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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U))
      ((ConcreteCategory.hom (Φ.obj.fiber.map t)) v) =
    (ConcreteCategory.hom (Φ.obj.presheafFiber.map (shrinkYoneda.{w, v, u}.map t)))
      ((ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app V)) v)

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U))
      ((ConcreteCategory.hom (Φ.obj.fiber.map t)) v) =
    (ConcreteCategory.hom (Φ.obj.presheafFiber.map (shrinkYoneda.{w, v, u}.map t)))
      ((ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app V)) v)
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.toPresheafFiber V v (shrinkYoneda.{w, v, u}.obj U))) (shrinkYonedaObjObjEquiv.symm t) =
    (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y)

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
F₁ : Cᵒᵖ ⥤ Type w
U : C
f : F₁ ⟶ shrinkYoneda.{w, v, u}.obj U
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
Φ : P.FullSubcategory
u : Φ.obj.fiber.obj U
V : C
v : Φ.obj.fiber.obj V
y : ToType (F₁.obj (op V))
hx₁ :
  (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y) =
    (ConcreteCategory.hom (Φ.obj.shrinkYonedaCompPresheafFiberIso.inv.app U)) u
t : unop (op V) ⟶ U
ht : shrinkYonedaObjObjEquiv.symm t = (ConcreteCategory.hom (f.app (op V))) y
⊢ (ConcreteCategory.hom (Φ.obj.toPresheafFiber V v (shrinkYoneda.{w, v, u}.obj U))) (shrinkYonedaObjObjEquiv.symm t) =
    (ConcreteCategory.hom (Φ.obj.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.obj.toPresheafFiber V v F₁)) y)
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : HasSheafify J (Type w)
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
x✝² x✝¹ : Sheaf J (Type w)
f : x✝² ⟶ x✝¹
x✝ : Mono f
hf : ∀ (i : P.FullSubcategory), IsIso (i.obj.sheafFiber.map f)
⊢ Sheaf.IsLocallySurjective f

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : HasSheafify J (Type w)
hP :
  ∀ ⦃X : C⦄ (S : Sieve X),
    (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),
        ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →
      S ∈ J X
x✝² x✝¹ : Sheaf J (Type w)
f : x✝² ⟶ x✝¹
x✝ : Mono f
hf : ∀ (i : P.FullSubcategory), IsIso (i.obj.sheafFiber.map f)
⊢ Sheaf.IsLocallySurjective f
Module: Mathlib.AlgebraicGeometry.Sites.EtalePoint

Goal state before:
pp: ι : Type u_1
S : Scheme
Ω : ι → Type u
inst✝¹ : (i : ι) → Field (Ω i)
inst✝ : ∀ (i : ι), IsSepClosed (Ω i)
s : (i : ι) → Spec (CommRingCat.of (Ω i)) ⟶ S
hs : ⋃ i, Set.range ⇑(s i) = Set.univ
X : S.Etale
α : Type (u + 1)
T : α → S.Etale
f : (i : α) → T i ⟶ X
hR :
  ∀ (Φ : (ObjectProperty.ofObj fun i ↦ pointSmallEtale (s i)).FullSubcategory) (x : Φ.obj.fiber.obj X),
    ∃ Y g, ∃ (_ : (Sieve.ofArrows T f).arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
⊢ ⋃ i, Set.range ⇑(f i).left = Set.univ

Tactic:
]

Goal state after:
ι : Type u_1
S : Scheme
Ω : ι → Type u
inst✝¹ : (i : ι) → Field (Ω i)
inst✝ : ∀ (i : ι), IsSepClosed (Ω i)
s : (i : ι) → Spec (CommRingCat.of (Ω i)) ⟶ S
hs : ⋃ i, Set.range ⇑(s i) = Set.univ
X : S.Etale
α : Type (u + 1)
T : α → S.Etale
f : (i : α) → T i ⟶ X
hR :
  ∀ (Φ : (ObjectProperty.ofObj fun i ↦ pointSmallEtale (s i)).FullSubcategory) (x : Φ.obj.fiber.obj X),
    ∃ Y g, ∃ (_ : (Sieve.ofArrows T f).arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
⊢ ⋃ i, Set.range ⇑(f i).left = Set.univ
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
q r : ℚ
heq : q = r
⊢ const (padicNorm p) r ≈ const (padicNorm p) r

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
q r : ℚ
h : q = r
⊢ ↑r = ↑r

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
m n : ℕ
⊢ ↑↑m = ↑n → m = n

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
m n : ℕ
⊢ ↑↑m = ↑n → m = n
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ_[p]
r : CauSeq ℚ (padicNorm p)
⊢ r ≈ 0 ↔ ⟦r⟧ = ⟦0⟧

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ_[p]
r : CauSeq ℚ (padicNorm p)
⊢ r ≈ 0 ↔ ⟦r⟧ = ⟦0⟧
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h✝ : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hge : ε ≤ 0
h : f - const (padicNorm p) (↑f i) ≈ 0
⊢ False

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h✝ : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hge : ε ≤ 0
h : f - const (padicNorm p) (↑f i) ≈ 0
⊢ False
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k : ι
i j : (glueData hf).J
⊢ yoneda.map (⋯.snd (f j)) ≫ f j = yoneda.map (⋯.snd (f j)) ≫ f j

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
S : Scheme
Ω : ι → Type u
inst✝¹ : (i : ι) → Field (Ω i)
inst✝ : ∀ (i : ι), IsSepClosed (Ω i)
s : (i : ι) → Spec (CommRingCat.of (Ω i)) ⟶ S
hs : ⋃ i, Set.range ⇑(s i) = Set.univ
X : S.Etale
α : Type (u + 1)
T : α → S.Etale
f : (i : α) → T i ⟶ X
hR :
  ∀ (Φ : (ObjectProperty.ofObj fun i ↦ pointSmallEtale (s i)).FullSubcategory) (x : Φ.obj.fiber.obj X),
    ∃ Y g, ∃ (_ : (Sieve.ofArrows T f).arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
x : ↥X.left
i : ι
hi : (s i) default = X.hom x
x' : (pointSmallEtale (s i)).fiber.obj X
hx' : ↑(pointSmallEtaleFiberObjToPreimage (s i) hi x') = ↑⟨x, ⋯⟩
⊢ ∃ i y, (f i).left y = x

Tactic:
]

Goal state after:
ι : Type u_1
S : Scheme
Ω : ι → Type u
inst✝¹ : (i : ι) → Field (Ω i)
inst✝ : ∀ (i : ι), IsSepClosed (Ω i)
s : (i : ι) → Spec (CommRingCat.of (Ω i)) ⟶ S
hs : ⋃ i, Set.range ⇑(s i) = Set.univ
X : S.Etale
α : Type (u + 1)
T : α → S.Etale
f : (i : α) → T i ⟶ X
hR :
  ∀ (Φ : (ObjectProperty.ofObj fun i ↦ pointSmallEtale (s i)).FullSubcategory) (x : Φ.obj.fiber.obj X),
    ∃ Y g, ∃ (_ : (Sieve.ofArrows T f).arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x
x : ↥X.left
i : ι
hi : (s i) default = X.hom x
x' : (pointSmallEtale (s i)).fiber.obj X
hx' : ↑(pointSmallEtaleFiberObjToPreimage (s i) hi x') = ↑⟨x, ⋯⟩
⊢ ∃ i y, (f i).left y = x
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
hge : ε ≤ padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
hngen : stationaryPoint hne ≤ N
this :
  padicNorm p (↑(f - const (padicNorm p) (↑f i)) N) =
    padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
⊢ padicNorm p (↑(f - const (padicNorm p) (↑f i)) N) < ε

Tactic:
]

Goal state after:
case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
ε : ℚ
hε : 0 < ε
h : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε
i : ℕ
hi : i ≥ N
hne : ¬f - const (padicNorm p) (↑f i) ≈ 0
hge : ε ≤ padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
hngen : stationaryPoint hne ≤ N
this :
  padicNorm p (↑(f - const (padicNorm p) (↑f i)) N) =
    padicNorm p (↑(f - const (padicNorm p) (↑f i)) (stationaryPoint hne))
⊢ padicNorm p (↑(f - const (padicNorm p) (↑f i)) N) < ε
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i : ι
⊢ (ConcreteCategory.hom
        ({ hom := yonedaEquiv.symm ((glueData hf).sheafValGluedMk (fun i ↦ yonedaEquiv (f i)) ⋯) }.hom.app (op (X i))))
      ((ConcreteCategory.hom (↾fun g ↦ g ≫ toGlued hf i)) (𝟙 (X i))) =
    (ConcreteCategory.hom ((f i).app (op (X i)))) (𝟙 (X i))

Tactic:
]

Goal state after:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i : ι
⊢ (ConcreteCategory.hom
        ({ hom := yonedaEquiv.symm ((glueData hf).sheafValGluedMk (fun i ↦ yonedaEquiv (f i)) ⋯) }.hom.app (op (X i))))
      ((ConcreteCategory.hom (↾fun g ↦ g ≫ toGlued hf i)) (𝟙 (X i))) =
    (ConcreteCategory.hom ((f i).app (op (X i)))) (𝟙 (X i))
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i : ι
⊢ (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op (X i)))) (toGlued hf i) =
    (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op (X i)))) (toGlued hf i)

Tactic:
]

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

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
U : Scheme
i j : ι
a : U ⟶ X i
b : U ⟶ X j
h : yoneda.map a ≫ f i = yoneda.map b ≫ f j
⊢ ⋯.lift' a b h ≫ ⋯.fst' (f j) ≫ toGlued hf i = b ≫ toGlued hf j

Tactic:
]

Goal state after:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
U : Scheme
i j : ι
a : U ⟶ X i
b : U ⟶ X j
h : yoneda.map a ≫ f i = yoneda.map b ≫ f j
⊢ ⋯.lift' a b h ≫ ⋯.fst' (f j) ≫ toGlued hf i = b ≫ toGlued hf j
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ P (⋯.lift (fst ≫ f) snd ⋯ ≫ ⋯.snd p)

Tactic:
]

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ P (⋯.lift (fst ≫ f) snd ⋯ ≫ ⋯.snd p)
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ IsPullback fst (F✝.map snd) (f ≫ g) p

Tactic:
]

Goal state after:
case refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F✝ : C ⥤ D
X✝ Y : D
P : MorphismProperty C
inst✝² : F✝.Faithful
inst✝¹ : F✝.Full
inst✝ : P.IsStableUnderComposition
F G H : D
f : F ⟶ G
g : G ⟶ H
hf : relative F✝ P f
hg : relative F✝ P g
Z X : C
p : F✝.obj Z ⟶ H
fst : F✝.obj X ⟶ F
snd : X ⟶ Z
h : IsPullback fst (F✝.map snd) (f ≫ g) p
⊢ IsPullback fst (F✝.map snd) (f ≫ g) p
Module: Mathlib.AlgebraicTopology.DoldKan.Compatibility

Goal state before:
pp: A : Type u_1
A' : Type u_2
B : Type u_3
B' : Type u_4
inst✝³ : Category.{v_1, u_1} A
inst✝² : Category.{v_2, u_2} A'
inst✝¹ : Category.{v_3, u_3} B
inst✝ : Category.{v_4, u_4} B'
eA : A ≌ A'
eB : B ≌ B'
e' : A' ≌ B'
F : A ⥤ B'
hF : eA.functor ⋙ e'.functor ≅ F
G : B ⥤ A
hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor
η : G ⋙ F ≅ eB.functor
hη : τ₀ = τ₁ hF hG η
Y : B
⊢ eB.inverse.map
        (hF.inv.app (G.obj Y) ≫
          (eA.functor ⋙ e'.functor).map (eA.unitIso.hom.app (G.obj Y)) ≫
            (eA.functor ⋙ e'.functor).map (eA.inverse.map (hG.inv.app Y)) ≫
              e'.functor.map (eA.counitIso.hom.app (e'.inverse.obj (eB.functor.obj Y))) ≫
                e'.functor.map (hG.hom.app Y) ≫ hF.hom.app (G.obj Y) ≫ η.hom.app Y) ≫
      eB.unitIso.inv.app Y =
    eB.inverse.map (η.hom.app Y) ≫ eB.unitIso.inv.app Y

Tactic:
]

Goal state after:
A : Type u_1
A' : Type u_2
B : Type u_3
B' : Type u_4
inst✝³ : Category.{v_1, u_1} A
inst✝² : Category.{v_2, u_2} A'
inst✝¹ : Category.{v_3, u_3} B
inst✝ : Category.{v_4, u_4} B'
eA : A ≌ A'
eB : B ≌ B'
e' : A' ≌ B'
F : A ⥤ B'
hF : eA.functor ⋙ e'.functor ≅ F
G : B ⥤ A
hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor
η : G ⋙ F ≅ eB.functor
hη : τ₀ = τ₁ hF hG η
Y : B
⊢ eB.inverse.map
        (hF.inv.app (G.obj Y) ≫
          (eA.functor ⋙ e'.functor).map (eA.unitIso.hom.app (G.obj Y)) ≫
            (eA.functor ⋙ e'.functor).map (eA.inverse.map (hG.inv.app Y)) ≫
              e'.functor.map (eA.counitIso.hom.app (e'.inverse.obj (eB.functor.obj Y))) ≫
                e'.functor.map (hG.hom.app Y) ≫ hF.hom.app (G.obj Y) ≫ η.hom.app Y) ≫
      eB.unitIso.inv.app Y =
    eB.inverse.map (η.hom.app Y) ≫ eB.unitIso.inv.app Y
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ ↑i * ε + 1 * ε

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ ↑i * ε + 1 * ε
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : MorphismProperty C
G : Cᵒᵖ ⥤ Type v₁
hP : P ≤ monomorphisms C
X : C
f : yoneda.obj X ⟶ G
hf : P.presheaf f
this : Mono f
⊢ yoneda.map (⋯.snd f) ≫ f = yoneda.map (⋯.snd f) ≫ f

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: j : ℕ
hj : j + 1 = 0
⊢ 0 ≤ 0

Tactic:
]

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

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k : ι
U : Scheme
α β : U ⟶ (glueData hf).glued
mem :
  Sieve.ofArrows (glueData hf).openCover.X (glueData hf).openCover.f ∈
    (grothendieckTopology IsOpenImmersion) (glueData hf).glued
V : Scheme
γ : V ⟶ U
Y✝ : Scheme
i : (glueData hf).openCover.I₀
a : V ⟶ (glueData hf).openCover.X i
fac₁ : a ≫ (glueData hf).openCover.f i = γ ≫ α
Y : Scheme
j : (glueData hf).openCover.I₀
b : V ⟶ (glueData hf).openCover.X j
fac₂ : b ≫ (glueData hf).openCover.f j = γ ≫ β
h :
  (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op V))) (a ≫ (glueData hf).openCover.f i) =
    (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op V))) (b ≫ (glueData hf).openCover.f j)
⊢ a ≫ (glueData hf).openCover.f i = b ≫ (glueData hf).openCover.f j

Tactic:
]

Goal state after:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k : ι
U : Scheme
α β : U ⟶ (glueData hf).glued
mem :
  Sieve.ofArrows (glueData hf).openCover.X (glueData hf).openCover.f ∈
    (grothendieckTopology IsOpenImmersion) (glueData hf).glued
V : Scheme
γ : V ⟶ U
Y✝ : Scheme
i : (glueData hf).openCover.I₀
a : V ⟶ (glueData hf).openCover.X i
fac₁ : a ≫ (glueData hf).openCover.f i = γ ≫ α
Y : Scheme
j : (glueData hf).openCover.I₀
b : V ⟶ (glueData hf).openCover.X j
fac₂ : b ≫ (glueData hf).openCover.f j = γ ≫ β
h :
  (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op V))) (a ≫ (glueData hf).openCover.f i) =
    (ConcreteCategory.hom ((yonedaGluedToSheaf hf).hom.app (op V))) (b ≫ (glueData hf).openCover.f j)
⊢ a ≫ (glueData hf).openCover.f i = b ≫ (glueData hf).openCover.f j
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n a : ℕ
ha : n = a + q
⊢ (-1) ^ a • X.σ ⟨a, ⋯⟩ = (-1) ^ a • X.σ ⟨a, ⋯⟩

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
⊢ hσ' q 0 1 ⋯ ≫ K[X].d 1 0 = 0

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
⊢ hσ' q 0 1 ⋯ ≫ K[X].d 1 0 = 0
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
⊢ (((-1) ^ 0 • X.σ ⟨0, ⋯⟩) ≫ eqToHom ⋯) ≫ K[X].d 1 0 = 0

Tactic:
]

Goal state after:
case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
⊢ (((-1) ^ 0 • X.σ ⟨0, ⋯⟩) ≫ eqToHom ⋯) ≫ K[X].d 1 0 = 0
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
this : padicNormE (↑(limSeq f j) - ↑(limSeq f (max N N2))) < ε
⊢ padicNormE ↑(limSeq f j - limSeq f (max N N2)) < ε

Tactic:
]

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ max N N2
this : padicNormE (↑(limSeq f j) - ↑(limSeq f (max N N2))) < ε
⊢ padicNormE ↑(limSeq f j - limSeq f (max N N2)) < ε
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
⊢ 𝟙 (X _⦋0⦌) + -𝟙 (X _⦋0⦌) = 0

Tactic:
]

Goal state after:
case zero
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
⊢ 𝟙 (X _⦋0⦌) + -𝟙 (X _⦋0⦌) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q : ℕ
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
inst✝ : Presheaf.IsLocallySurjective zariskiTopology (Sigma.desc f)
⊢ Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf) ∧ Sheaf.IsLocallySurjective (yonedaGluedToSheaf hf)

Tactic:
]

Goal state after:
F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
inst✝ : Presheaf.IsLocallySurjective zariskiTopology (Sigma.desc f)
⊢ Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf) ∧ Sheaf.IsLocallySurjective (yonedaGluedToSheaf hf)
Module: Mathlib.AlgebraicTopology.DoldKan.Homotopies

Goal state before:
pp: case pos
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
h✝ : n < q
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i j k : ι
inst✝ : Presheaf.IsLocallySurjective zariskiTopology (Sigma.desc f)
⊢ Sheaf.IsLocallyInjective (yonedaGluedToSheaf hf) ∧ Sheaf.IsLocallySurjective (yonedaGluedToSheaf hf)

Tactic:
constructor <;> infer_instance

Goal state after: