mathlib-initiative/mathlib-tactics huggingface dataset:
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
X : D
a : C
inst✝¹ : HasPullbacks C
inst✝ : PreservesLimitsOfShape WalkingCospan F
g : F.obj a ⟶ Limits.pullback (terminal.from X) (terminal.from X)
⊢ F.relativelyRepresentable (pullback.lift (𝟙 (F.obj a)) (𝟙 (F.obj a)) ⋯)

Tactic:
by
  let pbIso := pullback.congrHom
    (terminal.comp_from _ : (g ≫ pullback.fst _ _) ≫ terminal.from X = terminal.from _)
    (terminal.comp_from _ : (g ≫ pullback.snd _ _) ≫ terminal.from X = terminal.from _) ≪≫
    (prodIsoPullback _ _).symm ≪≫ (HasLimit.isoOfNatIso (pairComp _ _ _)).symm ≪≫
    (preservesLimitIso _ (pair _ _)).symm
  rw [← comp_id (pullback.lift _ _), ← pbIso.hom_inv_id, ← Category.assoc]
  apply (respectsIso F).toRespectsRight.postcomp _ (inferInstance : IsIso _) _
  exact map_preimage F (_ ≫ pbIso.hom) ▸ map F (F.preimage _)

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

Goal state before:
pp: case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n : ℕ
φ : Y ⟶ X _⦋n + 1⦌
q : ℕ
hq : HigherFacesVanish q φ → φ ≫ (P q).f (n + 1) = φ
v : HigherFacesVanish (q + 1) φ
⊢ φ ≫ (Hσ q).f (n + 1) = 0

Tactic:
by_cases! hqn : n < q

Goal state after:
case pos
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n : ℕ
φ : Y ⟶ X _⦋n + 1⦌
q : ℕ
hq : HigherFacesVanish q φ → φ ≫ (P q).f (n + 1) = φ
v : HigherFacesVanish (q + 1) φ
hqn : n < q
⊢ φ ≫ (Hσ q).f (n + 1) = 0

case neg
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n : ℕ
φ : Y ⟶ X _⦋n + 1⦌
q : ℕ
hq : HigherFacesVanish q φ → φ ≫ (P q).f (n + 1) = φ
v : HigherFacesVanish (q + 1) φ
hqn : q ≤ n
⊢ φ ≫ (Hσ q).f (n + 1) = 0
Module: Mathlib.AlgebraicTopology.DoldKan.Decomposition

Goal state before:
pp: case succ
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n q : ℕ
hq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev
⊢ (Q (q + 1)).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev

Tactic:
by_cases! hqn : n < q

Goal state after:
case pos
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n q : ℕ
hq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev
hqn : n < q
⊢ (Q (q + 1)).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev

case neg
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n q : ℕ
hq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev
hqn : q ≤ n
⊢ (Q (q + 1)).f (n + 1) = ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev
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 + 1)
hj₁ : n + 1 ≤ ↑j + (q + 1)
⊢ φ ≫ X.δ j.succ + (φ ≫ (Hσ q).f (n + 1)) ≫ X.δ j.succ = 0

Tactic:
by_cases! hqn : n < q

Goal state after:
case pos
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 + 1)
hj₁ : n + 1 ≤ ↑j + (q + 1)
hqn : n < q
⊢ φ ≫ X.δ j.succ + (φ ≫ (Hσ q).f (n + 1)) ≫ X.δ j.succ = 0

case neg
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 + 1)
hj₁ : n + 1 ≤ ↑j + (q + 1)
hqn : q ≤ n
⊢ φ ≫ X.δ j.succ + (φ ≫ (Hσ q).f (n + 1)) ≫ X.δ j.succ = 0
Module: Mathlib.AlgebraicTopology.DoldKan.Degeneracies

Goal state before:
pp: case H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
Y : C
X : SimplicialObject C
n b q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
v : HigherFacesVanish q φ
hnbq : n + 1 = b + q
j : Fin (n + 1 + 1)
hj : n + 1 + 1 ≤ ↑j + q
⊢ ⟨b, ⋯⟩.succ < j.succ

Tactic:
· dsimp
    rw [Fin.lt_def, Fin.val_succ]
    linarith

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

Goal state before:
pp: case neg.inl
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
h✝ : a < m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.δ j ≫ X.σ (⟨a, ⋯⟩.castLT ⋯)

Tactic:
rw [← X.δ_comp_δ''_assoc]

Goal state after:
case neg.inl
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
h✝ : a < m
⊢ φ ≫ X.δ j.succ = φ ≫ X.δ j.succ ≫ X.δ (⟨a + 1, ⋯⟩.castLT ⋯) ≫ X.σ (⟨a, ⋯⟩.castLT ⋯)

case neg.inl.H
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
q a m : ℕ
φ : Y ⟶ X _⦋m + 1 + 1⦌
v : HigherFacesVanish q φ
j : Fin (m + 1 + 1)
hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)
hqn : q ≤ m + 1
ha : q + a = m + 1
hj₂ : ¬a = ↑j
haj : a < ↑j
ham : a ≤ m
h✝ : a < m
⊢ ⟨a + 1, ⋯⟩ ≤ j.castSucc
Module: Mathlib.AlgebraicTopology.SimplicialObject.Split

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op ⦋n⦌) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op ⦋n⦌) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
n : ℕ
⊢ S₁.s.ι n ≫ F₁.app (op ⦋n⦌) = S₁.s.ι n ≫ F₂.app (op ⦋n⦌)

Tactic:
c₁,

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op ⦋n⦌) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op ⦋n⦌) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
n : ℕ
⊢ f₁ n ≫ S₂.s.ι n = S₁.s.ι n ≫ F₂.app (op ⦋n⦌)
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
K✝ K' : ChainComplex C ℕ
f : K✝ ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
Δ'' Δ' Δ : SimplexCategoryᵒᵖ
θ' : Δ'' ⟶ Δ'
θ : Δ' ⟶ Δ
x✝ : Discrete (Splitting.IndexSet Δ'')
A : Splitting.IndexSet Δ''
⊢ colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K (θ' ≫ θ) =
    colimit.ι (Discrete.functor fun A ↦ Obj.summand K Δ'' A) { as := A } ≫ Obj.map K θ' ≫ Obj.map K θ

Tactic:
have fac : θ.unop ≫ θ'.unop ≫ A.e = (θ' ≫ θ).unop ≫ A.e := by rw [unop_comp, assoc]
    rw [← image.fac (θ'.unop ≫ A.e), ← assoc, ←
      image.fac (θ.unop ≫ factorThruImage (θ'.unop ≫ A.e)), assoc] at fac
    simp only [Obj.map_on_summand₀'_assoc K A θ', Obj.map_on_summand₀' K _ θ,
      Obj.Termwise.mapMono_comp_assoc, Obj.map_on_summand₀ K A fac]
    rfl

Goal state after:
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
⊢ HigherFacesVanish (n + 1) (((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)))

Tactic:
intro j _

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.NReflectsIso

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X Y : SimplicialObject C
f : X ⟶ Y
inst✝ : IsIso (N₁.map f)
this✝ : ∀ (n : ℕ), IsIso (f.app (op ⦋n⦌))
this : ∀ (Δ : SimplexCategoryᵒᵖ), IsIso (f.app Δ)
⊢ IsIso f

Tactic:
apply NatIso.isIso_of_isIso_app

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

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

Tactic:
by_cases hi : Isδ₀ i

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' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : Isδ₀ i
⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ PInfty.f n = PInfty.f n' ≫ X.map i.op

case neg
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X : SimplicialObject C
n n' : ℕ
i : ⦋n⦌ ⟶ ⦋n'⦌
inst✝ : Mono i
h : ¬n = n'
hi : ¬Isδ₀ i
⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ PInfty.f n = PInfty.f n' ≫ X.map i.op
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
Δ Δ' : SimplexCategoryᵒᵖ
θ : Δ ⟶ Δ'
A : Splitting.IndexSet Δ
⊢ (((Γ₀.splitting K[X]).cofan Δ).inj A ≫
      ((N₁ ⋙ Γ₂).obj X).X.map θ ≫ (Γ₀.splitting K[X]).desc Δ' fun A ↦ PInfty.f (unop A.fst).len ≫ X.map A.e.op) =
    ((Γ₀.splitting K[X]).cofan Δ).inj A ≫
      ((Γ₀.splitting K[X]).desc Δ fun A ↦ PInfty.f (unop A.fst).len ≫ X.map A.e.op) ≫
        ((toKaroubi (SimplicialObject C)).obj X).X.map θ

Tactic:
change _ ≫ (Γ₀.obj K[X]).map θ ≫ _ = _

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
Δ Δ' : SimplexCategoryᵒᵖ
θ : Δ ⟶ Δ'
A : Splitting.IndexSet Δ
⊢ (((Γ₀.splitting K[X]).cofan Δ).inj A ≫
      (Γ₀.obj K[X]).map θ ≫ (Γ₀.splitting K[X]).desc Δ' fun A ↦ PInfty.f (unop A.fst).len ≫ X.map A.e.op) =
    ((Γ₀.splitting K[X]).cofan Δ).inj A ≫
      ((Γ₀.splitting K[X]).desc Δ fun A ↦ PInfty.f (unop A.fst).len ≫ X.map A.e.op) ≫
        ((toKaroubi (SimplicialObject C)).obj X).X.map θ
Module: Mathlib.Topology.Homotopy.TopCat.Basic

Goal state before:
pp: X Y Z : TopCat
f₀ f₁ : X ⟶ Y
g₀ g₁ : Y ⟶ Z
G : Homotopy g₀ g₁
F : Homotopy f₀ f₁
⊢ (G.comp F).h = X ◁ lift (𝟙 I) (𝟙 I) ≫ (α_ X I I).inv ≫ F.h ▷ I ≫ G.h

Tactic:
by
  ext
  simp

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    (Γ₂N₂ToKaroubiIso.app X).inv ≫
      𝟙 (Γ₂.obj (N₂.obj ((toKaroubi (SimplicialObject C)).obj X))) ≫
        (Γ₂N₂ToKaroubiIso.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫
            Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫
          𝟙
            { X := ((toKaroubi (SimplicialObject C)).obj X).X, p := 𝟙 ((toKaroubi (SimplicialObject C)).obj X).X,
              idem := ⋯ }

Tactic:
Iso.app_inv

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
⊢ Γ₂N₁.natTrans.app X =
    Γ₂N₂ToKaroubiIso.inv.app X ≫
      𝟙 (Γ₂.obj (N₂.obj ((toKaroubi (SimplicialObject C)).obj X))) ≫
        (Γ₂N₂ToKaroubiIso.hom.app ((toKaroubi (SimplicialObject C)).obj X).X ≫
            Γ₂N₁.natTrans.app ((toKaroubi (SimplicialObject C)).obj X).X) ≫
          𝟙
            { X := ((toKaroubi (SimplicialObject C)).obj X).X, p := 𝟙 ((toKaroubi (SimplicialObject C)).obj X).X,
              idem := ⋯ }
Module: Mathlib.AlgebraicTopology.DoldKan.NCompGamma

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
this : ∀ (P : Karoubi (SimplicialObject C)), IsIso (Γ₂N₂.natTrans.app P)
⊢ IsIso Γ₂N₂.natTrans

Tactic:
apply NatIso.isIso_of_isIso_app

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
this : ∀ (X : SimplicialObject C), IsIso (Γ₂N₁.natTrans.app X)
⊢ IsIso Γ₂N₁.natTrans

Tactic:
apply NatIso.isIso_of_isIso_app

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f₀ f₁ : C(X, Y)
F : f₀.Homotopy f₁
⊢ F.symm.symm = F

Tactic:
by
  ext
  simp

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p₀ q₀ : Path x₀ x₁
p₁ q₁ : Path x₁ x₂
F : p₀.Homotopy q₀
G : p₁.Homotopy q₁
x t : ↑I
ht : t ∈ {0, 1}
⊢ {
        toFun := fun x_1 ↦
          if ↑(x, x_1).2 ≤ 1 / 2 then (F.eval (x, x_1).1).extend (2 * ↑(x, x_1).2)
          else (G.eval (x, x_1).1).extend (2 * ↑(x, x_1).2 - 1),
        continuous_toFun := ⋯ }
      t =
    (p₀.trans p₁).toContinuousMap t

Tactic:
rcases ht with ht | ht

Goal state after:
case inl
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p₀ q₀ : Path x₀ x₁
p₁ q₁ : Path x₁ x₂
F : p₀.Homotopy q₀
G : p₁.Homotopy q₁
x t : ↑I
ht : t = 0
⊢ {
        toFun := fun x_1 ↦
          if ↑(x, x_1).2 ≤ 1 / 2 then (F.eval (x, x_1).1).extend (2 * ↑(x, x_1).2)
          else (G.eval (x, x_1).1).extend (2 * ↑(x, x_1).2 - 1),
        continuous_toFun := ⋯ }
      t =
    (p₀.trans p₁).toContinuousMap t

case inr
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p₀ q₀ : Path x₀ x₁
p₁ q₁ : Path x₁ x₂
F : p₀.Homotopy q₀
G : p₁.Homotopy q₁
x t : ↑I
ht : t ∈ {1}
⊢ {
        toFun := fun x_1 ↦
          if ↑(x, x_1).2 ≤ 1 / 2 then (F.eval (x, x_1).1).extend (2 * ↑(x, x_1).2)
          else (G.eval (x, x_1).1).extend (2 * ↑(x, x_1).2 - 1),
        continuous_toFun := ⋯ }
      t =
    (p₀.trans p₁).toContinuousMap t
Module: Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : ModelCategory C
X Y : CofibrantObject C
inst✝ : IsFibrant Y.obj
f g : X ⟶ Y
⊢ Relation.EqvGen (HomRel.CompClosure (homRel C)) f g ↔ homRel C f g

Tactic:
HomRel.compClosure_eq_self,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : ModelCategory C
X Y : CofibrantObject C
inst✝ : IsFibrant Y.obj
f g : X ⟶ Y
⊢ Relation.EqvGen (homRel C) f g ↔ homRel C f g
Module: Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ModelCategory C
inst✝ : LocallySmall.{w, v, u} C
⊢ LocallySmall.{w, v, u} (HoCat C)

Tactic:
dsimp [HoCat]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ModelCategory C
inst✝ : LocallySmall.{w, v, u} C
⊢ LocallySmall.{w, v, u} (CategoryTheory.Quotient (homRel C))
Module: Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic

Goal state before:
pp: case mk
X : Type u_1
inst✝ : TopologicalSpace X
x₀ x₁ : X
γ : Path x₀ x₁
⊢ (mk γ).trans (refl x₁) = mk γ

Tactic:
simpa [← mk_trans, ← mk_refl, eq] using Homotopic.trans_refl γ

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
n : ℕ
y : Δ[d] _⦋n⦌
hy₁ : y ∈ Δ[d].nonDegenerate n
hy₂ : ↑⟨y, hy₁⟩ ∈ ∂Δ[d].obj (op ⦋n⦌)
⊢ ⟨y, hy₂⟩ ∈ ∂Δ[d].toSSet.nonDegenerate n

Tactic:
simpa [Subcomplex.mem_nonDegenerate_iff]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Skeleton

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
x✝¹ : SimplexCategoryᵒᵖ
n : ℕ
x✝ : (fun X ↦ X) ↑(((skeletonOfMono i) d).obj (op ⦋n⦌))
y : (fun X ↦ X) (sigmaStdSimplex i d _⦋n⦌)
hx : ↑((ConcreteCategory.hom ((b i d).app (op ⦋n⦌))) y) ∈ ((skeletonOfMono i) d).obj (op ⦋n⦌)
⊢ ∃ x₁,
    (ConcreteCategory.hom ((t i d).app (op ⦋n⦌))) x₁ = ⟨↑((ConcreteCategory.hom ((b i d).app (op ⦋n⦌))) y), hx⟩ ∧
      (ConcreteCategory.hom ((l i d).app (op ⦋n⦌))) x₁ = y

Tactic:
obtain ⟨c, y, rfl⟩ := ιSigmaStdSimplex_jointly_surjective y

Goal state after:
X Y : SSet
i : X ⟶ Y
d : ℕ
x✝¹ : SimplexCategoryᵒᵖ
n : ℕ
x✝ : (fun X ↦ X) ↑(((skeletonOfMono i) d).obj (op ⦋n⦌))
c : Cell i d
y : Δ[d] _⦋n⦌
hx :
  ↑((ConcreteCategory.hom ((b i d).app (op ⦋n⦌))) ((ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) y)) ∈
    ((skeletonOfMono i) d).obj (op ⦋n⦌)
⊢ ∃ x₁,
    (ConcreteCategory.hom ((t i d).app (op ⦋n⦌))) x₁ =
        ⟨↑((ConcreteCategory.hom ((b i d).app (op ⦋n⦌))) ((ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) y)),
          hx⟩ ∧
      (ConcreteCategory.hom ((l i d).app (op ⦋n⦌))) x₁ = (ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) y
Module: Mathlib.AlgebraicTopology.SimplicialSet.Monoidal

Goal state before:
pp: X Y : SSet
A : X.Subcomplex
B : Y.Subcomplex
⊢ range ((A.prod B).ι ≫ fst X Y) ≤ A

Tactic:
by
      intro _ _ ⟨⟨_, ⟨_, _⟩⟩, _⟩
      cat_disch

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Monoidal

Goal state before:
pp: X Y : SSet
A : X.Subcomplex
B : Y.Subcomplex
⊢ range ((A.prod B).ι ≫ snd X Y) ≤ B

Tactic:
by
      intro _ _ ⟨⟨_, ⟨_, _⟩⟩, _⟩
      cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Enriched.Ordinary.Basic

Goal state before:
pp: V : Type u'
inst✝¹⁰ : Category.{v', u'} V
inst✝⁹ : MonoidalCategory V
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : EnrichedOrdinaryCategory V C
W✝ : Type u''
inst✝⁶ : Category.{v'', u''} W✝
inst✝⁵ : MonoidalCategory W✝
F✝ : V ⥤ W✝
inst✝⁴ : F✝.LaxMonoidal
W : Type u''
inst✝³ : Category.{v'', u''} W
inst✝² : MonoidalCategory W
F : V ⥤ W
inst✝¹ : F.LaxMonoidal
D : Type u
inst✝ : EnrichedCategory V D
e : (v : V) → (𝟙_ V ⟶ v) ≃ (𝟙_ W ⟶ F.obj v)
h : ∀ (v : V) (f : 𝟙_ V ⟶ v), (e v) f = Functor.LaxMonoidal.ε F ≫ F.map f
X✝ Y✝ Z✝ : TransportEnrichment F (ForgetEnrichment V D)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ ForgetEnrichment.homOf W
      (Functor.LaxMonoidal.ε F ≫
        (λ_ (F.obj (𝟙_ V))).inv ≫
          Functor.LaxMonoidal.ε F ▷ F.obj (𝟙_ V) ≫
            (F.map (ForgetEnrichment.homTo V f) ⊗ₘ F.map (ForgetEnrichment.homTo V g)) ≫
              eComp W (ForgetEnrichment.to V X✝) (ForgetEnrichment.to V Y✝) (ForgetEnrichment.to V Z✝)) =
    ForgetEnrichment.homOf W (Functor.LaxMonoidal.ε F ≫ F.map (ForgetEnrichment.homTo V f)) ≫
      ForgetEnrichment.homOf W (Functor.LaxMonoidal.ε F ≫ F.map (ForgetEnrichment.homTo V g))

Tactic:
← ForgetEnrichment.homOf_comp,

Goal state after:
V : Type u'
inst✝¹⁰ : Category.{v', u'} V
inst✝⁹ : MonoidalCategory V
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : EnrichedOrdinaryCategory V C
W✝ : Type u''
inst✝⁶ : Category.{v'', u''} W✝
inst✝⁵ : MonoidalCategory W✝
F✝ : V ⥤ W✝
inst✝⁴ : F✝.LaxMonoidal
W : Type u''
inst✝³ : Category.{v'', u''} W
inst✝² : MonoidalCategory W
F : V ⥤ W
inst✝¹ : F.LaxMonoidal
D : Type u
inst✝ : EnrichedCategory V D
e : (v : V) → (𝟙_ V ⟶ v) ≃ (𝟙_ W ⟶ F.obj v)
h : ∀ (v : V) (f : 𝟙_ V ⟶ v), (e v) f = Functor.LaxMonoidal.ε F ≫ F.map f
X✝ Y✝ Z✝ : TransportEnrichment F (ForgetEnrichment V D)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ ForgetEnrichment.homOf W
      (Functor.LaxMonoidal.ε F ≫
        (λ_ (F.obj (𝟙_ V))).inv ≫
          Functor.LaxMonoidal.ε F ▷ F.obj (𝟙_ V) ≫
            (F.map (ForgetEnrichment.homTo V f) ⊗ₘ F.map (ForgetEnrichment.homTo V g)) ≫
              eComp W (ForgetEnrichment.to V X✝) (ForgetEnrichment.to V Y✝) (ForgetEnrichment.to V Z✝)) =
    ForgetEnrichment.homOf W
      ((λ_ (𝟙_ W)).inv ≫
        (Functor.LaxMonoidal.ε F ≫ F.map (ForgetEnrichment.homTo V f) ⊗ₘ
            Functor.LaxMonoidal.ε F ≫ F.map (ForgetEnrichment.homTo V g)) ≫
          eComp W X✝ Y✝ Z✝)
Module: Mathlib.CategoryTheory.Monoidal.Center

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
A✝ B✝ : Center C
f : A✝ ⟶ B✝
i : IsIso ((forget C).map f)
⊢ IsIso f

Tactic:
dsimp at i; change IsIso (isoMk f).hom; infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Enriched.Ordinary.Basic

Goal state before:
pp: V : Type u'
inst✝¹⁰ : Category.{v', u'} V
inst✝⁹ : MonoidalCategory V
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : EnrichedOrdinaryCategory V C
W✝ : Type u''
inst✝⁶ : Category.{v'', u''} W✝
inst✝⁵ : MonoidalCategory W✝
F✝ : V ⥤ W✝
inst✝⁴ : F✝.LaxMonoidal
W : Type u''
inst✝³ : Category.{v'', u''} W
inst✝² : MonoidalCategory W
F : V ⥤ W
inst✝¹ : F.LaxMonoidal
D : Type u
inst✝ : EnrichedCategory V D
e : (v : V) → (𝟙_ V ⟶ v) ≃ (𝟙_ W ⟶ F.obj v)
h : ∀ (v : V) (f : 𝟙_ V ⟶ v), (e v) f = Functor.LaxMonoidal.ε F ≫ F.map f
X✝ Y✝ Z✝ : ForgetEnrichment W (TransportEnrichment F D)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ ForgetEnrichment.homOf V
      ((e (ForgetEnrichment.to W X✝ ⟶[V] ForgetEnrichment.to W Z✝)).symm (ForgetEnrichment.homTo W (f ≫ g))) =
    ForgetEnrichment.homOf V
        ((e (ForgetEnrichment.to W X✝ ⟶[V] ForgetEnrichment.to W Y✝)).symm (ForgetEnrichment.homTo W f)) ≫
      ForgetEnrichment.homOf V
        ((e (ForgetEnrichment.to W Y✝ ⟶[V] ForgetEnrichment.to W Z✝)).symm (ForgetEnrichment.homTo W g))

Tactic:
← ForgetEnrichment.homOf_comp

Goal state after:
V : Type u'
inst✝¹⁰ : Category.{v', u'} V
inst✝⁹ : MonoidalCategory V
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : EnrichedOrdinaryCategory V C
W✝ : Type u''
inst✝⁶ : Category.{v'', u''} W✝
inst✝⁵ : MonoidalCategory W✝
F✝ : V ⥤ W✝
inst✝⁴ : F✝.LaxMonoidal
W : Type u''
inst✝³ : Category.{v'', u''} W
inst✝² : MonoidalCategory W
F : V ⥤ W
inst✝¹ : F.LaxMonoidal
D : Type u
inst✝ : EnrichedCategory V D
e : (v : V) → (𝟙_ V ⟶ v) ≃ (𝟙_ W ⟶ F.obj v)
h : ∀ (v : V) (f : 𝟙_ V ⟶ v), (e v) f = Functor.LaxMonoidal.ε F ≫ F.map f
X✝ Y✝ Z✝ : ForgetEnrichment W (TransportEnrichment F D)
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ ForgetEnrichment.homOf V
      ((e (ForgetEnrichment.to W X✝ ⟶[V] ForgetEnrichment.to W Z✝)).symm (ForgetEnrichment.homTo W (f ≫ g))) =
    ForgetEnrichment.homOf V
      ((λ_ (𝟙_ V)).inv ≫
        ((e (ForgetEnrichment.to W X✝ ⟶[V] ForgetEnrichment.to W Y✝)).symm (ForgetEnrichment.homTo W f) ⊗ₘ
            (e (ForgetEnrichment.to W Y✝ ⟶[V] ForgetEnrichment.to W Z✝)).symm (ForgetEnrichment.homTo W g)) ≫
          eComp V (ForgetEnrichment.to W X✝) (ForgetEnrichment.to W Y✝) (ForgetEnrichment.to W Z✝))
Module: Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic

Goal state before:
pp: case of_comp
P : MorphismProperty SimplexCategoryGenRel
id : ∀ {n : ℕ}, P (𝟙 (mk n))
δ_comp : ∀ {n m : ℕ} (u : mk (m + 1) ⟶ mk n) (i : Fin (m + 2)), P u → P (δ i ≫ u)
σ_comp : ∀ {n m : ℕ} (u : mk m ⟶ mk n) (i : Fin (m + 1)), P u → P (σ i ≫ u)
a b : SimplexCategoryGenRel
f✝¹ : a ⟶ b
X✝ Y✝ : SimplexCategoryGenRel
f✝ : X✝ ⟶ Y✝
x✝ y✝ z✝ : SimplexCategoryGenRel
f : x✝ ⟶ y✝
g : y✝ ⟶ z✝
hf : generators f
hg : generators.multiplicativeClosure' g
hrec : P g
⊢ P (f ≫ g)

Tactic:
rcases hf with ⟨⟨i⟩⟩ | ⟨⟨i⟩⟩
    · simpa using! (δ_comp g i hrec)
    · simpa using! (σ_comp g i hrec)

Goal state after: