mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X✝ : C
⊢ ∀ ⦃Y : C⦄ (f : X✝ ⟶ Y), transfiniteCompositions.{w, v, u} W f → W.rlp.llp f

Tactic:
_

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
X✝ Y✝ : C
⊢ ∀ (f : X✝ ⟶ Y✝), transfiniteCompositions.{w, v, u} W f → W.rlp.llp f
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
this✝² : LocallySmall.{w, v, u} C
this✝¹ : IsSmall.{w, v, u} I
this✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this : HasPushouts C
X✝ : Arrow C ⥤ Arrow C
⊢ ∀ ⦃Y : Arrow C ⥤ Arrow C⦄ (f : X✝ ⟶ Y), (succStruct I κ).prop f → (propArrow I).functorCategory (Arrow C) f

Tactic:
_

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
this✝² : LocallySmall.{w, v, u} C
this✝¹ : IsSmall.{w, v, u} I
this✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this : HasPushouts C
X✝ Y✝ : Arrow C ⥤ Arrow C
⊢ ∀ (f : X✝ ⟶ Y✝), (succStruct I κ).prop f → (propArrow I).functorCategory (Arrow C) f
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
this✝² : LocallySmall.{w, v, u} C
this✝¹ : IsSmall.{w, v, u} I
this✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this : HasPushouts C
X✝ Y✝ : Arrow C ⥤ Arrow C
⊢ ∀ (f : X✝ ⟶ Y✝), (succStruct I κ).prop f → (propArrow I).functorCategory (Arrow C) f

Tactic:
_

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
this✝² : LocallySmall.{w, v, u} C
this✝¹ : IsSmall.{w, v, u} I
this✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this : HasPushouts C
X✝ Y✝ : Arrow C ⥤ Arrow C
f✝ : X✝ ⟶ Y✝
⊢ (succStruct I κ).prop f✝ → (propArrow I).functorCategory (Arrow C) f✝
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n₁ n₂ d₁ d₂ : ℤ
hn₁ : n₁ ≠ 0
hn₂ : n₂ ≠ 0
hd₁ : d₁ ≠ 0
hd₂ : d₂ ≠ 0
hf1 : FiniteMultiplicity (↑p) (n₁ * d₂)
hf2 : FiniteMultiplicity (↑p) (n₂ * d₁)
⊢ padicValRat p (n₁ /. d₁) ≤ padicValRat p (n₂ /. d₂) ↔ ∀ (n : ℕ), ↑p ^ n ∣ n₁ * d₂ → ↑p ^ n ∣ n₂ * d₁

Tactic:
conv =>
    lhs
    rw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn₁ hd₁) rfl,
      padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn₂ hd₂) rfl, sub_le_iff_le_add', ←
      add_sub_assoc, le_sub_iff_add_le]
    norm_cast
    rw [← multiplicity_mul (Nat.prime_iff_prime_int.1 hp.1) hf1, add_comm,
        ← multiplicity_mul (Nat.prime_iff_prime_int.1 hp.1) hf2,
        hf1.multiplicity_le_multiplicity_iff hf2]

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
⊢ (monomorphisms C).HasFunctorialFactorization (monomorphisms C).rlp

Tactic:
have hG := isSeparator_separator C
  rw [← generatingMonomorphisms_rlp hG, ← llp_rlp_monomorphisms hG,
    ← generatingMonomorphisms_rlp hG]
  infer_instance

Goal state after:
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 r ≤ padicValRat p q
⊢ min (padicValRat p q) (padicValRat p r) ≤ padicValRat p (q + r)

Tactic:
by rw [min_eq_right h, add_comm]; exact le_padicValRat_add_of_le (by rwa [add_comm]) h

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
r : ℕ
⊢ -padicValRat p (↑p ^ r) = -↑r

Tactic:
neg_inj,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
r : ℕ
⊢ padicValRat p (↑p ^ r) = ↑r
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < ↑k
⊢ ε⁻¹ < (↑p ^ ↑k)⁻¹⁻¹

Tactic:
inv_inv,

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
ε : ℝ
hε : 0 < ε
k : ℕ
hk : ε⁻¹ < ↑k
⊢ ε⁻¹ < ↑p ^ ↑k
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
T : C
hT : Limits.IsTerminal T
F G : Sheaf J AddCommGrpCat
f : F ⟶ G
x : ↑(F.obj.obj (op T))
⊢ (map f 0) ((equiv₀ F hT).symm x) = (equiv₀ G hT).symm ((ConcreteCategory.hom (f.hom.app (op T))) x)

Tactic:
by
  apply (H.equiv₀ G hT).injective
  simp [← H.equiv₀_naturality]

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

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : g ≈ 0
⊢ (f * g).norm = f.norm * g.norm

Tactic:
by
      have hf : f * g ≈ 0 := mul_equiv_zero _ hg
      simp only [hf, hg, norm, dif_pos, mul_zero]

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
P Q : Cᵒᵖ ⥤ A
M N : A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
f : Φ.presheafFiber.obj P ⟶ M
X Y : Cᵒᵖ
g : X ⟶ Y
⊢ (P.map g ≫ Pi.lift fun x ↦ Φ.toPresheafFiber (unop Y) x P ≫ f) =
    (Pi.lift fun x ↦ Φ.toPresheafFiber (unop X) x P ≫ f) ≫ (Φ.skyscraperPresheaf M).map g

Tactic:
by
        dsimp
        ext y
        have := Φ.toPresheafFiber_w g.unop y P
        dsimp at this
        simp [reassoc_of% this]

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

Goal state before:
pp: case inr
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 F₂ : Cᵒᵖ ⥤ Type w
f : F₁ ⟶ F₂
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
this✝ :
  ∀ {F₁ F₂ : Cᵒᵖ ⥤ Type w} (f : F₁ ⟶ F₂) [Mono f],
    (∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))) →
      (∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U) → Presheaf.IsLocallySurjective J f
hF₂ : ¬∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U
U : C
s : ToType (F₂.obj (op U))
f' : pullback f (shrinkYonedaEquiv.symm s) ⟶ shrinkYoneda.{w, v, u}.obj U := pullback.snd f (shrinkYonedaEquiv.symm s)
hf' : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f'))
this : Presheaf.IsLocallySurjective J f'
⊢ Presheaf.imageSieve f' (shrinkYonedaObjObjEquiv.symm (𝟙 U)) ≤ Presheaf.imageSieve f s

Tactic:
rintro V g ⟨v, hv⟩

Goal state after:
case inr
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 F₂ : Cᵒᵖ ⥤ Type w
f : F₁ ⟶ F₂
inst✝ : Mono f
hf : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))
this✝ :
  ∀ {F₁ F₂ : Cᵒᵖ ⥤ Type w} (f : F₁ ⟶ F₂) [Mono f],
    (∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f))) →
      (∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U) → Presheaf.IsLocallySurjective J f
hF₂ : ¬∃ U, F₂ = shrinkYoneda.{w, v, u}.obj U
U : C
s : ToType (F₂.obj (op U))
f' : pullback f (shrinkYonedaEquiv.symm s) ⟶ shrinkYoneda.{w, v, u}.obj U := pullback.snd f (shrinkYonedaEquiv.symm s)
hf' : ∀ (Φ : P.FullSubcategory), Function.Surjective ⇑(ConcreteCategory.hom (Φ.obj.presheafFiber.map f'))
this : Presheaf.IsLocallySurjective J f'
V : C
g : V ⟶ U
v : ToType ((pullback f (shrinkYonedaEquiv.symm s)).obj (op V))
hv :
  (ConcreteCategory.hom (f'.app (op V))) v =
    (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.obj U).map g.op)) (shrinkYonedaObjObjEquiv.symm (𝟙 U))
⊢ (Presheaf.imageSieve f s).arrows g
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case inl
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
heq : q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε

pp: case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε

Tactic:
· simpa only [heq, PadicSeq.norm, dif_pos]

Goal state after:
case inr
p : ℕ
inst✝ : Fact (Nat.Prime p)
q : ℚ_[p]
ε : ℚ
hε : 0 < ε
q' : CauSeq ℚ (padicNorm p)
this : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
N : ℕ
hN : ∀ m ≥ N, ∀ n ≥ N, padicNorm p (↑q' m - ↑q' n) < ε
hne' : ¬q' - const (padicNorm p) (↑q' N) ≈ 0
⊢ (q' - const (padicNorm p) (↑q' N)).norm < ε
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
x✝ : ℚ_[p]
⊢ ∀ {y : ℚ_[p]}, ↑(padicNormE (x✝ - y)) = 0 → x✝ = y

Tactic:
_

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
x✝ y✝ : ℚ_[p]
⊢ ↑(padicNormE (x✝ - y✝)) = 0 → x✝ = y✝
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ_[p]
f : CauSeq ℚ (padicNorm p)
hf : ⟦f⟧ ≠ 0
this : ¬f ≈ 0
n : ℤ
hn : PadicSeq.norm f = ↑p ^ (-n)
⊢ ‖⟦f⟧‖ = ↑(↑p ^ (-n))

Tactic:
rw [← hn]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ_[p]
f : CauSeq ℚ (padicNorm p)
hf : ⟦f⟧ ≠ 0
this : ¬f ≈ 0
n : ℤ
hn : PadicSeq.norm f = ↑p ^ (-n)
⊢ ‖⟦f⟧‖ = ↑(PadicSeq.norm f)
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hxy : ¬x + y = 0
hx : ¬x = 0
hy : ¬y = 0
⊢ min ↑x.valuation ↑y.valuation ≤ ↑(x + y).valuation

Tactic:
← WithTop.coe_min,

Goal state after:
case neg
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℚ_[p]
hxy : ¬x + y = 0
hx : ¬x = 0
hy : ¬y = 0
⊢ ↑(min x.valuation y.valuation) ≤ ↑(x + y).valuation
Module: Mathlib.AlgebraicTopology.DoldKan.Decomposition

Goal state before:
pp: case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ PInfty.f (n + 1) ≫ (id X n).a = (P (n + 1)).f (n + 1)

pp: case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ (id X n).b i.rev = (Q (n + 1)).f (n + 1)

Tactic:
· simp only [id, PInfty_f, P_f_idem]

Goal state after:
case e_a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ (id X n).b i.rev = (Q (n + 1)).f (n + 1)
Module: Mathlib.AlgebraicTopology.DoldKan.Decomposition

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X X' : SimplicialObject C
n : ℕ
Z : C
f : MorphComponents X n Z
g : X' ⟶ X
⊢ (f.preComp g).φ = g.app (op ⦋n + 1⦌) ≫ f.φ

Tactic:
unfold φ preComp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X X' : SimplicialObject C
n : ℕ
Z : C
f : MorphComponents X n Z
g : X' ⟶ X
⊢ PInfty.f (n + 1) ≫ { a := g.app (op ⦋n + 1⦌) ≫ f.a, b := fun i ↦ g.app (op ⦋n⦌) ≫ f.b i }.a +
      ∑ i,
        (P ↑i).f (n + 1) ≫
          X'.δ i.rev.succ ≫ { a := g.app (op ⦋n + 1⦌) ≫ f.a, b := fun i ↦ g.app (op ⦋n⦌) ≫ f.b i }.b i.rev =
    g.app (op ⦋n + 1⦌) ≫ (PInfty.f (n + 1) ≫ f.a + ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ f.b i.rev)
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : SimplicialObject C
s : X.Splitting
f : X ⟶ Y
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ (s.cofan Δ).inj A ≫ f.app Δ = s.φ f (unop A.fst).len ≫ Y.map A.e.op

Tactic:
simp only [cofan_inj_eq_assoc, φ, assoc]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : SimplicialObject C
s : X.Splitting
f : X ⟶ Y
Δ : SimplexCategoryᵒᵖ
A : IndexSet Δ
⊢ s.ι (unop A.fst).len ≫ X.map A.e.op ≫ f.app Δ = s.ι (unop A.fst).len ≫ f.app (op ⦋(unop A.fst).len⦌) ≫ Y.map A.e.op
Module: Mathlib.AlgebraicTopology.DoldKan.Normalized

Goal state before:
pp: case succ
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
n : ℕ
i : Fin (n + 1)
⊢ i ∈ Finset.univ → (kernelSubobject (X.δ i.succ)).Factors ((P (n + 1)).f (n + 1))

Tactic:
_

Goal state after:
case succ
A : Type u_1
inst✝¹ : Category.{v_1, u_1} A
inst✝ : Abelian A
X : SimplicialObject A
n : ℕ
i : Fin (n + 1)
a✝ : i ∈ Finset.univ
⊢ (kernelSubobject (X.δ i.succ)).Factors ((P (n + 1)).f (n + 1))
Module: Mathlib.CategoryTheory.Idempotents.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
α : Type u_3
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : IsIdempotentComplete C
⊢ IsIdempotentComplete (HomologicalComplex C c)

Tactic:
rw [isIdempotentComplete_iff_of_equivalence
      ((toKaroubiEquivalence C).mapHomologicalComplex c),
    ← isIdempotentComplete_iff_of_equivalence (karoubiHomologicalComplexEquivalence C c)]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
α : Type u_3
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : IsIdempotentComplete C
⊢ IsIdempotentComplete (Karoubi (HomologicalComplex C c))
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Δ : SimplexCategoryᵒᵖ
A B : IndexSet Δ
⊢ B ∈ Finset.univ → B ≠ A → (s.cofan Δ).inj A ≫ s.πSummand B ≫ (s.cofan Δ).inj B = 0

Tactic:
_

Goal state after:
case h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Δ : SimplexCategoryᵒᵖ
A B : IndexSet Δ
a✝ : B ∈ Finset.univ
⊢ B ≠ A → (s.cofan Δ).inj A ≫ s.πSummand B ≫ (s.cofan Δ).inj B = 0
Module: Mathlib.AlgebraicTopology.DoldKan.FunctorGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
j : Fin (n + 1)
⊢ n + 1 ≤ ↑j + (n + 1) →
    ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫ (Γ₀.obj K).δ j.succ = 0

Tactic:
_

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
j : Fin (n + 1)
a✝ : n + 1 ≤ ↑j + (n + 1)
⊢ ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫ (Γ₀.obj K).δ j.succ = 0
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case mp.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Z : C
n : ℕ
f : Z ⟶ X _⦋n + 1⦌
h : f ≫ PInfty.f (n + 1) = 0
h' : f ≫ QInfty.f (n + 1) = f
i : Fin (n + 1)
⊢ i ∈ {i | ↑i < n + 1} → f ≫ ((P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev) ≫ s.πSummand (IndexSet.id (op ⦋n + 1⦌)) = 0

Tactic:
_

Goal state after:
case mp.succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
Z : C
n : ℕ
f : Z ⟶ X _⦋n + 1⦌
h : f ≫ PInfty.f (n + 1) = 0
h' : f ≫ QInfty.f (n + 1) = f
i : Fin (n + 1)
a✝ : i ∈ {i | ↑i < n + 1}
⊢ f ≫ ((P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev) ≫ s.πSummand (IndexSet.id (op ⦋n + 1⦌)) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n : ℕ
i : ⦋n⦌ ⟶ ⦋n⦌
inst✝ : Mono i
⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ PInfty.f n = PInfty.f n ≫ X.map i.op

Tactic:
simp only [SimplexCategory.eq_id_of_mono i, Γ₀.Obj.Termwise.mapMono_id, op_id, X.map_id]

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n : ℕ
i : ⦋n⦌ ⟶ ⦋n⦌
inst✝ : Mono i
⊢ 𝟙 (K[X].X n) ≫ PInfty.f n = PInfty.f n ≫ 𝟙 (X _⦋n⦌)
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: case pos.h₀
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n : ℕ
i : ⦋n⦌ ⟶ ⦋n + 1⦌
inst✝ : Mono i
h : ¬n = n + 1
hi : Isδ₀ i
b : Fin (n + 2)
⊢ b ∈ Finset.univ → b ≠ 0 → PInfty.f (n + 1) ≫ ((-1) ^ ↑b • X.δ b) = 0

Tactic:
_

Goal state after:
case pos.h₀
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n : ℕ
i : ⦋n⦌ ⟶ ⦋n + 1⦌
inst✝ : Mono i
h : ¬n = n + 1
hi : Isδ₀ i
b : Fin (n + 2)
a✝ : b ∈ Finset.univ
⊢ b ≠ 0 → PInfty.f (n + 1) ≫ ((-1) ^ ↑b • X.δ b) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject

Goal state before:
pp: case h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
A : IndexSet (op ⦋i⦌)
⊢ A ∈ Finset.univ →
    A ≠ IndexSet.id (op ⦋i⦌) →
      ((s.πSummand A ≫ (s.cofan (op ⦋i⦌)).inj A) ≫ K[X].d i j) ≫ s.πSummand (IndexSet.id (op ⦋j⦌)) = 0

Tactic:
_

Goal state after:
case h₀
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X : SimplicialObject C
s : X.Splitting
inst✝ : Preadditive C
i j : ℕ
x✝ : (ComplexShape.down ℕ).Rel i j
A : IndexSet (op ⦋i⦌)
a✝ : A ∈ Finset.univ
⊢ A ≠ IndexSet.id (op ⦋i⦌) →
    ((s.πSummand A ≫ (s.cofan (op ⦋i⦌)).inj A) ≫ K[X].d i j) ≫ s.πSummand (IndexSet.id (op ⦋j⦌)) = 0
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.Product

Goal state before:
pp: case h_map
I : Type u
X : I → TopCat
X✝ : ↑(π.obj (TopCat.of ((i : I) → ↑(X i))))
⊢ ∀ (Y : ↑(π.obj (TopCat.of ((i : I) → ↑(X i))))) (f : X✝ ⟶ Y),
    (CategoryTheory.Functor.pi' (proj X) ⋙ piToPiTop X).map f =
      CategoryTheory.eqToHom ⋯ ≫ (𝟭 ↑(π.obj (TopCat.of ((i : I) → ↑(X i))))).map f ≫ CategoryTheory.eqToHom ⋯

Tactic:
_

Goal state after:
case h_map
I : Type u
X : I → TopCat
X✝ Y✝ : ↑(π.obj (TopCat.of ((i : I) → ↑(X i))))
⊢ ∀ (f : X✝ ⟶ Y✝),
    (CategoryTheory.Functor.pi' (proj X) ⋙ piToPiTop X).map f =
      CategoryTheory.eqToHom ⋯ ≫ (𝟭 ↑(π.obj (TopCat.of ((i : I) → ↑(X i))))).map f ≫ CategoryTheory.eqToHom ⋯
Module: Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X : C
⊢ ∃ X', ∃ (_ : IsCofibrant X'), ∃ p, Fibration p ∧ WeakEquivalence p

Tactic:
have h := MorphismProperty.factorizationData (cofibrations C) (trivialFibrations C)
    (initial.to X)
  refine ⟨h.Z, ?_, h.p, inferInstance, inferInstance⟩
  rw [isCofibrant_iff_of_isInitial h.i initialIsInitial]
  infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: case mp.inl_inl
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
x₀✝ y₀✝ : S
h : (ConcreteCategory.hom g) x₀✝ = (ConcreteCategory.hom g) y₀✝
⊢ (ConcreteCategory.hom f) x₀✝ = (ConcreteCategory.hom f) y₀✝ ∨
    ∃ x₀ y₀,
      ∃ (_ : (ConcreteCategory.hom g) x₀ = (ConcreteCategory.hom g) y₀),
        (ConcreteCategory.hom f) x₀✝ = (ConcreteCategory.hom f) x₀ ∧
          (ConcreteCategory.hom f) y₀✝ = (ConcreteCategory.hom f) y₀

Tactic:
· exact Or.inr ⟨_, _, h, rfl, rfl⟩

Goal state after:
Module: Mathlib.CategoryTheory.Types.Monomorphisms

Goal state before:
pp: T : Type u'
X₁ X₂ : T → Type u
⊢ ∀ [inst : HasCoproduct X₁] [inst_1 : HasCoproduct X₂] (f : (j : T) → X₁ j ⟶ X₂ j),
    (∀ (j : T), monomorphisms (Type u) (f j)) → monomorphisms (Type u) (Limits.Sigma.map f)

Tactic:
_

Goal state after:
T : Type u'
X₁ X₂ : T → Type u
inst✝ : HasCoproduct X₁
⊢ ∀ [inst : HasCoproduct X₂] (f : (j : T) → X₁ j ⟶ X₂ j),
    (∀ (j : T), monomorphisms (Type u) (f j)) → monomorphisms (Type u) (Limits.Sigma.map f)
Module: Mathlib.CategoryTheory.Types.Monomorphisms

Goal state before:
pp: T : Type u'
X₁ X₂ : T → Type u
inst✝ : HasCoproduct X₁
⊢ ∀ [inst : HasCoproduct X₂] (f : (j : T) → X₁ j ⟶ X₂ j),
    (∀ (j : T), monomorphisms (Type u) (f j)) → monomorphisms (Type u) (Limits.Sigma.map f)

Tactic:
_

Goal state after:
T : Type u'
X₁ X₂ : T → Type u
inst✝¹ : HasCoproduct X₁
inst✝ : HasCoproduct X₂
⊢ ∀ (f : (j : T) → X₁ j ⟶ X₂ j), (∀ (j : T), monomorphisms (Type u) (f j)) → monomorphisms (Type u) (Limits.Sigma.map f)
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex

Goal state before:
pp: Z : SSet
h :
  ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 2)⦄ (f : (j : Fin (n + 2)) → j ≠ i → (Δ[n] ⟶ Z)),
    horn.IsCompatible f → ∃ φ, ∀ (j : Fin (n + 2)) (hj : j ≠ i), stdSimplex.δ j ≫ φ = f j hj
X✝ : SSet
⊢ ∀ ⦃Y : SSet⦄ (g : X✝ ⟶ Y), J g → HasLiftingProperty g (terminal.from Z)

Tactic:
_

Goal state after:
Z : SSet
h :
  ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 2)⦄ (f : (j : Fin (n + 2)) → j ≠ i → (Δ[n] ⟶ Z)),
    horn.IsCompatible f → ∃ φ, ∀ (j : Fin (n + 2)) (hj : j ≠ i), stdSimplex.δ j ≫ φ = f j hj
X✝ Y✝ : SSet
⊢ ∀ (g : X✝ ⟶ Y✝), J g → HasLiftingProperty g (terminal.from Z)
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex

Goal state before:
pp: Z : SSet
h :
  ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 2)⦄ (f : (j : Fin (n + 2)) → j ≠ i → (Δ[n] ⟶ Z)),
    horn.IsCompatible f → ∃ φ, ∀ (j : Fin (n + 2)) (hj : j ≠ i), stdSimplex.δ j ≫ φ = f j hj
X✝ Y✝ : SSet
⊢ ∀ (g : X✝ ⟶ Y✝), J g → HasLiftingProperty g (terminal.from Z)

Tactic:
_

Goal state after:
Z : SSet
h :
  ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 2)⦄ (f : (j : Fin (n + 2)) → j ≠ i → (Δ[n] ⟶ Z)),
    horn.IsCompatible f → ∃ φ, ∀ (j : Fin (n + 2)) (hj : j ≠ i), stdSimplex.δ j ≫ φ = f j hj
X✝ Y✝ : SSet
g✝ : X✝ ⟶ Y✝
⊢ J g✝ → HasLiftingProperty g✝ (terminal.from Z)
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: case pos
X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h : IsPushout t l r b
x₄ : X₄
x₃ : X₃
hx₃ : (ConcreteCategory.hom b) x₃ = x₄
h₂ : x₃ ∈ Set.range ⇑(ConcreteCategory.hom l)
⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨
    ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)

pp: case neg
X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h : IsPushout t l r b
x₄ : X₄
x₃ : X₃
hx₃ : (ConcreteCategory.hom b) x₃ = x₄
h₂ : x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)
⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨
    ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)

Tactic:
· obtain ⟨x₁, rfl⟩ := h₂
      exact Or.inl ⟨t x₁, by simpa only [← hx₃] using! ConcreteCategory.congr_hom h.w x₁⟩

Goal state after:
case neg
X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h : IsPushout t l r b
x₄ : X₄
x₃ : X₃
hx₃ : (ConcreteCategory.hom b) x₃ = x₄
h₂ : x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)
⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨
    ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X : SSet
d n : ℕ
h : d < n
x : X _⦋d⦌
⊢ x ∈ ⊤ → x ∈ (X.skeleton n).obj (op ⦋d⦌)

Tactic:
_

Goal state after:
X : SSet
d n : ℕ
h : d < n
x : X _⦋d⦌
a✝ : x ∈ ⊤
⊢ x ∈ (X.skeleton n).obj (op ⦋d⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X : SSet
n : ℕ
x : ↑(X.nonDegenerate n)
⊢ ↑x ∈ ⊤.obj (op ⦋n⦌) → ↑x ∈ (⨆ n, X.skeleton n).obj (op ⦋n⦌)

Tactic:
_

Goal state after:
X : SSet
n : ℕ
x : ↑(X.nonDegenerate n)
a✝ : ↑x ∈ ⊤.obj (op ⦋n⦌)
⊢ ↑x ∈ (⨆ n, X.skeleton n).obj (op ⦋n⦌)