mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Sites.ConstantSheaf

Goal state before:
pp: case refine_2
T : Type v
inst✝ : TopologicalSpace T
R S : CommRingCat
f : R ⟶ S
hf₁ : Flat (Spec.map f)
hf₂ : Surjective (Spec.map f)
⊢ ∀ (x : (continuousMapPresheaf T).obj (Opposite.op (Spec S))),
    (∀ {Z : Scheme} (p₁ p₂ : Z ⟶ Spec S),
        p₁ ≫ Spec.map f = p₂ ≫ Spec.map f →
          (ConcreteCategory.hom ((continuousMapPresheaf T).map p₁.op)) x =
            (ConcreteCategory.hom ((continuousMapPresheaf T).map p₂.op)) x) →
      ∃! y, (ConcreteCategory.hom ((continuousMapPresheaf T).map (Spec.map f).op)) y = x

Tactic:
have : Topology.IsQuotientMap (Spec.map f) := Flat.isQuotientMap_of_surjective _

Goal state after:
case refine_2
T : Type v
inst✝ : TopologicalSpace T
R S : CommRingCat
f : R ⟶ S
hf₁ : Flat (Spec.map f)
hf₂ : Surjective (Spec.map f)
this : Topology.IsQuotientMap ⇑(Spec.map f)
⊢ ∀ (x : (continuousMapPresheaf T).obj (Opposite.op (Spec S))),
    (∀ {Z : Scheme} (p₁ p₂ : Z ⟶ Spec S),
        p₁ ≫ Spec.map f = p₂ ≫ Spec.map f →
          (ConcreteCategory.hom ((continuousMapPresheaf T).map p₁.op)) x =
            (ConcreteCategory.hom ((continuousMapPresheaf T).map p₂.op)) x) →
      ∃! y, (ConcreteCategory.hom ((continuousMapPresheaf T).map (Spec.map f).op)) y = x
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
⊢ Presheaf.IsSheaf J (presheaf data G₀)

Tactic:
rw [isSheaf_iff data]

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
⊢ Presheaf.IsSheaf J₀ (F.op ⋙ presheaf data G₀) ∧
    ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork (presheaf data G₀)))
Module: Mathlib.CategoryTheory.Sites.Subcanonical

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X Y : C
F : Sheaf J (Type (max v v'))
f : (uliftYoneda.{v', v, u} J).obj X ⟶ F
g : Y ⟶ X
⊢ (ConcreteCategory.hom (F.obj.map g.op)) (J.uliftYonedaEquiv f) =
    (ConcreteCategory.hom (f.hom.app (op Y))) { down := g }

Tactic:
uliftYonedaEquiv_naturality,

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : J.Subcanonical
X Y : C
F : Sheaf J (Type (max v v'))
f : (uliftYoneda.{v', v, u} J).obj X ⟶ F
g : Y ⟶ X
⊢ J.uliftYonedaEquiv ((uliftYoneda.{v', v, u} J).map g ≫ f) = (ConcreteCategory.hom (f.hom.app (op Y))) { down := g }
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
inst✝³ : IsGrothendieckAbelian.{w, v, u} C
X : C
J : Type w
inst✝² : SmallCategory J
F : J ⥤ MonoOver X
inst✝¹ : IsFiltered J
c : Cocone (F ⋙ MonoOver.forget X)
inst✝ : Mono c.pt.hom
h : Subobject.mk c.pt.hom = ⨆ j, Subobject.mk (F.obj j).obj.hom
f : colimit (F ⋙ MonoOver.forget X ⋙ Over.forget X) ⟶ X :=
  colimit.desc (F ⋙ MonoOver.forget X ⋙ Over.forget X)
    { pt := X, ι := { app := fun j ↦ (F.obj j).obj.hom, naturality := ⋯ } }
this✝ : Mono f
this : Subobject.mk f = Subobject.mk c.pt.hom
j : J
⊢ colimit.ι ((F ⋙ MonoOver.forget X) ⋙ Over.forget X) j ≫ f = (F.obj j).obj.hom

Tactic:
apply colimit.ι_desc

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
W : MorphismProperty C
J : Type w
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
inst✝¹ : OrderBot J
inst✝ : WellFoundedLT J
⊢ W.transfiniteCompositionsOfShape J ≤ transfiniteCompositions.{w, v, u} W

Tactic:
intro A B f hf
  rw [transfiniteCompositions_iff]
  exact ⟨_, _, _, _, _, hf⟩

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : Type u
inst✝ : LinearOrder J
j : J
F : ↑(Set.Iio j) ⥤ C
c : Cocone F
i : J
hi : i < j
⊢ arrowMap (ofCocone c) i j ⋯ ⋯ = Arrow.mk (c.ι.app ⟨i, hi⟩)

Tactic:
arrowMap,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : Type u
inst✝ : LinearOrder J
j : J
F : ↑(Set.Iio j) ⥤ C
c : Cocone F
i : J
hi : i < j
⊢ Arrow.mk ((ofCocone c).map (homOfLE ⋯)) = Arrow.mk (c.ι.app ⟨i, hi⟩)
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteIteration

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
j : J
hj : ¬IsMax j
⊢ Φ.prop ((Φ.iterationFunctor J).map (homOfLE ⋯))

Tactic:
have := (Φ.iter (Order.succ j)).prop_map_succ j (Order.lt_succ_of_not_isMax hj)

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
j : J
hj : ¬IsMax j
this : Φ.prop ((Φ.iter (Order.succ j)).F.map (homOfLE ⋯))
⊢ Φ.prop ((Φ.iterationFunctor J).map (homOfLE ⋯))
Module: Mathlib.CategoryTheory.SmallObject.WellOrderInductionData

Goal state before:
pp: case e'_3
J : Type u
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
F : Jᵒᵖ ⥤ Type v
d : F.WellOrderInductionData
inst✝¹ : OrderBot J
val₀ : F.obj (op ⊥)
inst✝ : WellFoundedLT J
j : J
hj : Order.IsSuccLimit j
e : (i : J) → i < j → d.Extension val₀ i
i : J
hi : i < j
⊢ d.succ i ⋯
      ((ConcreteCategory.hom (F.map (homOfLE ⋯).op))
        (d.lift j hj
          ⟨fun x ↦
            match x with
            | op ⟨i, hi⟩ => (e i hi).val,
            ⋯⟩)) =
    d.succ i ⋯ ((ConcreteCategory.hom (F.map (homOfLE ⋯).op)) (e (Order.succ i) ⋯).val)

Tactic:
· congr 1
      rw [d.map_lift _ _ _ _ hi]
      symm
      apply compatibility

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: case h
C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X A✝ : C
f : A✝ ⟶ X
inst✝ : Mono f
hA : Subobject.mk f ≠ ⊤
X' : C
i : A✝ ⟶ X'
p' : X' ⟶ X
hi : (generatingMonomorphisms G).pushouts i
hi' : ¬IsIso i
hp' : Mono p'
fac : i ≫ p' = f
⊢ Arrow.mk ((Subobject.mk f).ofLE (Subobject.mk p') ⋯) ≅ Arrow.mk i

Tactic:
refine Arrow.isoMk (Subobject.underlyingIso f) (Subobject.underlyingIso p') ?_

Goal state after:
case h
C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
hG : IsSeparator G
X A✝ : C
f : A✝ ⟶ X
inst✝ : Mono f
hA : Subobject.mk f ≠ ⊤
X' : C
i : A✝ ⟶ X'
p' : X' ⟶ X
hi : (generatingMonomorphisms G).pushouts i
hi' : ¬IsIso i
hp' : Mono p'
fac : i ≫ p' = f
⊢ (Subobject.underlyingIso f).hom ≫ (Arrow.mk i).hom =
    (Arrow.mk ((Subobject.mk f).ofLE (Subobject.mk p') ⋯)).hom ≫ (Subobject.underlyingIso p').hom
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
h : padicValRat p q ≤ padicValRat p r
hq : ¬q = 0
hr : ¬r = 0
hqn : q.num ≠ 0
hqd : ↑q.den ≠ 0
hrn : r.num ≠ 0
hrd : ↑r.den ≠ 0
hqreq : q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)
hqrd : q.num * ↑r.den + ↑q.den * r.num ≠ 0
| padicValRat p q

Tactic:
← q.num_divInt_den

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
h : padicValRat p q ≤ padicValRat p r
hq : ¬q = 0
hr : ¬r = 0
hqn : q.num ≠ 0
hqd : ↑q.den ≠ 0
hrn : r.num ≠ 0
hrd : ↑r.den ≠ 0
hqreq : q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)
hqrd : q.num * ↑r.den + ↑q.den * r.num ≠ 0
| padicValRat p (q.num /. ↑q.den)
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hval : padicValRat p q < padicValRat p r
⊢ padicValRat p (q + r) = padicValRat p q

Tactic:
add_eq_min hqr hq hr (ne_of_lt hval),

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hval : padicValRat p q < padicValRat p r
⊢ min (padicValRat p q) (padicValRat p r) = padicValRat p q
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
⊢ 1 ≤ ‖x‖ ↔ ‖x‖ = 1

Tactic:
by
  simp [le_antisymm_iff, ← padic_norm_e_of_padicInt, x.prop]

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
c : ℤ_[p]
hc : c ≠ 0
⊢ (↑p ^ n * c).valuation = n + c.valuation

Tactic:
by
  rw [valuation_mul (NeZero.ne _) hc, valuation_pow, valuation_p, mul_one]

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

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this✝² : ∃ ε > 0, ∃ N1, ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p (↑f j - ↑f k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this✝¹ : padicNorm p (↑f n - ↑f m) < ε
this✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)
this : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))
hne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)
⊢ False

Tactic:
have hnam := add_eq_max_of_ne hne

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ (padicNorm p)
hf : ¬f ≈ 0
this✝² : ∃ ε > 0, ∃ N1, ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
ε : ℚ
hε : ε > 0
N1 : ℕ
hN1 : ∀ j ≥ N1, ε ≤ padicNorm p (↑f j)
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNorm p (↑f j - ↑f k) < ε
n m : ℕ
hn : max N1 N2 ≤ n
hm : max N1 N2 ≤ m
this✝¹ : padicNorm p (↑f n - ↑f m) < ε
this✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)
this : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))
hne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)
hnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (-↑f m)) (padicNorm p (↑f n))
⊢ False
Module: Mathlib.CategoryTheory.Functor.TypeValuedFlat

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
F : C ⥤ Type w
inst✝¹ : HasFiniteLimits C
inst✝ : PreservesFiniteLimits F
X : C
x : F.obj X
Y : C
y : F.obj Y
f : X ⟶ Y
hf : (ConcreteCategory.hom (F.map f)) x = y
g : X ⟶ Y
hg : (ConcreteCategory.hom (F.map g)) x = y
⊢ ∃ W h, h ≫ ⟨f, hf⟩ = h ≫ ⟨g, hg⟩

Tactic:
rw [← hg] at hf

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
F : C ⥤ Type w
inst✝¹ : HasFiniteLimits C
inst✝ : PreservesFiniteLimits F
X : C
x : F.obj X
Y : C
y : F.obj Y
f : X ⟶ Y
hf✝ : (ConcreteCategory.hom (F.map f)) x = y
g : X ⟶ Y
hf : (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map g)) x
hg : (ConcreteCategory.hom (F.map g)) x = y
⊢ ∃ W h, h ≫ ⟨f, hf✝⟩ = h ≫ ⟨g, hg⟩
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
hf : c.fst = c.snd
Z : C
⊢ ∀ (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h

Tactic:
g

Goal state after:
case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
hf : c.fst = c.snd
Z : C
g : Z ⟶ X
⊢ ∀ (h : Z ⟶ X), g ≫ f = h ≫ f → g = h
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
hf : c.inl = c.inr
Z : C
⊢ ∀ (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h

Tactic:
g

Goal state after:
case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
hf : c.inl = c.inr
Z : C
g : Y ⟶ Z
⊢ ∀ (h : Y ⟶ Z), f ≫ g = f ≫ h → g = h
Module: Mathlib.CategoryTheory.Sites.Point.Skyscraper

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
F : Sheaf J A
M : A
f : Φ.presheafFiber.obj F.obj ⟶ M
⊢ ((Φ.skyscraperSheafAdjunction.homEquiv F M) f).hom = Φ.skyscraperPresheafHomEquiv f

Tactic:
by
  simp [skyscraperSheafAdjunction, Functor.FullyFaithful.homEquiv]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Skyscraper

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
F : Sheaf J A
M : A
f : F ⟶ Φ.skyscraperSheaf M
⊢ (Φ.skyscraperSheafAdjunction.homEquiv F M).symm f = Φ.skyscraperPresheafHomEquiv.symm f.hom

Tactic:
by
  simp [skyscraperSheafAdjunction, Functor.FullyFaithful.homEquiv]

Goal state after:
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))
this : Presheaf.imageSieve f (shrinkYonedaObjObjEquiv.symm (𝟙 U)) ∈ J U
V : C
⊢ ∀ (s : ToType ((shrinkYoneda.{w, v, u}.obj U).obj (op V))), Presheaf.imageSieve f s ∈ J V

Tactic:
g

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))
this : Presheaf.imageSieve f (shrinkYonedaObjObjEquiv.symm (𝟙 U)) ∈ J U
V : C
g : ToType ((shrinkYoneda.{w, v, u}.obj U).obj (op V))
⊢ Presheaf.imageSieve f g ∈ J V
Module: Mathlib.NumberTheory.Padics.PadicNumbers

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

Tactic:
PadicSeq.norm_zero_iff r

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

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
F : C ⥤ D
P : MorphismProperty C
inst✝⁴ : F.Faithful
inst✝³ : F.Full
inst✝² : HasPullbacks C
inst✝¹ : P.IsStableUnderBaseChange
a b : C
f : a ⟶ b
inst✝ : ∀ (c : C) (g : c ⟶ b), PreservesLimit (cospan f g) F
hf : P f
Y' : C
⊢ ∀ (g : F.obj Y' ⟶ F.obj b), ∃ b_1 fst snd, ∃ (_ : IsPullback fst (F.map snd) (F.map f) g), P snd

Tactic:
g

Goal state after:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
F : C ⥤ D
P : MorphismProperty C
inst✝⁴ : F.Faithful
inst✝³ : F.Full
inst✝² : HasPullbacks C
inst✝¹ : P.IsStableUnderBaseChange
a b : C
f : a ⟶ b
inst✝ : ∀ (c : C) (g : c ⟶ b), PreservesLimit (cospan f g) F
hf : P f
Y' : C
g : F.obj Y' ⟶ F.obj b
⊢ ∃ b_1 fst snd, ∃ (_ : IsPullback fst (F.map snd) (F.map f) g), P snd
Module: Mathlib.AlgebraicTopology.DoldKan.Faces

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
j : Fin (n + 2)
hj₁ : j ≠ 0
hj₂ : n + 2 ≤ ↑j + q
⊢ φ ≫ X.δ j = 0

Tactic:
obtain ⟨i, rfl⟩ := Fin.eq_succ_of_ne_zero hj₁

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
i : Fin (n + 1)
hj₁ : i.succ ≠ 0
hj₂ : n + 2 ≤ ↑i.succ + q
⊢ φ ≫ X.δ i.succ = 0
Module: Mathlib.CategoryTheory.MorphismProperty.Representable

Goal state before:
pp: C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝⁷ : HasBinaryProducts C
inst✝⁶ : HasPullbacks D
inst✝⁵ : HasBinaryProducts D
inst✝⁴ : HasTerminal D
inst✝³ : F.Full
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : HasPullbacks C
inst✝ : PreservesLimitsOfShape WalkingCospan F
X : D
h : ∀ ⦃a : C⦄ (g : F.obj a ⟶ X), F.relativelyRepresentable g
a : C
⊢ ∀ (g : F.obj a ⟶ Limits.pullback (terminal.from X) (terminal.from X)),
    ∃ b snd fst, IsPullback fst (F.map snd) (pullback.lift (𝟙 X) (𝟙 X) ⋯) g

Tactic:
g

Goal state after:
C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝⁷ : HasBinaryProducts C
inst✝⁶ : HasPullbacks D
inst✝⁵ : HasBinaryProducts D
inst✝⁴ : HasTerminal D
inst✝³ : F.Full
inst✝² : PreservesLimitsOfShape (Discrete WalkingPair) F
inst✝¹ : HasPullbacks C
inst✝ : PreservesLimitsOfShape WalkingCospan F
X : D
h : ∀ ⦃a : C⦄ (g : F.obj a ⟶ X), F.relativelyRepresentable g
a : C
g : F.obj a ⟶ Limits.pullback (terminal.from X) (terminal.from X)
⊢ ∃ b snd fst, IsPullback fst (F.map snd) (pullback.lift (𝟙 X) (𝟙 X) ⋯) g
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case mpr
p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ
⊢ ↑p ∣ k → ‖↑k‖ < 1

Tactic:
rintro ⟨x, rfl⟩
    push_cast
    rw [padicNormE.mul]
    calc
      _ ≤ ‖(p : ℚ_[p])‖ * 1 :=
        mul_le_mul le_rfl (by simpa using norm_int_le_one _) (norm_nonneg _) (norm_nonneg _)
      _ < 1 := by
        rw [mul_one, norm_p]
        exact inv_lt_one_of_one_lt₀ <| mod_cast hp.1.one_lt

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
⊢ ∃ b, f ≈ const norm b

Tactic:
have cau_seq_norm_e : IsCauSeq padicNormE f := fun ε hε => by
      have h := isCauSeq f ε (mod_cast hε)
      dsimp [norm] at h
      exact mod_cast h

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
cau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f
⊢ ∃ b, f ≈ const norm b
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: case succ.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
T : C
n : ℕ
f : X _⦋n + 1⦌ ⟶ T
hf : QInfty.f (n + 1) ≫ f = 0
this : (PInfty.f (n + 1) + QInfty.f (n + 1)) ≫ f = 𝟙 ((AlternatingFaceMapComplex.obj X).X (n + 1)) ≫ f
⊢ DegeneraciesVanish f

Tactic:
Category.id_comp

Goal state after:
case succ.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
T : C
n : ℕ
f : X _⦋n + 1⦌ ⟶ T
hf : QInfty.f (n + 1) ≫ f = 0
this : (PInfty.f (n + 1) + QInfty.f (n + 1)) ≫ f = f
⊢ DegeneraciesVanish f
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
n : ℕ
⊢ s.πSummand (IndexSet.id (op ⦋n⦌)) ≫ (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n =
    𝟙 (X _⦋n⦌) ≫ PInfty.f n

Tactic:
rw [s.decomposition_id, Preadditive.sum_comp]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
n : ℕ
⊢ s.πSummand (IndexSet.id (op ⦋n⦌)) ≫ (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n =
    ∑ j, (s.πSummand j ≫ (s.cofan (op ⦋n⦌)).inj j) ≫ PInfty.f n
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: case mk.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n m : ℕ
i : ⦋m⦌ ⟶ ⦋n⦌
hi : Mono i
h₁ : ⦋m⦌.len ≠ n
h₂ : ¬Isδ₀ i
k : ℕ
hk : ⦋n⦌.len = ⦋m⦌.len + (k + 1) + 1
⊢ PInfty.f n ≫ X.map i.op = 0

Tactic:
· simp only [← add_assoc] at hk
    clear h₂ hi
    subst hk
    obtain ⟨j₁ : Fin (_ + 1), i, rfl⟩ :=
      eq_comp_δ_of_not_surjective i fun h => by
        rw [← SimplexCategory.epi_iff_surjective] at h
        grind [→ le_of_epi]
    obtain ⟨j₂, i, rfl⟩ :=
      eq_comp_δ_of_not_surjective i fun h => by
        rw [← SimplexCategory.epi_iff_surjective] at h
        grind [→ le_of_epi]
    by_cases hj₁ : j₁ = 0
    · subst hj₁
      rw [assoc, ← SimplexCategory.δ_comp_δ'' (Fin.zero_le _)]
      simp only [op_comp, X.map_comp, assoc, PInfty_f]
      erw [(HigherFacesVanish.of_P _ _).comp_δ_eq_zero_assoc _ j₂.succ_ne_zero, zero_comp]
      simp only [Fin.succ]
      lia
    · simp only [op_comp, X.map_comp, assoc, PInfty_f]
      erw [(HigherFacesVanish.of_P _ _).comp_δ_eq_zero_assoc _ hj₁, zero_comp]
      by_contra
      exact hj₁ (by simp only [Fin.ext_iff, Fin.val_zero]; lia)

Goal state after:
Module: Mathlib.Topology.Homotopy.Contractible

Goal state before:
pp: case h
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : Nonempty Y
inst✝ : IndiscreteTopology Y
f : C(X, Y)
inhabited_h : Inhabited Y
⊢ f.Homotopic (const X default)

Tactic:
exact homotopic_of_indiscrete _ _

Goal state after:
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps

Goal state before:
pp: X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
⊢ (π.map (TopCat.ofHom f)).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧
    ⟦H.evalAt x₀⟧ ≫ (π.map (TopCat.ofHom g)).map p = H.diagonalPath' p

Tactic:
H.apply_zero_path,

Goal state after:
X Y : TopCat
f g : C(↑X, ↑Y)
H : f.Homotopy g
x₀ x₁ : ↑X
p : fromTop x₀ ⟶ fromTop x₁
⊢ (hcast ⋯ ≫ (π.map (TopCat.ofHom H.uliftMap)).map (prodToProdTopI (𝟙 (fromTop { down := 0 })) p) ≫ hcast ⋯) ≫
        ⟦H.evalAt x₁⟧ =
      H.diagonalPath' p ∧
    ⟦H.evalAt x₀⟧ ≫ (π.map (TopCat.ofHom g)).map p = H.diagonalPath' p
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
p : Path x₀ x₁
x : ↑I
x✝ : ↑x ≤ 2⁻¹ ∨ 2⁻¹ < ↑x
⊢ p ⟨if ↑x ≤ 1 / 2 then ↑1 * 2 * ↑x else ↑1 * (2 - 2 * ↑x), ⋯⟩ =
    { toFun := (fun t ↦ if t ≤ 1 / 2 then p.extend (2 * t) else p.symm.extend (2 * t - 1)) ∘ Subtype.val,
        continuous_toFun := ⋯ }
      x

Tactic:
cases le_or_gt (x : ℝ) 2⁻¹ with

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ModelCategory C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
F : BifibrantObject C ⥤ D
h : ∀ ⦃K L : BifibrantObject C⦄ (f g : K ⟶ L), homRel C f g → F.map f = F.map g
X Y : BifibrantObject C
f : X ⟶ Y
hf : WeakEquivalence f.hom
⊢ IsIso (F.map f)

Tactic:
obtain ⟨g', h₁, h₂⟩ := RightHomotopyClass.whitehead f.hom

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ModelCategory C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
F : BifibrantObject C ⥤ D
h : ∀ ⦃K L : BifibrantObject C⦄ (f g : K ⟶ L), homRel C f g → F.map f = F.map g
X Y : BifibrantObject C
f : X ⟶ Y
hf : WeakEquivalence f.hom
g' : Y.obj ⟶ X.obj
h₁ : RightHomotopyRel (f.hom ≫ g') (𝟙 X.obj)
h₂ : RightHomotopyRel (g' ≫ f.hom) (𝟙 Y.obj)
⊢ IsIso (F.map f)
Module: Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ModelCategory C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
K L : BifibrantObject C
f : K ⟶ L
⊢ ∀ (g : K ⟶ L), homRel C f g → toHoCat.map f = toHoCat.map g

Tactic:
g

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ModelCategory C
D : Type u_1
inst✝ : Category.{v_1, u_1} D
K L : BifibrantObject C
f g : K ⟶ L
⊢ homRel C f g → toHoCat.map f = toHoCat.map g
Module: Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
X Y : CofibrantObject C
f : X ⟶ Y
⊢ ∀ (f₂ : X ⟶ Y), homRel C f f₂ → bifibrantResolution'.map f = bifibrantResolution'.map f₂

Tactic:
g

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
X Y : CofibrantObject C
f g : X ⟶ Y
⊢ homRel C f g → bifibrantResolution'.map f = bifibrantResolution'.map g