory.hom (P.map (F.map (pullback.fst (f a) (f b))).op)) (x a) =
        (ConcreteCategory.hom (P.map (F.map (pullback.snd (f a) (f b))).op)) (x b) ↔
      (ConcreteCategory.hom (P.map (pullback.fst (F.map (f a)) (F.map (f b))).op)) (x a) =
        (ConcreteCategory.hom (P.map (pullback.snd (F.map (f a)) (F.map (f b))).op)) (x b)

Tactic:
intro i

Goal state after:
C : Type u_4
inst✝³ : Category.{u_3, u_4} C
D : Type u_1
inst✝² : Category.{v_1, u_1} D
F : C ⥤ D
P : Dᵒᵖ ⥤ Type u_2
X : C
ι : Type (max u_4 u_3)
Y : ι → C
f : (i : ι) → Y i ⟶ X
inst✝¹ : (ofArrows Y f).HasPairwisePullbacks
inst✝ : F.PreservesPairwisePullbacks (ofArrows Y f)
this : (ofArrows (fun i ↦ F.obj (Y i)) fun i ↦ F.map (f i)).HasPairwisePullbacks
x : (i : ι) → P.obj (Opposite.op (F.obj (Y i)))
i : ι
⊢ ∀ (b : ι),
    (ConcreteCategory.hom (P.map (F.map (pullback.fst (f i) (f b))).op)) (x i) =
        (ConcreteCategory.hom (P.map (F.map (pullback.snd (f i) (f b))).op)) (x b) ↔
      (ConcreteCategory.hom (P.map (pullback.fst (F.map (f i)) (F.map (f b))).op)) (x i) =
        (ConcreteCategory.hom (P.map (pullback.snd (F.map (f i)) (F.map (f b))).op)) (x b)
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 : PreOneHypercover S
c : Cofan E.X
hc : IsColimit c
d : Cofan E.Y'
hd : IsColimit d
F : Cᵒᵖ ⥤ A
⊢ ∀ (i : E.I₁'),
    d.inj i ≫ (Cofan.IsColimit.desc hd fun x ↦ E.p₁ x.snd ≫ c.inj x.fst.1) ≫ Cofan.IsColimit.desc hc E.f =
      d.inj i ≫ (Cofan.IsColimit.desc hd fun x ↦ E.p₂ x.snd ≫ c.inj x.fst.2) ≫ Cofan.IsColimit.desc hc E.f

Tactic:
simp [E.w]

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 : PreOneHypercover S
c : Cofan E.X
hc : IsColimit c
d : Cofan E.Y'
hd : IsColimit d
x✝² x✝¹ : (E.sigmaOfIsColimit hc).I₀
x✝ : PUnit.{w + 1}
⊢ ∀ (i : E.I₁'),
    d.inj i ≫ (Cofan.IsColimit.desc hd fun i ↦ E.p₁ i.snd ≫ c.inj i.fst.1) ≫ (E.sigmaOfIsColimit hc).f x✝² =
      d.inj i ≫ (Cofan.IsColimit.desc hd fun i ↦ E.p₂ i.snd ≫ c.inj i.fst.2) ≫ (E.sigmaOfIsColimit hc).f x✝¹

Tactic:
simp [E.w]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
X : C
T : ↑(K (G.obj X))
Y : D
iYX : Y ⟶ G.obj X
hiYX : (↑T).arrows iYX
Z : C
iZY : G.obj Z ⟶ Y
W : D
V : C
iVZ : V ⟶ Z
iWV : W ⟶ G.obj V
iVX : V ⟶ X
e : G.map iVX = G.map iVZ ≫ iZY ≫ iYX
⊢ (Sieve.functorPullback G ↑T).arrows iVX

Tactic:
by simpa [e] using T.1.downward_closed hiYX (G.map iVZ ≫ iZY)

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u_1
inst✝² : Category.{v_1, u_1} A
S : C
E✝ : PreOneHypercover S
E : PreOneHypercover S
F : PreOneHypercover S
inst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
i j : __spread✝⁻⁰.I₀
k : E.I₁ i.1 j.1 × F.I₁ i.2 j.2
⊢ (E.p₁ k.1 ≫ E.f i.1) ≫ 𝟙 S = E.p₂ k.1 ≫ E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm j).fst

Tactic:
simp [E.w]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
A : Type u_1
inst✝² : Category.{v_1, u_1} A
S : C
E✝ : PreOneHypercover S
E : PreOneHypercover S
F : PreOneHypercover S
inst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)
inst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullback (E.p₁ k ≫ E.f i) (F.p₁ l ≫ F.f a)
⊢ ∀ ⦃i₁ i₂ : (E.inter F.toPreZeroHypercover).I₀⦄ (j : E.I₁ i₁.1 i₂.1 × F.I₁ i₁.2 i₂.2),
    pullback.map (E.p₁ j.1 ≫ E.f i₁.1) (F.p₁ j.2 ≫ F.f i₁.2)
          (E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i₁).fst)
          (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i₁).snd) (E.p₁ j.1) (F.p₁ j.2) (𝟙 S) ⋯ ⋯ ≫
        (E.inter F.toPreZeroHypercover).f i₁ =
      pullback.map (E.p₁ j.1 ≫ E.f i₁.1) (F.p₁ j.2 ≫ F.f i₁.2)
          (E.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i₂).fst)
          (F.f ((Equiv.sigmaEquivProd E.I₀ F.toPreZeroHypercover.1).symm i₂).snd) (E.p₂ j.1) (F.p₂ j.2) (𝟙 S) ⋯ ⋯ ≫
        (E.inter F.toPreZeroHypercover).f i₂

Tactic:
simp [E.w]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : Z ⟶ X
⊢ mapPreimage K G F f ≫ F.obj.map g.op = mapPreimage K G F f ≫ mapPreimage K G F (G.map g)

Tactic:
mapPreimage_map

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
J : GrothendieckTopology C
K : GrothendieckTopology D
G : C ⥤ D
A : Type u_4
inst✝¹ : Category.{v_4, u_4} A
inst✝ : IsDenseSubsite J K G
F : Sheaf J A
X Y Z : C
f : G.obj X ⟶ G.obj Y
g : Z ⟶ X
⊢ mapPreimage K G F f ≫ F.obj.map g.op = mapPreimage K G F f ≫ F.obj.map g.op
Module: Mathlib.CategoryTheory.Sites.Hypercover.One

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
toHomf : E.Hom F.toPreZeroHypercover
fs₁ : {i j : E.I₀} → E.I₁ i j → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁✝¹ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂✝¹ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
gh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁✝ : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂✝ : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
hs₀ :
  { toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁✝¹, w₁₂ := w₁₂✝¹ }.s₀ =
    { toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁✝, w₁₂ := w₁₂✝ }.s₀
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁✝¹, w₁₂ := w₁₂✝¹ }.s₁ k =
      (congrIndexOneOfEq ⋯ ⋯) ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁✝, w₁₂ := w₁₂✝ }.s₁ k)
hh₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), fh₁ k = gh₁ k ≫ 𝟙 (F.Y ((congrIndexOneOfEq ⋯ ⋯) (fs₁ k)))
⊢ fh₁ = gh₁

Tactic:
ext i j k

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
toHomf : E.Hom F.toPreZeroHypercover
fs₁ : {i j : E.I₀} → E.I₁ i j → F.I₁ (toHomf.s₀ i) (toHomf.s₀ j)
fh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁✝¹ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂✝¹ : ∀ {i j : E.I₀} (k : E.I₁ i j), fh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
gh₁ : {i j : E.I₀} → (k : E.I₁ i j) → E.Y k ⟶ F.Y (fs₁ k)
w₁₁✝ : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₁ (fs₁ k) = E.p₁ k ≫ toHomf.h₀ i
w₁₂✝ : ∀ {i j : E.I₀} (k : E.I₁ i j), gh₁ k ≫ F.p₂ (fs₁ k) = E.p₂ k ≫ toHomf.h₀ j
hs₀ :
  { toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁✝¹, w₁₂ := w₁₂✝¹ }.s₀ =
    { toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁✝, w₁₂ := w₁₂✝ }.s₀
hs₁ :
  ∀ (i j : E.I₀) (k : E.I₁ i j),
    { toHom := toHomf, s₁ := fs₁, h₁ := fh₁, w₁₁ := w₁₁✝¹, w₁₂ := w₁₂✝¹ }.s₁ k =
      (congrIndexOneOfEq ⋯ ⋯) ({ toHom := toHomf, s₁ := fs₁, h₁ := gh₁, w₁₁ := w₁₁✝, w₁₂ := w₁₂✝ }.s₁ k)
hh₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), fh₁ k = gh₁ k ≫ 𝟙 (F.Y ((congrIndexOneOfEq ⋯ ⋯) (fs₁ k)))
i j : E.I₀
k : E.I₁ i j
⊢ fh₁ k = gh₁ k