mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case pos
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
H : ⟨i, isLt✝²⟩ ≤ ⟨j, isLt✝¹⟩
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
h✝³ : ¬⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩
h✝² : ¬⟨k, isLt✝⟩.succ.castSucc < ⟨j, isLt✝¹⟩.succ
h✝¹ : ⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩
h✝ : ⟨k, isLt✝⟩.castSucc.castSucc < ⟨i, isLt✝²⟩.castSucc
⊢ ↑⟨k, isLt✝⟩.succ.succ = ↑⟨k, isLt✝⟩.castSucc.castSucc

Tactic:
simp at *

Goal state after:
case pos
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
H : i ≤ j
h✝³ : i ≤ k
h✝² : j ≤ k
h✝¹ : k < j
h✝ : k < i
⊢ k + 1 + 1 = k
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case neg
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
H : ⟨i, isLt✝²⟩ ≤ ⟨j, isLt✝¹⟩
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
h✝³ : ¬⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩
h✝² : ¬⟨k, isLt✝⟩.succ.castSucc < ⟨j, isLt✝¹⟩.succ
h✝¹ : ⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩
h✝ : ¬⟨k, isLt✝⟩.castSucc.castSucc < ⟨i, isLt✝²⟩.castSucc
⊢ ↑⟨k, isLt✝⟩.succ.succ = ↑⟨k, isLt✝⟩.castSucc.succ

Tactic:
simp at *

Goal state after:
case neg
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
H : i ≤ j
h✝³ : i ≤ k
h✝² : j ≤ k
h✝¹ : k < j
h✝ : i ≤ k
⊢ False
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case pos
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
H : ⟨i, isLt✝²⟩ ≤ ⟨j, isLt✝¹⟩
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
h✝³ : ¬⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩
h✝² : ¬⟨k, isLt✝⟩.succ.castSucc < ⟨j, isLt✝¹⟩.succ
h✝¹ : ¬⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩
h✝ : ⟨k, isLt✝⟩.succ.castSucc < ⟨i, isLt✝²⟩.castSucc
⊢ ↑⟨k, isLt✝⟩.succ.succ = ↑⟨k, isLt✝⟩.succ.castSucc

Tactic:
simp at *

Goal state after:
case pos
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
H : i ≤ j
h✝³ : i ≤ k
h✝² h✝¹ : j ≤ k
h✝ : k + 1 < i
⊢ False
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case neg
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
H : ⟨i, isLt✝²⟩ ≤ ⟨j, isLt✝¹⟩
k : ℕ
isLt✝ : k < ⦋n⦌.len + 1
h✝³ : ¬⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩
h✝² : ¬⟨k, isLt✝⟩.succ.castSucc < ⟨j, isLt✝¹⟩.succ
h✝¹ : ¬⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩
h✝ : ¬⟨k, isLt✝⟩.succ.castSucc < ⟨i, isLt✝²⟩.castSucc
⊢ ↑⟨k, isLt✝⟩.succ.succ = ↑⟨k, isLt✝⟩.succ.succ

Tactic:
simp at *

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case pos
n i : ℕ
isLt✝¹ : i < n + 1
j : ℕ
isLt✝ : j < ⦋n⦌.len + 1
h✝¹ : ⟨j, ⋯⟩ < ⟨i + 1, ⋯⟩
h✝ : ⟨i, ⋯⟩ < ⟨j, ⋯⟩
⊢ j - 1 = j

Tactic:
simp at *

Goal state after:
case pos
n i : ℕ
isLt✝¹ : i < n + 1
j : ℕ
isLt✝ : j < ⦋n⦌.len + 1
h✝¹ : j < i + 1
h✝ : i < j
⊢ j - 1 = j
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: case neg
n i : ℕ
isLt✝¹ : i < n + 1
j : ℕ
isLt✝ : j < ⦋n⦌.len + 1
h✝¹ : ¬⟨j, ⋯⟩ < ⟨i + 1, ⋯⟩
h✝ : ¬⟨i, ⋯⟩ < ⟨j + 1, ⋯⟩
⊢ False

Tactic:
simp at *

Goal state after:
case neg
n i : ℕ
isLt✝¹ : i < n + 1
j : ℕ
isLt✝ : j < ⦋n⦌.len + 1
h✝¹ : i + 1 ≤ j
h✝ : j + 1 ≤ i
⊢ False
Module: Mathlib.AlgebraicTopology.SimplicialObject.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X : SimplicialObject C
n : ℕ
inst✝¹ : ∀ (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C), (SimplexCategory.Truncated.inclusion n).op.HasRightKanExtension F
inst✝ : ∀ (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C), (SimplexCategory.Truncated.inclusion n).op.HasLeftKanExtension F
⊢ IsLeftKanExtension ((sk n).obj X) ((skAdj n).unit.app ((truncation n).obj X))

Tactic:
dsimp [sk, skAdj]
  rw [lanAdjunction_unit]
  infer_instance

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: n : ℕ
i : Fin n
j : Fin (n + 2)
h : j = i.succ.castSucc
⊢ mkOfSucc i ≫ δ j = intervalEdge (↑i) 2 ⋯

Tactic:
ext x
  fin_cases x
  · subst h
    simp only [δ, len_mk, Nat.reduceAdd, mkHom, comp_toOrderHom, Hom.toOrderHom_mk,
      Fin.zero_eta, OrderHom.comp_coe, OrderEmbedding.toOrderHom_coe, Function.comp_apply,
      mkOfSucc_homToOrderHom_zero, Fin.succAboveOrderEmb_apply,
      Fin.castSucc_succAbove_castSucc, Fin.succAbove_succ_self]
    rfl
  · simp only [δ, len_mk, Nat.reduceAdd, mkHom, comp_toOrderHom, Hom.toOrderHom_mk, Fin.mk_one,
      OrderHom.comp_coe, OrderEmbedding.toOrderHom_coe, Function.comp_apply,
      mkOfSucc_homToOrderHom_one, Fin.succAboveOrderEmb_apply]
    subst h
    rw [Fin.succAbove_castSucc_self]
    rfl

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: f : ⦋1⦌ ⟶ ⦋2⦌
⊢ (∃ i, f = δ i) ∨ ∃ a, f = ⦋1⦌.const ⦋2⦌ a

Tactic:
have : f.toOrderHom 0 ≤ f.toOrderHom 1 := f.toOrderHom.monotone (by decide : (0 : Fin 2) ≤ 1)
  match e0 : f.toOrderHom 0, e1 : f.toOrderHom 1 with
  | 1, 2 =>
    refine .inl ⟨0, ?_⟩
    ext i : 3
    match i with
    | 0 => exact e0
    | 1 => exact e1
  | 0, 2 =>
    refine .inl ⟨1, ?_⟩
    ext i : 3
    match i with
    | 0 => exact e0
    | 1 => exact e1
  | 0, 1 =>
    refine .inl ⟨2, ?_⟩
    ext i : 3
    match i with
    | 0 => exact e0
    | 1 => exact e1
  | 0, 0 | 1, 1 | 2, 2 =>
    refine .inr ⟨f.toOrderHom 0, ?_⟩
    ext i : 3
    match i with
    | 0 => rfl
    | 1 => exact e1.trans e0.symm
  | 1, 0 | 2, 0 | 2, 1 =>
    rw [e0, e1] at this
    exact Not.elim (by decide) this

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: X : NonemptyFinLinOrd
aux : Fintype.card ↑X.toLinOrd = Fintype.card ↑X.toLinOrd - 1 + 1
f : Fin (Fintype.card ↑X.toLinOrd - 1 + 1) ≃o ↑X.toLinOrd := monoEquivOfFin (↑X.toLinOrd) aux
hf : StrictMono ⇑(Finset.univ.orderEmbOfFin aux)
⊢ skeletalFunctor.obj ⦋Fintype.card ↑X.toLinOrd - 1⦌ ≅ X

Tactic:
refine
          { hom := InducedCategory.homMk (LinOrd.ofHom ⟨f, hf.monotone⟩)
            inv := InducedCategory.homMk (LinOrd.ofHom ⟨f.symm, ?_⟩)
            hom_inv_id := by ext; apply f.symm_apply_apply
            inv_hom_id := by ext; apply f.apply_symm_apply }

Goal state after:
X : NonemptyFinLinOrd
aux : Fintype.card ↑X.toLinOrd = Fintype.card ↑X.toLinOrd - 1 + 1
f : Fin (Fintype.card ↑X.toLinOrd - 1 + 1) ≃o ↑X.toLinOrd := monoEquivOfFin (↑X.toLinOrd) aux
hf : StrictMono ⇑(Finset.univ.orderEmbOfFin aux)
⊢ Monotone ⇑f.symm
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: X : NonemptyFinLinOrd
⊢ skeletalFunctor.obj ⦋Fintype.card ↑X.toLinOrd - 1⦌ ≅ X

Tactic:
have aux : Fintype.card X = Fintype.card X - 1 + 1 :=
          (Nat.succ_pred_eq_of_pos <| Fintype.card_pos_iff.mpr ⟨⊥⟩).symm
        let f := monoEquivOfFin X aux
        have hf := (Finset.univ.orderEmbOfFin aux).strictMono
        refine
          { hom := InducedCategory.homMk (LinOrd.ofHom ⟨f, hf.monotone⟩)
            inv := InducedCategory.homMk (LinOrd.ofHom ⟨f.symm, ?_⟩)
            hom_inv_id := by ext; apply f.symm_apply_apply
            inv_hom_id := by ext; apply f.apply_symm_apply }
        intro i j h
        change f.symm i ≤ f.symm j
        rw [← hf.le_iff_le]
        change f (f.symm i) ≤ f (f.symm j)
        simpa only [OrderIso.apply_symm_apply]

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

Goal state before:
pp: case mp
X : SSet
n : ℕ
x : X _⦋n + 1⦌
hx : ∃ m, ∃ (_ : m < n + 1), ∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))
⊢ x ∈ ⋃ i, Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ X i))

Tactic:
obtain ⟨m, hm, f, hf, y, rfl⟩ := hx

Goal state after:
case mp
X : SSet
n m : ℕ
hm : m < n + 1
f : ⦋n + 1⦌ ⟶ ⦋m⦌
hf : Epi f
y : X _⦋m⦌
⊢ (ConcreteCategory.hom (X.map f.op)) y ∈ ⋃ i, Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ X i))
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasTerminal C
ι : Type w
X Y : C
⊢ Unique (Y ⟶ (wideCospan ι X).obj none)

Tactic:
by
  dsimp [wideCospan]
  infer_instance

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

Goal state before:
pp: X : SSet
x₁ x₂ : X.N
h : x₁ ≤ x₂
h' : x₂ ≤ x₁
⊢ x₁ = x₂

Tactic:
by
    obtain ⟨n₁, ⟨x₁, hx₁⟩, rfl⟩ := x₁.mk_surjective
    obtain ⟨n₂, ⟨x₂, hx₂⟩, rfl⟩ := x₂.mk_surjective
    obtain rfl : n₁ = n₂ := le_antisymm (dim_le_of_le h) (dim_le_of_le h')
    rw [le_iff_exists_mono] at h
    obtain ⟨f, hf, h⟩ := h
    obtain rfl := SimplexCategory.eq_id_of_mono f
    aesop

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

Goal state before:
pp: X Y : SSet
f : X ⟶ Y
B : Y.Subcomplex
inst✝ : IsIso f
⊢ B ≤ (B.preimage f).image f

Tactic:
intro n y hy
    exact ⟨(inv f).app _ y, by simpa [← NatIso.isIso_inv_app, ← NatTrans.comp_app_apply]⟩

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

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋n⦌
m₁ m₂ : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m₁⦌
inst✝¹ : Epi f₁
y₁ : ↑(X.nonDegenerate m₁)
hy₁ : x = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m₂⦌
inst✝ : Epi f₂
y₂ : ↑(X.nonDegenerate m₂)
hy₂ : x = (ConcreteCategory.hom (X.map f₂.op)) ↑y₂
⊢ m₁ = m₂

Tactic:
by
  obtain ⟨⟨hf₁⟩⟩ := isSplitEpi_of_epi f₁
  obtain ⟨⟨hf₂⟩⟩ := isSplitEpi_of_epi f₂
  exact le_antisymm (le hf₁ hy₁ hy₂) (le hf₂ hy₂ hy₁)

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

Goal state before:
pp: X : SSet
ι : Type u_1
inst✝ : Finite ι
A : ι → X.Subcomplex
⊢ (⨆ i, A i).toSSet.Finite ↔ ∀ (i : ι), (A i).toSSet.Finite

Tactic:
by
  refine ⟨fun h i ↦ finite_of_mono (Subcomplex.homOfLE (le_iSup A i)), fun h ↦ ⟨?_⟩⟩
  refine Finite.of_surjective (f := fun (⟨i, s⟩ : Σ (i : ι), (A i).toSSet.N) ↦
    N.mk ((Subcomplex.homOfLE (le_iSup A i)).app _ s.simplex)
      (by simpa only [nonDegenerate_iff_of_mono] using s.nonDegenerate)) ?_
  intro s
  obtain ⟨d, ⟨⟨s, h₁⟩, h₂⟩, rfl⟩ := s.mk_surjective
  simp only [Subfunctor.iSup_obj, Set.mem_iUnion] at h₁
  obtain ⟨i, hi⟩ := h₁
  rw [Subcomplex.mem_nonDegenerate_iff] at h₂
  exact ⟨⟨i, N.mk ⟨s, hi⟩ (by rwa [Subcomplex.mem_nonDegenerate_iff])⟩, rfl⟩

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

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : PartialOrder X
n : ℕ
s : nerve X _⦋n + 1⦌
i : Fin (n + 1)
⊢ s.obj i.castSucc = s.obj i.succ → s ∈ Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ (nerve X) i))

Tactic:
· intro h
    refine ⟨(nerve X).δ i.castSucc s, ?_⟩
    ext j
    rw [nerve.σ_obj, nerve.δ_obj]
    by_cases h₁ : i.castSucc < j
    · obtain ⟨j, rfl⟩ := Fin.eq_succ_of_ne_zero (Fin.ne_zero_of_lt h₁)
      rw [Fin.predAbove_of_castSucc_lt _ _ h₁, Fin.pred_succ,
        Fin.succAbove_of_le_castSucc _ _ (Fin.le_castSucc_iff.2 h₁)]
    · simp only [not_lt] at h₁
      grind [→ Fin.succAbove_of_castSucc_lt,
        → Fin.predAbove_of_le_castSucc, Fin.castSucc_castPred, Fin.castPred_castSucc,
        Fin.succAbove_castSucc_self, → LE.le.lt_or_eq]

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

Goal state before:
pp: case left
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
⊢ Function.Injective fun i ↦ ⟨↑x i, ⋯⟩

Tactic:
have := (mem_nonDegenerate_iff_mono x.val).1 x.property
      rw [SimplexCategory.mono_iff_injective] at this
      exact fun _ _ h ↦ this (by simpa using h)

Goal state after:
Module: Mathlib.Algebra.Homology.CommSq

Goal state before:
pp: case h₁
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
fst : X₁ ⟶ X₂
snd : X₁ ⟶ X₃
f : X₂ ⟶ X₄
g : X₃ ⟶ X₄
sq : CommSq fst snd f g
h : IsLimit (PullbackCone.mk fst snd ⋯)
s : Fork (biprod.desc f (-g)) 0
m : s.pt ⟶ sq.kernelFork.pt
hm : m ≫ Fork.ι sq.kernelFork = s.ι
⊢ m ≫ (PullbackCone.mk fst snd ⋯).snd =
    PullbackCone.IsLimit.lift h (s.ι ≫ biprod.fst) (s.ι ≫ biprod.snd) ⋯ ≫ (PullbackCone.mk fst snd ⋯).snd

Tactic:
replace hm := hm =≫ biprod.snd

Goal state after:
case h₁
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
X₁ X₂ X₃ X₄ : C
inst✝ : HasBinaryBiproduct X₂ X₃
fst : X₁ ⟶ X₂
snd : X₁ ⟶ X₃
f : X₂ ⟶ X₄
g : X₃ ⟶ X₄
sq : CommSq fst snd f g
h : IsLimit (PullbackCone.mk fst snd ⋯)
s : Fork (biprod.desc f (-g)) 0
m : s.pt ⟶ sq.kernelFork.pt
hm : (m ≫ Fork.ι sq.kernelFork) ≫ biprod.snd = s.ι ≫ biprod.snd
⊢ m ≫ (PullbackCone.mk fst snd ⋯).snd =
    PullbackCone.IsLimit.lift h (s.ι ≫ biprod.fst) (s.ι ≫ biprod.snd) ⋯ ≫ (PullbackCone.mk fst snd ⋯).snd
Module: Mathlib.CategoryTheory.Triangulated.Orthogonal

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
P : ObjectProperty C
M : Type u_1
inst✝³ : AddGroup M
inst✝² : HasShift C M
inst✝¹ : HasZeroMorphisms C
inst✝ : P.IsStableUnderShift M
n : M
Y : C
hY : P.rightOrthogonal Y
X : C
f : X ⟶ (shiftFunctor C n).obj Y
hX : P X
⊢ f = 0

Tactic:
by
    obtain ⟨g, rfl⟩ := ((shiftEquiv C n).symm.toAdjunction.homEquiv _ _).surjective f
    simp [hY g (P.le_shift (-n) _ hX), Adjunction.homEquiv_unit]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPushout s f g t
Z' W' : C
g' : Z' ⟶ W'
inst✝ : HasLiftingProperty f g'
u : Z ⟶ Z'
v : W ⟶ W'
sq : CommSq u g g' v
⊢ (s ≫ u) ≫ g' = f ≫ t ≫ v

Tactic:
← Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPushout s f g t
Z' W' : C
g' : Z' ⟶ W'
inst✝ : HasLiftingProperty f g'
u : Z ⟶ Z'
v : W ⟶ W'
sq : CommSq u g g' v
⊢ (s ≫ u) ≫ g' = (f ≫ t) ≫ v
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ u ≫ f ≫ t = f' ≫ v ≫ t

Tactic:
← Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ (u ≫ f) ≫ t = f' ≫ v ≫ t
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ (u ≫ f) ≫ t = f' ≫ v ≫ t

Tactic:
← Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ (u ≫ f) ≫ t = (f' ≫ v) ≫ t
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
X✝ Y✝ Z W : C
f✝ : X✝ ⟶ Y✝
s : X✝ ⟶ Z
g : Z ⟶ W
t✝ : Y✝ ⟶ W
J : Type u_2
A B : J → C
inst✝² : HasProduct A
inst✝¹ : HasProduct B
f : (j : J) → A j ⟶ B j
X Y : C
p : X ⟶ Y
inst✝ : ∀ (j : J), HasLiftingProperty p (f j)
t : X ⟶ ∏ᶜ A
b : Y ⟶ ∏ᶜ B
sq : CommSq t p (Limits.Pi.map f) b
j : J
⊢ (t ≫ Pi.π A j) ≫ f j = p ≫ b ≫ Pi.π B j

Tactic:
← Category.assoc,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
X✝ Y✝ Z W : C
f✝ : X✝ ⟶ Y✝
s : X✝ ⟶ Z
g : Z ⟶ W
t✝ : Y✝ ⟶ W
J : Type u_2
A B : J → C
inst✝² : HasProduct A
inst✝¹ : HasProduct B
f : (j : J) → A j ⟶ B j
X Y : C
p : X ⟶ Y
inst✝ : ∀ (j : J), HasLiftingProperty p (f j)
t : X ⟶ ∏ᶜ A
b : Y ⟶ ∏ᶜ B
sq : CommSq t p (Limits.Pi.map f) b
j : J
⊢ (t ≫ Pi.π A j) ≫ f j = (p ≫ b) ≫ Pi.π B j
Module: Mathlib.Algebra.Homology.HomotopyCategory.KInjective

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
n : ℤ
z : Cocycle K L n
inst✝ : L.IsKInjective
hL : HomologicalComplex.Acyclic L
m : ℤ
hm : m + 1 = n
⊢ ∃ α, δ m n α = ↑z

Tactic:
by
  obtain ⟨β, hβ⟩ :=
    IsKInjective.eq_δ_of_cocycle (Cocycle.ofHom (𝟙 L)) hL (-1) (by simp)
  exact ⟨z.1.comp β (by lia), by simp [δ_comp z.1 β _ _ 0 _ hm rfl (by simp), hβ]⟩

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
W₁ W₂ : MorphismProperty C
inst✝¹ : W₁.HasFactorization W₂
inst✝ : W₂.IsStableUnderRetracts
h₂ : W₂ ≤ W₁.rlp
X Y : C
p : X ⟶ Y
hp : W₁.rlp p
⊢ W₂ p

Tactic:
have h := factorizationData W₁ W₂ p
    have := hp _ h.hi
    simpa using of_retract (RetractArrow.ofRightLiftingProperty h.fac) h.hp

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
⊢ (trivialFibrations C).IsStableUnderBaseChange

Tactic:
← cofibrations_rlp

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
⊢ (cofibrations C).rlp.IsStableUnderBaseChange
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
⊢ (trivialFibrations C).IsMultiplicative

Tactic:
← cofibrations_rlp

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
⊢ (cofibrations C).rlp.IsMultiplicative
Module: Mathlib.AlgebraicTopology.ModelCategory.Instances

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
J : Type w
⊢ (trivialFibrations C).IsStableUnderProductsOfShape J

Tactic:
← cofibrations_rlp

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : CategoryWithWeakEquivalences C
inst✝² : CategoryWithCofibrations C
inst✝¹ : CategoryWithFibrations C
inst✝ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
J : Type w
⊢ (cofibrations C).rlp.IsStableUnderProductsOfShape J
Module: Mathlib.AlgebraicTopology.ModelCategory.Basic

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝³ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
inst✝² : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
inst✝¹ : Fibration f
inst✝ : WeakEquivalence w
h : RetractArrow f w
hw : (trivialCofibrations C).MapFactorizationData (fibrations C) w
⊢ (trivialFibrations C).IsStableUnderRetracts

Tactic:
← cofibrations_rlp

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : CategoryWithFibrations C
inst✝⁶ : CategoryWithCofibrations C
inst✝⁵ : CategoryWithWeakEquivalences C
inst✝⁴ : (weakEquivalences C).HasTwoOutOfThreeProperty
inst✝³ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)
inst✝² : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)
A B X Y : C
f : A ⟶ B
w : X ⟶ Y
inst✝¹ : Fibration f
inst✝ : WeakEquivalence w
h : RetractArrow f w
hw : (trivialCofibrations C).MapFactorizationData (fibrations C) w
⊢ (cofibrations C).rlp.IsStableUnderRetracts
Module: Mathlib.CategoryTheory.Localization.Quotient

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
homRel : HomRel C
W : MorphismProperty C
h : homRel.FactorsThroughLocalization W
W' : MorphismProperty (CategoryTheory.Quotient homRel)
hW : W = W'.inverseImage (Quotient.functor homRel)
E : Type u_3
inst✝ : Category.{v_3, u_3} E
L' : CategoryTheory.Quotient homRel ⥤ D
univ : StrictUniversalPropertyFixedTarget L' W' E
F : C ⥤ E
hF : W.IsInvertedBy F
⊢ Quotient.functor homRel ⋙ L' ⋙ univ.lift (CategoryTheory.Quotient.lift homRel F ⋯) ⋯ = F

Tactic:
univ.fac,

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
homRel : HomRel C
W : MorphismProperty C
h : homRel.FactorsThroughLocalization W
W' : MorphismProperty (CategoryTheory.Quotient homRel)
hW : W = W'.inverseImage (Quotient.functor homRel)
E : Type u_3
inst✝ : Category.{v_3, u_3} E
L' : CategoryTheory.Quotient homRel ⥤ D
univ : StrictUniversalPropertyFixedTarget L' W' E
F : C ⥤ E
hF : W.IsInvertedBy F
⊢ Quotient.functor homRel ⋙ CategoryTheory.Quotient.lift homRel F ⋯ = F
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₃ :
  ((shiftFunctor C 1).map o.m₃).op ≫
      ((shiftFunctor C 1).map w₁₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁ =
    ((shiftFunctor C 1).map w₁₂.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁
⊢ o.m₃.op ≫ w₁₃ = w₁₂

Tactic:
← Category.assoc,

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₃ :
  (((shiftFunctor C 1).map o.m₃).op ≫ ((shiftFunctor C 1).map w₁₃.unop).op) ≫
      (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁ =
    ((shiftFunctor C 1).map w₁₂.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁
⊢ o.m₃.op ≫ w₁₃ = w₁₂