mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₃ : t.IsGE T.obj₃ n
Z : C
g : T.obj₂ ⟶ Z
k : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj T.obj₁)
mem : Triangle.mk ((t.truncLTι n).app T.obj₁ ≫ T.mor₁) g k ∈ distinguishedTriangles
⊢ ∃ Z g h, ∃ (_ : Triangle.mk ((t.truncLTι n).app T.obj₁ ≫ T.mor₁) g h ∈ distinguishedTriangles), t.IsGE Z n

Tactic:
refine ⟨_, _, _, mem, ?_⟩

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
n : ℤ
h₃ : t.IsGE T.obj₃ n
Z : C
g : T.obj₂ ⟶ Z
k : Z ⟶ (shiftFunctor C 1).obj ((t.truncLT n).obj T.obj₁)
mem : Triangle.mk ((t.truncLTι n).app T.obj₁ ≫ T.mor₁) g k ∈ distinguishedTriangles
⊢ t.IsGE Z n
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
h₁ : t.IsLE T.obj₁ n₀
X : C
f : X ⟶ T.obj₂
k : (t.truncGE n₁).obj T.obj₃ ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (T.mor₂ ≫ (t.truncGEπ n₁).app T.obj₃) k ∈ distinguishedTriangles
⊢ ∃ X f h, ∃ (_ : Triangle.mk f (T.mor₂ ≫ (t.truncGEπ n₁).app T.obj₃) h ∈ distinguishedTriangles), t.IsLE X n₀

Tactic:
refine ⟨_, _, _, mem, ?_⟩

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
h₁ : t.IsLE T.obj₁ n₀
X : C
f : X ⟶ T.obj₂
k : (t.truncGE n₁).obj T.obj₃ ⟶ (shiftFunctor C 1).obj X
mem : Triangle.mk f (T.mor₂ ≫ (t.truncGEπ n₁).app T.obj₃) k ∈ distinguishedTriangles
⊢ t.IsLE X n₀
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
inst✝¹ : IsTriangulated C
X : C
a b : ℤ
inst✝ : t.IsLE X b
⊢ t.IsLE ((t.truncGE a).obj X) b

Tactic:
by
  rw [t.isLE_iff_isZero_truncGE_obj b (b + 1) rfl]
  have := t.isIso₂_truncGE_map_of_isLE _ (t.triangleLTGE_distinguished a X) b _ rfl
    (by dsimp; infer_instance)
  dsimp at this
  refine IsZero.of_iso ?_ (asIso ((t.truncGE (b + 1)).map ((t.truncGEπ a).app X))).symm
  rwa [← isLE_iff_isZero_truncGE_obj _ _ _ rfl]

Goal state after:
Module: Mathlib.Algebra.Homology.ModelCategory.Lifting

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
α : Cochain Q K 0
hα : δ 0 1 α = ↑(cocycle₁ sq hsq hQ hK)
⊢ (Cochain.ofHom π).comp ((δ 0 1 α).comp (Cochain.ofHom ι) ⋯) ⋯ = ↑(cocycle₁' sq hsq)

Tactic:
rw [← comp_coe_cocycle₁_comp sq hsq hQ hK, hα]

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ModelCategory C
X Y Z : C
inst✝³ : IsFibrant Z
f : X ⟶ Y
inst✝² : IsCofibrant X
inst✝¹ : IsCofibrant Y
inst✝ : WeakEquivalence f
⊢ Function.Bijective fun g ↦ g.precomp f

Tactic:
let h : CofibrantBrownFactorization f := Classical.arbitrary _
  have hj : Function.Bijective (fun (g : RightHomotopyClass Y Z) ↦ g.precomp h.p) := by
    rw [← Function.Bijective.of_comp_iff'
      (precomp_bijective_of_cofibration_of_weakEquivalence Z h.s)]
    convert! Function.bijective_id
    ext φ
    obtain ⟨φ, rfl⟩ := φ.mk_surjective
    simp
  convert! (precomp_bijective_of_cofibration_of_weakEquivalence Z h.i).comp hj using 1
  ext φ
  obtain ⟨φ, rfl⟩ := φ.mk_surjective
  simp

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
inst✝⁴ : Category.{v_3, u_3} D₁
inst✝³ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₁' : MorphismProperty D₁
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝² : R.functor.EssSurj
inst✝¹ : W₂'.RespectsIso
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : T.HasLeftResolutions
Y₂ : D₂
X₂ : C₂
e : R.functor.obj X₂ ≅ Y₂
ρ : T.LeftResolution X₂ := Classical.arbitrary (T.LeftResolution X₂)
⊢ Nonempty (B.LeftResolution Y₂)

Tactic:
exact ⟨{
    X₁ := L.functor.obj ρ.X₁
    w := iso.inv.app _ ≫ R.functor.map ρ.w ≫ e.hom
    hw := (W₂'.arrow_mk_iso_iff (Arrow.isoMk (iso.app _) e)).1 (R.map _ ρ.hw) }⟩

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
R R' : Φ.RightResolution X₂
φ : R ⟶ R'
⊢ ((TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv).structuredArrowDownwards X₂).map
        (StructuredArrow.homMk φ.f ⋯) ≫
      (CostructuredArrow.mk (StructuredArrow.homMk ((isoOfHom L W₂ R'.w ⋯).inv ≫ y) ⋯)).hom =
    (CostructuredArrow.mk (StructuredArrow.homMk ((isoOfHom L W₂ R.w ⋯).inv ≫ y) ⋯)).hom

Tactic:
by
    ext
    dsimp
    rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.Quotient

Goal state before:
pp: C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H✝ : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H✝
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H✝
B : C ⥤ H✝
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
X : C
g : R.obj Y₀ ⟶ B.obj X
X₀ : C₀ := T.objPreimage Y₀
e₀ : T.obj X₀ ≅ Y₀ := T.objObjPreimageIso Y₀
S : Type (max 0 v_2) := { f // B.map f = e.inv.app X₀ ≫ R.map e₀.hom ≫ g }
Z : S → CostructuredArrowDownwards e.hom g := fun s ↦ CostructuredArrowDownwards.mk e.hom g X₀ e₀.inv ↑s ⋯
this : Nonempty (CostructuredArrowDownwards e.hom g)
A₀ A₁ : CostructuredArrowDownwards e.hom g
H : ∀ (A : CostructuredArrowDownwards e.hom g), ∃ s, Nonempty (Z s ⟶ A)
s₀ : S
f₀ : Z s₀ ⟶ A₀
⊢ Zigzag A₀ A₁

Tactic:
obtain ⟨s₁, ⟨f₁⟩⟩ := H A₁

Goal state after:
C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H✝ : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H✝
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H✝
B : C ⥤ H✝
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
X : C
g : R.obj Y₀ ⟶ B.obj X
X₀ : C₀ := T.objPreimage Y₀
e₀ : T.obj X₀ ≅ Y₀ := T.objObjPreimageIso Y₀
S : Type (max 0 v_2) := { f // B.map f = e.inv.app X₀ ≫ R.map e₀.hom ≫ g }
Z : S → CostructuredArrowDownwards e.hom g := fun s ↦ CostructuredArrowDownwards.mk e.hom g X₀ e₀.inv ↑s ⋯
this : Nonempty (CostructuredArrowDownwards e.hom g)
A₀ A₁ : CostructuredArrowDownwards e.hom g
H : ∀ (A : CostructuredArrowDownwards e.hom g), ∃ s, Nonempty (Z s ⟶ A)
s₀ : S
f₀ : Z s₀ ⟶ A₀
s₁ : S
f₁ : Z s₁ ⟶ A₁
⊢ Zigzag A₀ A₁
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
⊢ Mono ((ι f n₀).f i)

Tactic:
by
    obtain ⟨q, _⟩ : ∃ (q : ℕ), IsIso ((midπ f n₀ q).f i) :=
      ⟨(i - n₀).natAbs, isIso_midπ_f f n₀ _ i⟩
    exact mono_of_mono_fac (ι_midπ_f f n₀ q i)

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = ↑n
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology L n
this : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology n
⊢ (ComplexShape.up ℕ).prev n = n - 1

Tactic:
cases n <;> simp

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = ↑n
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology L n
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology n
this : n ≠ 0
⊢ (ComplexShape.embeddingUpIntGE 0).f (n - 1) = m - 1

Tactic:
cases n <;> simp

Goal state after:
case zero
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
inst✝¹ : NeZero 0
hm : m = ↑0
inst✝ : HasHomology L 0
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology 0
this : 0 ≠ 0
⊢ 0 = m - 1

case succ
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
n✝ : ℕ
inst✝¹ : NeZero (n✝ + 1)
hm : m = ↑(n✝ + 1)
inst✝ : HasHomology L (n✝ + 1)
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology (n✝ + 1)
this : n✝ + 1 ≠ 0
⊢ ↑n✝ = m - 1
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = -↑(n + 1)
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology K n
this : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology n
⊢ (ComplexShape.down ℕ).next n = n - 1

Tactic:
cases n <;> simp

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = -↑(n + 1)
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology K n
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology n
this : n ≠ 0
⊢ (ComplexShape.embeddingUpIntLE (-1)).f (n - 1) = m + 1

Tactic:
cases n <;> simp

Goal state after:
case zero
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
inst✝¹ : NeZero 0
hm : m = -↑(0 + 1)
inst✝ : HasHomology K 0
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology 0
this : 0 ≠ 0
⊢ -1 = m + 1

case succ
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
n✝ : ℕ
inst✝¹ : NeZero (n✝ + 1)
hm : m = -↑(n✝ + 1 + 1)
inst✝ : HasHomology K (n✝ + 1)
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology (n✝ + 1)
this : n✝ + 1 ≠ 0
⊢ -1 - ↑n✝ = m + 1
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ i₆ i₇ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₆ : i₅ ⟶ i₆
f₇ : i₆ ⟶ i₇
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
⊢ X.d f₃ f₄ f₅ f₆ f₇ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₁ n₂ n₃ n₄ hn₂ hn₃ hn₄ = 0

Tactic:
← cancel_epi (X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂),

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ i₆ i₇ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₆ : i₅ ⟶ i₆
f₇ : i₆ ⟶ i₇
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
⊢ X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂ ≫
      X.d f₃ f₄ f₅ f₆ f₇ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₁ n₂ n₃ n₄ hn₂ hn₃ hn₄ =
    X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂ ≫ 0
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ i₆ i₇ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₆ : i₅ ⟶ i₆
f₇ : i₆ ⟶ i₇
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
⊢ X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂ ≫
      X.d f₃ f₄ f₅ f₆ f₇ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₁ n₂ n₃ n₄ hn₂ hn₃ hn₄ =
    X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂ ≫ 0

Tactic:
← cancel_epi (X.toCycles f₅ f₆ _ rfl n₁),

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ i₆ i₇ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₆ : i₅ ⟶ i₆
f₇ : i₆ ⟶ i₇
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
⊢ X.toCycles f₅ f₆ (f₅ ≫ f₆) ⋯ n₁ ≫
      X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂ ≫
        X.d f₃ f₄ f₅ f₆ f₇ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₁ n₂ n₃ n₄ hn₂ hn₃ hn₄ =
    X.toCycles f₅ f₆ (f₅ ≫ f₆) ⋯ n₁ ≫ X.πE f₅ f₆ f₇ n₀ n₁ n₂ hn₁ hn₂ ≫ 0
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
n₀ n₁ n₂ : ℤ
h₀ : IsIso ((X.H n₀).map ((functorArrows ι 2 3 3 isIso_map._proof_6 shortComplexMap._proof_3).map α))
h₁ : IsIso ((X.H n₁).map ((functorArrows ι 1 2 3 isIso_map._proof_8 isIso_map._proof_6).map α))
h₂ : IsIso ((X.H n₂).map ((functorArrows ι 0 1 3 isIso_map._proof_10 isIso_map._proof_12).map α))
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯)

Tactic:
apply +allowSynthFailures ShortComplex.isIso_of_isIso

Goal state after:
case inst
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
n₀ n₁ n₂ : ℤ
h₀ : IsIso ((X.H n₀).map ((functorArrows ι 2 3 3 isIso_map._proof_6 shortComplexMap._proof_3).map α))
h₁ : IsIso ((X.H n₁).map ((functorArrows ι 1 2 3 isIso_map._proof_8 isIso_map._proof_6).map α))
h₂ : IsIso ((X.H n₂).map ((functorArrows ι 0 1 3 isIso_map._proof_10 isIso_map._proof_12).map α))
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯).τ₁

case inst
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
n₀ n₁ n₂ : ℤ
h₀ : IsIso ((X.H n₀).map ((functorArrows ι 2 3 3 isIso_map._proof_6 shortComplexMap._proof_3).map α))
h₁ : IsIso ((X.H n₁).map ((functorArrows ι 1 2 3 isIso_map._proof_8 isIso_map._proof_6).map α))
h₂ : IsIso ((X.H n₂).map ((functorArrows ι 0 1 3 isIso_map._proof_10 isIso_map._proof_12).map α))
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯).τ₂

case inst
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
i' j' k' l' : ι
f₁' : i' ⟶ j'
f₂' : j' ⟶ k'
f₃' : k' ⟶ l'
α : mk₃ f₁ f₂ f₃ ⟶ mk₃ f₁' f₂' f₃'
n₀ n₁ n₂ : ℤ
h₀ : IsIso ((X.H n₀).map ((functorArrows ι 2 3 3 isIso_map._proof_6 shortComplexMap._proof_3).map α))
h₁ : IsIso ((X.H n₁).map ((functorArrows ι 1 2 3 isIso_map._proof_8 isIso_map._proof_6).map α))
h₂ : IsIso ((X.H n₂).map ((functorArrows ι 0 1 3 isIso_map._proof_10 isIso_map._proof_12).map α))
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯).τ₃
Module: Mathlib.CategoryTheory.Square

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
sq₁ sq₂ : Square C
e₁ : sq₁.X₁ ≅ sq₂.X₁
e₂ : sq₁.X₂ ≅ sq₂.X₂
e₃ : sq₁.X₃ ≅ sq₂.X₃
e₄ : sq₁.X₄ ≅ sq₂.X₄
comm₁₂ : sq₁.f₁₂ ≫ e₂.hom = e₁.hom ≫ sq₂.f₁₂
comm₁₃ : sq₁.f₁₃ ≫ e₃.hom = e₁.hom ≫ sq₂.f₁₃
comm₂₄ : sq₁.f₂₄ ≫ e₄.hom = e₂.hom ≫ sq₂.f₂₄
comm₃₄ : sq₁.f₃₄ ≫ e₄.hom = e₃.hom ≫ sq₂.f₃₄
⊢ sq₂.f₁₂ ≫ e₂.inv = e₁.inv ≫ sq₁.f₁₂

Tactic:
simp only [← cancel_mono e₂.hom, assoc, Iso.inv_hom_id,
                      comp_id, comm₁₂, Iso.inv_hom_id_assoc]

Goal state after:
Module: Mathlib.Order.Filter.EventuallyConst

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
f : α → β
ι : Sort u_5
p : ι → Prop
s : ι → Set α
h : l.HasBasis p s
⊢ (∃ i, p i ∧ (f '' s i).Subsingleton) ↔ ∃ i, p i ∧ ∀ x ∈ s i, ∀ y ∈ s i, f x = f y

Tactic:
simp only [Set.Subsingleton, forall_mem_image]

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r r' : ℤ
hrr' : r + 1 = r'
hr : r₀ ≤ r
pq' pq'' : κ
hpq' : (c r).next pq' = pq''
i₀' i₀ i₁ i₂ i₃ : ι
hi₀' : i₀' = data.i₀ r' pq' ⋯
hi₀ : i₀ = data.i₀ r pq' ⋯
hi₁ : i₁ = data.i₁ pq'
hi₂ : i₂ = data.i₂ pq'
hi₃ : i₃ = data.i₃ r pq' ⋯
n₀ n₁ n₂ : ℤ
hn₁' : n₁ = data.deg pq'
inst✝ : X.HasSpectralSequence data
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).Exact

Tactic:
by_cases h : (c r).Rel pq' pq''
  · refine ShortComplex.exact_of_iso (Iso.symm ?_)
      (X.dKernelSequence_exact
        (homOfLE (show data.i₀ r pq'' ≤ i₀' by
          simpa only [hi₀', data.i₀_prev r r' _ _ h] using data.le₀₁ r pq''))
        (homOfLE (data.i₀_le' hrr' hr pq' hi₀' hi₀)) (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
        (homOfLE (data.le₁₂' pq' hi₁ hi₂)) (homOfLE (data.le₂₃' r hr pq' hi₂ hi₃)) _ rfl
        n₀ n₁ n₂ (n₂ + 1) hn₁ hn₂ rfl)
    refine ShortComplex.isoMk (Iso.refl _)
      (pageXIso X data _ hr _ _ _ _ _ hi₀ hi₁ hi₂ hi₃ _ _ _ hn₁')
      (pageXIso X data _ hr _ _ _ _ _ rfl (by rw [hi₀', data.i₀_prev r r' _ _ h])
      (by rw [hi₀, data.hc₀₂ r _ _ h]) (by rw [hi₁, data.hc₁₃ r _ _ h]) _ _ _
      (by have := data.hc r _ _ h; lia)) ?_ ?_
    · simp
    · dsimp
      rw [pageD_eq X data r hr pq' pq'' h
        (homOfLE (data.le₀₁' r hr pq'' rfl (by simpa [← data.i₀_prev r r' _ _ h])))
        (homOfLE (data.i₀_le' hrr' hr pq' hi₀' hi₀)) (homOfLE (data.le₀₁' r hr pq' hi₀ hi₁))
        (homOfLE (data.le₁₂' pq' hi₁ hi₂)) (homOfLE (data.le₂₃' r hr pq' hi₂ hi₃))
        rfl (by rw [hi₀', data.i₀_prev r r' _ _ h]) hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ (n₂ + 1) hn₁',
        Category.assoc, Category.assoc, Iso.inv_hom_id, Category.comp_id]
  · rw [ShortComplex.exact_iff_epi _ ((page X data r hr).shape _ _ h)]
    have := isIso_mapFourδ₁Toδ₀' X data r r' hrr' hr pq' pq'' hpq'
      i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' h
    dsimp
    infer_instance

Goal state after:
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsArtinian R M
inst✝ : IsNoetherian R M
f : Module.End R M
n : ℕ
⊢ Disjoint (f ^ n).ker (f ^ n).range ∧ Codisjoint (f ^ n).ker (f ^ n).range → IsCompl (f ^ n).ker (f ^ n).range

Tactic:
hn

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsArtinian R M
inst✝ : IsNoetherian R M
f : Module.End R M
n : ℕ
hn : Disjoint (f ^ n).ker (f ^ n).range ∧ Codisjoint (f ^ n).ker (f ^ n).range
⊢ IsCompl (f ^ n).ker (f ^ n).range
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
r : ℤ
hr : r₀ ≤ r
pq pq' : κ
hpq : (c r).Rel pq pq'
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
h₀ : i₀ = data.i₀ r pq' ⋯
h₁ : i₁ = data.i₁ pq'
h₂ : i₂ = data.i₀ r pq ⋯
h₃ : i₃ = data.i₁ pq
h₄ : i₄ = data.i₂ pq
h₅ : i₅ = data.i₃ r pq ⋯
n₀ n₁ n₂ n₃ : ℤ
hn₁' : n₁ = data.deg pq
hn₁ : autoParam (n₀ + 1 = n₁) spectralSequence_page_d_eq._auto_1
hn₂ : autoParam (n₁ + 1 = n₂) spectralSequence_page_d_eq._auto_3
hn₃ : autoParam (n₂ + 1 = n₃) spectralSequence_page_d_eq._auto_5
⊢ i₂ = data.i₂ pq'

Tactic:
by rw [h₂, ← data.hc₀₂ r pq pq' hpq]

Goal state after:
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : IsArtinianRing R
x : R
⊢ IsUnit x ↔ IsRightRegular x

Tactic:
by
  rw [IsRightRegular, IsUnit.isUnit_iff_mulRight_bijective, Bijective, and_iff_left_of_imp]
  exact IsArtinian.surjective_of_injective_endomorphism (.toSpanSingleton R R x)

Goal state after:
Module: Mathlib.RingTheory.Artinian.Module

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsArtinianRing R
⊢ range PrimeSpectrum.asIdeal = range MaximalSpectrum.asIdeal

Tactic:
by
  simp only [PrimeSpectrum.range_asIdeal, MaximalSpectrum.range_asIdeal,
    isPrime_iff_isMaximal]

Goal state after: