mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝¹ : LinearOrder ι
f : P.RankFunction ι
inst✝ : P.IsProper
j : ι
c : f.Cell j
s : X.N
hs : s.subcomplex ≤ c.horn.image c.map
hs' : s.subcomplex ≤ A
⊢ A ≤ f.filtration j

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.ExtremalEpi

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
f : X ⟶ Y
inst✝ : StrongEpi f
Z : C
p : X ⟶ Z
i : Z ⟶ Y
fac : p ≫ i = f
x✝ : Mono i
sq : CommSq p f i (𝟙 Y)
⊢ sq.lift ≫ i = 𝟙 Y

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.ExtremalEpi

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
f : X ⟶ Y
inst✝ : HasPullbacks C
x✝¹ : ExtremalEpi f
A B : C
i : A ⟶ B
x✝ : Mono i
t : X ⟶ A
b : Y ⟶ B
sq : CommSq t f i b
⊢ pullback.lift t f ⋯ ≫ pullback.snd i b = f

Tactic:
by simp

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u'
inst✝³ : Category.{v', u'} K
F : K ⥤ C ⥤ Type w'
c : Cone F
hc : (Y : C) → IsLimit (((evaluation C (Type w')).obj Y).mapCone c)
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
hK : HasCardinalLT (Arrow K) κ
J : Type w
inst✝¹ : SmallCategory J
inst✝ : IsCardinalFiltered J κ
X : J ⥤ C
cX : Cocone X
hF : (k : K) → IsColimit ((F.obj k).mapCocone cX)
j : J
x₁ x₂ : c.pt.obj (X.obj j)
h : (ConcreteCategory.hom (c.pt.map (cX.ι.app j))) x₁ = (ConcreteCategory.hom (c.pt.map (cX.ι.app j))) x₂
⊢ ∃ j' α, (ConcreteCategory.hom (c.pt.map (X.map α))) x₁ = (ConcreteCategory.hom (c.pt.map (X.map α))) x₂

Tactic:
by
  have := isFiltered_of_isCardinalFiltered J κ
  let y₁ := Types.isLimitEquivSections (hc (X.obj j)) x₁
  let y₂ := Types.isLimitEquivSections (hc (X.obj j)) x₂
  have hy₁ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₁ = x₁ := by simp [y₁]
  have hy₂ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₂ = x₂ := by simp [y₂]
  have H (k : K) := (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k)
    (x := y₁.1 k) (y := y₂.1 k)).1 (by
      simp only [y₁, y₂, Types.isLimitEquivSections_apply]
      dsimp at h ⊢
      simp only [← NatTrans.naturality_apply, h])
  let j₁ (k : K) : J := (H k).choose
  let f (k : K) : j ⟶ j₁ k := (H k).choose_spec.choose
  have hf (k : K) : (F.obj k).map (X.map (f k)) (y₁.1 k) =
      (F.obj k).map (X.map (f k)) (y₂.1 k) :=
    (H k).choose_spec.choose_spec
  have hK' := hasCardinalLT_of_hasCardinalLT_arrow hK
  let ψ (k : K) : j ⟶ IsCardinalFiltered.max j₁ hK' :=
    f k ≫ IsCardinalFiltered.toMax j₁ hK' k
  refine ⟨IsCardinalFiltered.coeq ψ hK', IsCardinalFiltered.toCoeq ψ hK', ?_⟩
  apply (Types.isLimitEquivSections (hc _)).injective
  ext k
  simp only [Types.isLimitEquivSections_apply, ← hy₁, ← hy₂]
  have h₁ := Types.isLimitEquivSections_symm_apply (hc (X.obj j)) y₁ k
  have h₂ := Types.isLimitEquivSections_symm_apply (hc (X.obj j)) y₂ k
  dsimp at h₁ h₂ ⊢
  simp [h₁, h₂, ← IsCardinalFiltered.coeq_condition ψ hK' k, ψ, hf]

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : LocallySmall.{w, v, u} C
K : Type u'
inst✝³ : Category.{v', u'} K
inst✝² : HasLimitsOfShape Kᵒᵖ (Type w)
Y : K ⥤ C
c : Cocone Y
hc : IsColimit c
κ : Cardinal.{w}
inst✝¹ : Fact κ.IsRegular
hK : HasCardinalLT (Arrow K) κ
inst✝ : ∀ (k : K), IsCardinalPresentable (Y.obj k) κ
⊢ IsCardinalPresentable c.pt κ

Tactic:
let e := ShrinkHoms.equivalence.{w} C
  have (k : K) : IsCardinalPresentable ((Y ⋙ e.functor).obj k) κ := by
    dsimp; infer_instance
  rw [← isCardinalPresentable_iff_of_isEquivalence c.pt κ e.functor]
  exact isCardinalPresentable_of_isColimit' _
    (isColimitOfPreserves e.functor hc) κ hK

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝³ : LinearOrder ι
f : P.RankFunction ι
inst✝² : P.IsProper
inst✝¹ : SuccOrder ι
inst✝ : NoMaxOrder ι
j : ι
s : (range (f.m j)).N
⊢ ∀ ⦃a₂ : (range (f.m j)).N⦄, f.mapN s = f.mapN a₂ → s = a₂

Tactic:
t

Goal state after:
X : SSet
A : X.Subcomplex
P : A.Pairing
ι : Type v
inst✝³ : LinearOrder ι
f : P.RankFunction ι
inst✝² : P.IsProper
inst✝¹ : SuccOrder ι
inst✝ : NoMaxOrder ι
j : ι
s t : (range (f.m j)).N
⊢ f.mapN s = f.mapN t → s = t
Module: Mathlib.CategoryTheory.Presentable.StrongGenerator

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : LocallySmall.{w, v, u} C
J : Type u'
inst✝⁵ : Category.{v', u'} J
inst✝⁴ : EssentiallySmall.{w, v', u'} J
F : J ⥤ C
inst✝³ : F.IsDense
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : ∀ (j : J), IsCardinalPresentable (F.obj j) κ
inst✝ : ∀ (X : C), IsCardinalFiltered (CostructuredArrow F X) κ
X : C
ip : F.DenseAt X
e : CostructuredArrow F X ≌ SmallModel.{w, v', max u' v} (CostructuredArrow F X) :=
  equivSmallModel (CostructuredArrow F X)
j : SmallModel.{w, v', max u' v} (CostructuredArrow F X)
⊢ ⊤ ((CostructuredArrow.proj F X).obj (e.symm.functor.obj j))

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Dense

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : F.IsDense
Y : D
⊢ ∀ (j : ShrinkHoms.{max (max u₁ v₁) v₂} (Shrink.{max (max u₁ v₁) v₂, max u₁ v₂} (CostructuredArrow F Y))),
    ofObj F.obj
      ((((ShrinkHoms.equivalence (Shrink.{max (max u₁ v₁) v₂, max u₁ v₂} (CostructuredArrow F Y))).symm.trans
                (Shrink.equivalence (CostructuredArrow F Y)).symm).functor ⋙
            CostructuredArrow.proj F Y ⋙ F).obj
        j)

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Presentable.Presheaf

Goal state before:
pp: A : Type u'
inst✝³ : Category.{v', u'} A
P : ObjectProperty A
inst✝² : HasCoproducts A
inst✝¹ : HasPullbacks A
C : Type w
inst✝ : SmallCategory C
hP₁ : P.IsSeparating
hP₂ : ∀ ⦃X Y : A⦄ (i : X ⟶ Y) [Mono i], (∀ (G : A), P G → Function.Surjective fun f ↦ f ≫ i) → IsIso i
P₁ P₂ : Cᵒᵖ ⥤ A
i : P₁ ⟶ P₂
x✝ : Mono i
hi : ∀ (G : Cᵒᵖ ⥤ A), ObjectProperty.ofObj (fun T ↦ freeYoneda T.1 ↑T.2) G → Function.Surjective fun f ↦ f ≫ i
⊢ IsIso i

Tactic:
rw [NatTrans.isIso_iff_isIso_app]

Goal state after:
A : Type u'
inst✝³ : Category.{v', u'} A
P : ObjectProperty A
inst✝² : HasCoproducts A
inst✝¹ : HasPullbacks A
C : Type w
inst✝ : SmallCategory C
hP₁ : P.IsSeparating
hP₂ : ∀ ⦃X Y : A⦄ (i : X ⟶ Y) [Mono i], (∀ (G : A), P G → Function.Surjective fun f ↦ f ≫ i) → IsIso i
P₁ P₂ : Cᵒᵖ ⥤ A
i : P₁ ⟶ P₂
x✝ : Mono i
hi : ∀ (G : Cᵒᵖ ⥤ A), ObjectProperty.ofObj (fun T ↦ freeYoneda T.1 ↑T.2) G → Function.Surjective fun f ↦ f ≫ i
⊢ ∀ (X : Cᵒᵖ), IsIso (i.app X)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic

Goal state before:
pp: X : SSet
A : X.Subcomplex
left✝ : Mono A.ι
P : (Subcomplex.range A.ι).Pairing
h✝ : P.IsRegular
⊢ Subcomplex.range A.ι = A

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.Basic

Goal state before:
pp: X : SSet
A : X.Subcomplex
left✝ : Mono A.ι
P₁ : (Subcomplex.range A.ι).Pairing
w✝ : P₁.IsRegular
P₂ : P₁.IsInner
⊢ Subcomplex.range A.ι = A

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.Basic

Goal state before:
pp: X✝ : SSet
A : X✝.Subcomplex
P : A.Pairing
inst✝¹ : P.IsRegular
inst✝ : P.IsInner
j : ℕ
hj : ¬IsMax j
X Y : SSet
c : P.rankFunction.Cell j
h0 : 0 < ⋯.index ⋯
⊢ SSet.innerAnodyneExtensions (P.rankFunction.basicCell j c)

Tactic:
have hn := Fin.lt_last_iff_ne_last.mpr (IsInner.ne_last c.s rfl)

Goal state after:
X✝ : SSet
A : X✝.Subcomplex
P : A.Pairing
inst✝¹ : P.IsRegular
inst✝ : P.IsInner
j : ℕ
hj : ¬IsMax j
X Y : SSet
c : P.rankFunction.Cell j
h0 : 0 < ⋯.index ⋯
hn : ⋯.index ⋯ < Fin.last ((↑c.s).dim + 1)
⊢ SSet.innerAnodyneExtensions (P.rankFunction.basicCell j c)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
⊢ ∀ (a : A.op.N), a ∈ ⇑N.opEquiv ⁻¹' P.II ↔ N.opEquiv.toEquiv a ∈ P.II

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
⊢ ∀ (a : A.N), a ∈ P.I ↔ N.opEquiv.symm.toEquiv a ∈ ⇑N.opEquiv ⁻¹' P.I

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
x : ↑P.II
⊢ N.opEquiv.symm ↑(P.p x) ∈ P.op.I

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Op

Goal state before:
pp: X : SSet
A : X.Subcomplex
P : A.Pairing
x y : ↑P.II
⊢ ↑⟨N.opEquiv.symm ↑x, ⋯⟩ < ↑(P.op.p ⟨N.opEquiv.symm ↑y, ⋯⟩) ↔ ↑x < ↑(P.p y)

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H✝ : Homotopy f g
n : ℕ
α : Fin (n + 1) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • X.δ x.2 ≫ H✝.h x.1
β : Fin (n + 3) × Fin (n + 2) → (X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌) := fun x ↦ (-1) ^ (↑x.1 + ↑x.2) • H✝.h x.2 ≫ Y.δ x.1
h₂✝ : ∀ (x : Fin (n + 1) × Fin (n + 2)), x.1.castSucc < x.2 → α x = -β (x.2.succ, x.1.castSucc)
S : Finset (Fin (n + 1) × Fin (n + 2)) := {x | x.1.castSucc < x.2}
γ₁ : Fin (n + 1) × Fin (n + 2) → Fin (n + 2 + 1) × Fin (n + 1 + 1) := fun x ↦ (x.2.castSucc, x.1.succ)
γ₂ : Fin (n + 1) × Fin (n + 2) → Fin (n + 2 + 1) × Fin (n + 1 + 1) := fun x ↦ (x.2.succ, x.1.castSucc)
γ₃ : Fin (n + 1) → Fin (n + 1 + 1 + 1) × Fin (n + 1 + 1) := fun i ↦ (i.castSucc.succ, i.succ)
γ₄ : Fin (n + 1) → Fin (n + 1 + 1 + 1) × Fin (n + 1 + 1) := fun i ↦ (i.castSucc.succ, i.castSucc)
hγ₁ : Function.Injective γ₁
hγ₂ : Function.Injective γ₂
hγ₃ : Function.Injective γ₃
hγ₄ : Function.Injective γ₄
eq₁ : H✝.h 0 ≫ Y.δ 0 = β (0, 0)
eq₂ : H✝.h (Fin.last (n + 1)) ≫ Y.δ (Fin.last (n + 1 + 1)) = -β (Fin.last (n + 2), Fin.last (n + 1))
eq₃ : ∑ x ∈ Sᶜ, α x = -∑ y ∈ Finset.image γ₁ Sᶜ, β y
eq₄ : ∑ x ∈ S, α x = -∑ y ∈ Finset.image γ₂ S, β y
eq₅ : ∑ x, β (γ₄ x) = -∑ x, β (γ₃ x)
h₁ : Disjoint (Finset.image γ₁ Sᶜ) (Finset.image γ₂ S)
h₂ : Disjoint (Finset.image γ₃ Finset.univ) (Finset.image γ₄ Finset.univ)
h₃ :
  Disjoint ((Finset.image γ₃ Finset.univ).disjUnion (Finset.image γ₄ Finset.univ) h₂)
    {(0, 0), (Fin.last (n + 2), Fin.last (n + 1))}
h₄ :
  Disjoint ((Finset.image γ₁ Sᶜ).disjUnion (Finset.image γ₂ S) h₁)
    (((Finset.image γ₃ Finset.univ).disjUnion (Finset.image γ₄ Finset.univ) h₂).disjUnion
      {(0, 0), (Fin.last (n + 2), Fin.last (n + 1))} h₃)
H :
  ((Finset.image γ₁ Sᶜ).disjUnion (Finset.image γ₂ S) h₁)ᶜ =
    ((Finset.image γ₃ Finset.univ).disjUnion (Finset.image γ₄ Finset.univ) h₂).disjUnion
      {(0, 0), (Fin.last (n + 2), Fin.last (n + 1))} h₃
⊢ (0, 0) ∉ {(Fin.last (n + 2), Fin.last (n + 1))}

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H✝ H : Homotopy f g
f✝ : (i j : ℕ) → ((alternatingFaceMapComplex C).obj X).X i ⟶ ((alternatingFaceMapComplex C).obj Y).X j
⊢ (ComplexShape.down ℕ).Rel 1 0

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H✝ H : Homotopy f g
⊢ ¬(ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H✝ H : Homotopy f g
n : ℕ
f✝ : (i j : ℕ) → ((alternatingFaceMapComplex C).obj X).X i ⟶ ((alternatingFaceMapComplex C).obj Y).X j
⊢ (ComplexShape.down ℕ).Rel (n + 1) n

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X Y : SimplicialObject C
f g : X ⟶ Y
H✝ H : Homotopy f g
n : ℕ
f✝ : (i j : ℕ) → ((alternatingFaceMapComplex C).obj X).X i ⟶ ((alternatingFaceMapComplex C).obj Y).X j
⊢ (ComplexShape.down ℕ).Rel (n + 2) (n + 1)

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.LiftingProperties.ParametrizedAdjunction

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
X₃ Y₃ : C₃
f₃ : X₃ ⟶ Y₃
sq₁₂ : F.PushoutObjObj f₁ f₂
sq₁₃ : G.PullbackObjObj f₁ f₃
β : Arrow.mk f₂ ⟶ Arrow.mk sq₁₃.π
⊢ ⋯.desc (adj₂.homEquiv.symm (Arrow.Hom.left β)) (adj₂.homEquiv.symm (Arrow.Hom.right β ≫ sq₁₃.fst)) ⋯ ≫
      (Arrow.mk f₃).hom =
    (Arrow.mk sq₁₂.ι).hom ≫ adj₂.homEquiv.symm (Arrow.Hom.right β ≫ sq₁₃.snd)

Tactic:
by
        apply sq₁₂.isPushout.hom_ext
        · have := Arrow.w β =≫ sq₁₃.snd
          dsimp at this
          simp only [Category.assoc, sq₁₃.π_snd] at this
          simp [← adj₂.homEquiv_symm_naturality_two,
            ← adj₂.homEquiv_symm_naturality_three, this]
        · simp [← adj₂.homEquiv_symm_naturality_one,
            ← adj₂.homEquiv_symm_naturality_three, sq₁₃.isPullback.w]

Goal state after:
Module: Mathlib.CategoryTheory.Adjunction.ParametrizedLimits

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C₁
inst✝³ : Category.{v_2, u_2} C₂
inst✝² : Category.{v_3, u_3} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
J : Type u_4
inst✝¹ : Category.{v_4, u_4} J
P : J ⥤ C₁ᵒᵖ
inst✝ : ∀ (X₂ : C₂), PreservesColimit P.leftOp (F.flip.obj X₂)
X₃ : C₃
c : Cone P
hc : IsLimit c
cocone : (s : Cone (P ⋙ G.flip.obj X₃)) → Cocone (P.leftOp ⋙ F.flip.obj s.pt) :=
  fun s ↦ { pt := X₃, ι := { app := fun j ↦ adj₂.homEquiv.symm (s.π.app (unop j)), naturality := ⋯ } }
hc' : (s : Cone (P ⋙ G.flip.obj X₃)) → IsColimit ((F.flip.obj s.pt).mapCocone (coconeLeftOpOfCone c)) :=
  fun s ↦ isColimitOfPreserves (F.flip.obj s.pt) (isColimitCoconeLeftOpOfCone P hc)
s : Cone (P ⋙ G.flip.obj X₃)
j : J
⊢ adj₂.homEquiv ((hc' s).desc (cocone s)) ≫ (G.map (c.π.app j)).app X₃ = s.π.app j

Tactic:
rw [← dsimp% adj₂.homEquiv_naturality_one (c.π.app j).unop,
          dsimp% (hc' s).fac (cocone s) (op j)]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C₁
inst✝³ : Category.{v_2, u_2} C₂
inst✝² : Category.{v_3, u_3} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
adj₂ : F ⊣₂ G
J : Type u_4
inst✝¹ : Category.{v_4, u_4} J
P : J ⥤ C₁ᵒᵖ
inst✝ : ∀ (X₂ : C₂), PreservesColimit P.leftOp (F.flip.obj X₂)
X₃ : C₃
c : Cone P
hc : IsLimit c
cocone : (s : Cone (P ⋙ G.flip.obj X₃)) → Cocone (P.leftOp ⋙ F.flip.obj s.pt) :=
  fun s ↦ { pt := X₃, ι := { app := fun j ↦ adj₂.homEquiv.symm (s.π.app (unop j)), naturality := ⋯ } }
hc' : (s : Cone (P ⋙ G.flip.obj X₃)) → IsColimit ((F.flip.obj s.pt).mapCocone (coconeLeftOpOfCone c)) :=
  fun s ↦ isColimitOfPreserves (F.flip.obj s.pt) (isColimitCoconeLeftOpOfCone P hc)
s : Cone (P ⋙ G.flip.obj X₃)
j : J
⊢ adj₂.homEquiv ((cocone s).ι.app (op j)) = s.π.app j
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
⊢ (φ x hd (min x hd).castSucc).1 < (φ x hd (min x hd).succ).1

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
⊢ (φ x hd (min x hd).castSucc).2 ≤ (φ x hd (min x hd).succ).2

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.PushoutProduct

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left) (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom) ≫
      pushout.desc
          ((α_ X₁.right X₂.right X₃.left).hom ≫
            X₁.right ◁
                pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
              pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
          (Arrow.Hom.left (whiskerRightIso X₁ X₂).hom ≫
            pushout.desc
              (X₁.right ◁
                  pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left)
                    (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
              (pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
              ⋯)
          ⋯ ≫
        pushout.desc
          (Arrow.Hom.left (whiskerLeftIso X₂ X₃).hom ≫
            pushout.desc
              (pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
              (pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                  X₃.right ≫
                pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
              ⋯)
          ((α_ X₁.left X₂.right X₃.right).inv ≫
            pushout.inr (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                X₃.right ≫
              pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
          ⋯ =
    pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
        (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom) ≫
      𝟙 ((X₁ □ X₂) □ X₃).left

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.PushoutProduct

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left) (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
      pushout.desc
          (Arrow.Hom.left (whiskerLeftIso X₂ X₃).hom ≫
            pushout.desc
              (pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
              (pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                  X₃.right ≫
                pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                  (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
              ⋯)
          ((α_ X₁.left X₂.right X₃.right).inv ≫
            pushout.inr (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷
                X₃.right ≫
              pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
                (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom))
          ⋯ ≫
        pushout.desc
          ((α_ X₁.right X₂.right X₃.left).hom ≫
            X₁.right ◁
                pushout.inl (((curriedTensor C).map X₂.hom).app X₃.left) (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
              pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
          (Arrow.Hom.left (whiskerRightIso X₁ X₂).hom ≫
            pushout.desc
              (X₁.right ◁
                  pushout.inr (((curriedTensor C).map X₂.hom).app X₃.left)
                    (((curriedTensor C).obj X₂.left).map X₃.hom) ≫
                pushout.inl (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                  (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
              (pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
                (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom))
              ⋯)
          ⋯ =
    pushout.inr (((curriedTensor C).map X₁.hom).app (X₂ □ X₃).left)
        (((curriedTensor C).obj X₁.left).map (X₂ □ X₃).hom) ≫
      𝟙 (X₁ □ X₂ □ X₃).left

Tactic:
by simp

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
d : ℕ
hd : x.dim = d + 1
l : Fin (d + 1)
hl : IsIndex x hd l.succ
⊢ ∀ (l_1 : Fin (hl.δ.dim + 1)), ¬IsIndex hl.δ ⋯ l_1

Tactic:
t

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
d : ℕ
hd : x.dim = d + 1
l : Fin (d + 1)
hl : IsIndex x hd l.succ
t : Fin (hl.δ.dim + 1)
⊢ ¬IsIndex hl.δ ⋯ t
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
pt : C₃
inl : (F.obj Y₁).obj X₂ ⟶ pt
inr : (F.obj X₁).obj Y₂ ⟶ pt
isPushout : IsPushout ((F.map f₁).app X₂) ((F.obj X₁).map f₂) inl inr
⊢ (F.map f₁).app X₂ ≫ (F.obj Y₁).map f₂ = (F.obj X₁).map f₂ ≫ (F.map f₁).app Y₂

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
inst✝ : HasPushout ((F.map f₁).app X₂) ((F.obj X₁).map f₂)
⊢ (F.map f₁).app X₂ ≫ (F.obj Y₁).map f₂ = (F.obj X₁).map f₂ ≫ (F.map f₁).app Y₂

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
sq : F.PushoutObjObj f₁ f₂
F' : C₁ ⥤ C₂ ⥤ C₃
e : F ≅ F'
⊢ (F.map f₁).app X₂ ≫ ((e.app Y₁).app X₂).hom = ((e.app X₁).app X₂).hom ≫ (F'.map f₁).app X₂

Tactic:
by simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
inst✝ : Category.{v₃, u₃} C₃
F : C₁ ⥤ C₂ ⥤ C₃
G : C₁ᵒᵖ ⥤ C₃ ⥤ C₂
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
sq : F.PushoutObjObj f₁ f₂
F' : C₁ ⥤ C₂ ⥤ C₃
e : F ≅ F'
⊢ (F.obj X₁).map f₂ ≫ ((e.app X₁).app Y₂).hom = ((e.app X₁).app X₂).hom ≫ (F'.obj X₁).map f₂

Tactic:
by simp

Goal state after: