mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝ : LeftHomotopyRel f g
P : Cylinder X
left✝ : P.IsGood
h : P.LeftHomotopy f g
fac : (trivialCofibrations C).MapFactorizationData (fibrations C) P.π :=
  (trivialCofibrations C).factorizationData (fibrations C) P.π
P' : Cylinder X :=
  { I := fac.Z, i₀ := P.i₀ ≫ fac.i, i₁ := P.i₁ ≫ fac.i, π := fac.p, i₀_π := ⋯, i₁_π := ⋯, weakEquivalence_π := ⋯ }
⊢ Cofibration P'.i

Tactic:
show P'.i = P.i ≫ fac.i by cat_disch

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝ : LeftHomotopyRel f g
P : Cylinder X
left✝ : P.IsGood
h : P.LeftHomotopy f g
fac : (trivialCofibrations C).MapFactorizationData (fibrations C) P.π :=
  (trivialCofibrations C).factorizationData (fibrations C) P.π
P' : Cylinder X :=
  { I := fac.Z, i₀ := P.i₀ ≫ fac.i, i₁ := P.i₁ ≫ fac.i, π := fac.p, i₀_π := ⋯, i₁_π := ⋯, weakEquivalence_π := ⋯ }
⊢ Cofibration (P.i ≫ fac.i)
Module: Mathlib.AlgebraicTopology.ModelCategory.LeftHomotopy

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : ModelCategory C
f g : X ⟶ Y
inst✝ : IsFibrant Y
h✝¹ : LeftHomotopyRel f g
Z : C
i : Z ⟶ X
P : Cylinder X
left✝ : P.IsVeryGood
h : P.LeftHomotopy f g
Q : Cylinder Z
h✝ : Q.IsVeryGood
sq : CommSq (coprod.desc (i ≫ P.i₀) (i ≫ P.i₁)) Q.i P.π (Q.π ≫ i)
⊢ Q.i₀ ≫ sq.lift ≫ h.h = i ≫ f

Tactic:
have := coprod.inl ≫= sq.fac_left
        simp only [Q.inl_i_assoc, coprod.inl_desc] at this
        simp [reassoc_of% this]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a + 1 = b
⊢ t.truncLE a = t.truncLT b

Tactic:
rw [← h]

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a + 1 = b
⊢ t.truncLE a = t.truncLT (a + 1)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a + 1 = b
⊢ t.truncGT a = t.truncGE b

Tactic:
rw [← h]

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a + 1 = b
⊢ t.truncGT a = t.truncGE (a + 1)
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : A.IsTriangulated
inst✝¹ : B.IsTriangulated
inst✝ : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ Z s' a, A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
X Y : Cᵒᵖ
s : X ⟶ Y
hX : A.op X
hs : B.op.trW s
⊢ ∃ Z s' b, A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'

Tactic:
· obtain ⟨Z', s', b, hZ', hs', fac⟩ := hA s.unop hX (trW_of_op _ hs)
    exact ⟨_, s'.op, b.op, hZ', trW_of_unop _ hs', by cat_disch⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
⊢ t.truncLTι n ≫ t.truncGEπ n = 0

Tactic:
by
  cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
⊢ t.truncGEδLT n ≫ Functor.whiskerRight (t.truncLTι n) (shiftFunctor C 1) = 0

Tactic:
by
  cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a ≤ b
⊢ t.natTransTruncLTOfLE a b h ≫ t.truncLTι b = t.truncLTι a

Tactic:
by
  cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a b : ℤ
h : a ≤ b
⊢ t.truncGEπ a ≫ t.natTransTruncGEOfLE a b h = t.truncGEπ b

Tactic:
by
  cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
f : X ⟶ Y
inst✝ : IsIso f
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f_1, ∃ (w : f_1 ≫ f = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Y, f := f_1, g := f, zero := w }.Splitting

Tactic:
exact ⟨0, inferInstance, 0, by simp,
    ⟨ShortComplex.Splitting.ofIsZeroOfIsIso _ (isZero_zero C) (by assumption)⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: case mpr
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
a✝ : IsIso ((t.truncGEπ n).app X)
⊢ t.IsGE X n

Tactic:
exact t.isGE_of_iso (asIso ((truncGEπ t n).app X)).symm n

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0
⊢ t.IsLE ((t.truncLT n₁).obj X) n₀

Tactic:
rw [← h]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0
⊢ t.IsLE ((t.truncLT (n₀ + 1)).obj X) n₀
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
n m : ℤ
hnm : n + 1 = m
fac_right : ∀ (k : ℤ), (hsq k).l ≫ p.f k = b.f k
⊢ (↑(cocycle₁' sq hsq)).v n m hnm ≫ p.f m = 0

Tactic:
simp [cocycle₁', -HomologicalComplex.Hom.comm,
    ← p.comm, fac_right, reassoc_of% fac_right, b.comm]

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)
n m : ℤ
hnm : n + 1 = m
⊢ ∃ φ, π.f n ≫ φ ≫ ι.f m = (↑(cocycle₁' sq hsq)).v n m hnm

Tactic:
have : Epi π := Cofork.IsColimit.epi hQ

Goal state after:
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)
n m : ℤ
hnm : n + 1 = m
this : Epi π
⊢ ∃ φ, π.f n ≫ φ ≫ ι.f m = (↑(cocycle₁' sq hsq)).v n m hnm
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)
⊢ δ 1 2 (cochain₁ sq hsq hQ hK) = 0

Tactic:
have : Epi π := Cofork.IsColimit.epi hQ

Goal state after:
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)
this : Epi π
⊢ δ 1 2 (cochain₁ sq hsq hQ hK) = 0
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
a b : ℤ
X : C
⊢ inv ((t.truncLTι b).app ((t.truncGE a).obj ((t.truncLT b).obj X))) ≫
      (t.truncLTι b).app ((t.truncGE a).obj ((t.truncLT b).obj X)) ≫ (t.truncGELTToLTGE a b).app X =
    inv ((t.truncLTι b).app ((t.truncGE a).obj ((t.truncLT b).obj X))) ≫
      (t.truncLT b).map ((t.truncGE a).map ((t.truncLTι b).app X))

Tactic:
IsIso.inv_hom_id_assoc

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
a b : ℤ
X : C
⊢ (t.truncGELTToLTGE a b).app X =
    inv ((t.truncLTι b).app ((t.truncGE a).obj ((t.truncLT b).obj X))) ≫
      (t.truncLT b).map ((t.truncGE a).map ((t.truncLTι b).app X))
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 : IsCofibrant X
g : Y ⟶ Z
inst✝² : IsFibrant Y
inst✝¹ : IsFibrant Z
inst✝ : WeakEquivalence g
⊢ Function.Bijective fun f ↦ f.postcomp g

Tactic:
let h : FibrantBrownFactorization g := Classical.arbitrary _

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ModelCategory C
X Y Z : C
inst✝³ : IsCofibrant X
g : Y ⟶ Z
inst✝² : IsFibrant Y
inst✝¹ : IsFibrant Z
inst✝ : WeakEquivalence g
h : FibrantBrownFactorization g := Classical.arbitrary (FibrantBrownFactorization g)
⊢ Function.Bijective fun f ↦ f.postcomp g
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
x✝² x✝¹ : FibrantObject C
x✝ : x✝² ⟶ x✝¹
h : WeakEquivalence x✝
⊢ (weakEquivalences (HoCat C)).inverseImage toHoCat x✝

Tactic:
simpa only [MorphismProperty.inverseImage_iff, ← weakEquivalence_iff,
      weakEquivalence_toHoCat_map_iff]

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
X₁ : C₁
b : L.obj X₁ ⟶ X₃
a : (CostructuredArrow.mk g).left ⟶ ((w.costructuredArrowRightwards X₃).obj (CostructuredArrow.mk b)).left
ha : R.map a ≫ ((w.costructuredArrowRightwards X₃).obj (CostructuredArrow.mk b)).hom = (CostructuredArrow.mk g).hom
⊢ ∃ X₁_1 a_1 b_1,
    ∃ (comm : R.map a_1 ≫ w.app X₁_1 ≫ B.map b_1 = g),
      StructuredArrow.mk (CostructuredArrow.homMk a ha) = mk w g X₁_1 a_1 b_1 comm

Tactic:
exact ⟨X₁, a, b, by simpa using ha, rfl⟩

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
f₁ f₂ : w.StructuredArrowRightwards g
φ : f₁ ⟶ f₂
⊢ (StructuredArrow.hom f₁).left ≫ T.map (StructuredArrow.Hom.right φ).left = (StructuredArrow.hom f₂).left

Tactic:
← StructuredArrow.w φ

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
f₁ f₂ : w.StructuredArrowRightwards g
φ : f₁ ⟶ f₂
⊢ (StructuredArrow.hom f₁).left ≫ T.map (StructuredArrow.Hom.right φ).left =
    (StructuredArrow.hom f₁ ≫ (w.costructuredArrowRightwards X₃).map (StructuredArrow.Hom.right φ)).left
Module: Mathlib.CategoryTheory.GuitartExact.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ X₂' : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
g' : R.obj X₂' ⟶ B.obj X₃
γ : X₂' ⟶ X₂
hγ : R.map γ ≫ g = g'
A A' : w.CostructuredArrowDownwards g
φ : A ⟶ A'
⊢ L.map (StructuredArrow.Hom.right φ.left) ≫ StructuredArrow.Hom.right (CostructuredArrow.hom A') =
    StructuredArrow.Hom.right (CostructuredArrow.hom A)

Tactic:
← CostructuredArrow.w φ,

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₂ X₂' : C₂
X₃ : C₃
g : R.obj X₂ ⟶ B.obj X₃
g' : R.obj X₂' ⟶ B.obj X₃
γ : X₂' ⟶ X₂
hγ : R.map γ ≫ g = g'
A A' : w.CostructuredArrowDownwards g
φ : A ⟶ A'
⊢ L.map (StructuredArrow.Hom.right φ.left) ≫ StructuredArrow.Hom.right (CostructuredArrow.hom A') =
    StructuredArrow.Hom.right ((w.structuredArrowDownwards X₂).map φ.left ≫ CostructuredArrow.hom A')
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝⁵ : Category.{v₁, u₁} C₁
inst✝⁴ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝³ : Category.{v_1, u_1} D₁
inst✝² : Category.{v_2, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
h : Φ.IsRightDerivabilityStructure
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
⊢ TwoSquare.GuitartExact e.hom

Tactic:
by
  simpa only [Φ.isRightDerivabilityStructure_iff L₁ L₂ F e] using h

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
RF RF' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
α' : F ⟶ L ⋙ RF'
W : MorphismProperty C
inst✝ : L.IsLocalization W
e : RF ≅ RF'
comm : α ≫ L.whiskerLeft e.hom = α'
⊢ RF.IsRightDerivedFunctor α W ↔ RF'.IsRightDerivedFunctor α' W

Tactic:
simp only [isRightDerivedFunctor_iff_isLeftKanExtension]

Goal state after:
C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
RF RF' : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
α' : F ⟶ L ⋙ RF'
W : MorphismProperty C
inst✝ : L.IsLocalization W
e : RF ≅ RF'
comm : α ≫ L.whiskerLeft e.hom = α'
⊢ RF.IsLeftKanExtension α ↔ RF'.IsLeftKanExtension α'
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
inst✝¹ : EnoughInjectives C
n : ℤ
inst✝ : K.IsStrictlyGE n
this : HasDerivedCategory C
L : CochainComplex C ℤ
i : K ⟶ L
w✝¹ : Mono i
w✝ : QuasiIso i
hL : ∀ (n : ℤ), Injective (L.X n)
h✝ : L.IsStrictlyGE (n - 1)
hK : K.IsGE n
⊢ L.IsGE n

Tactic:
rw [← DerivedCategory.isGE_Q_obj_iff] at hK ⊢

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
inst✝¹ : EnoughInjectives C
n : ℤ
inst✝ : K.IsStrictlyGE n
this : HasDerivedCategory C
L : CochainComplex C ℤ
i : K ⟶ L
w✝¹ : Mono i
w✝ : QuasiIso i
hL : ∀ (n : ℤ), Injective (L.X n)
h✝ : L.IsStrictlyGE (n - 1)
hK : (DerivedCategory.Q.obj K).IsGE n
⊢ (DerivedCategory.Q.obj L).IsGE n
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

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
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
X : C
P : StructuredArrow (L.obj X) L → Prop
hP₀ : P (StructuredArrow.mk (𝟙 (L.obj X)))
hP₁ :
  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂) (φ : L.obj X ⟶ L.obj Y₁), P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ L.map f))
hP₂ :
  ∀ ⦃Y₁ Y₂ : C⦄ (w : Y₁ ⟶ Y₂) (hw : W w) (φ : L.obj X ⟶ L.obj Y₂),
    P (StructuredArrow.mk φ) → P (StructuredArrow.mk (φ ≫ (isoOfHom L W w hw).inv))
g : StructuredArrow (L.obj X) L
⊢ P g

Tactic:
by
  let P' : StructuredArrow (W.Q.obj X) W.Q → Prop :=
    fun g ↦ P (structuredArrowEquiv W W.Q L g)
  rw [← (structuredArrowEquiv W W.Q L).apply_symm_apply g]
  apply induction_structuredArrow' W P'
  · convert! hP₀
    simp
  · intro Y₁ Y₂ f φ hφ
    convert! hP₁ f (homEquiv W W.Q L φ) hφ
    simp [homEquiv_comp]
  · intro Y₁ Y₂ w hw φ hφ
    convert! hP₂ w hw (homEquiv W W.Q L φ) hφ
    simp [homEquiv_comp, homEquiv_isoOfHom_inv]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.KProjective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
d : ℤ
inst✝¹ : K.IsStrictlyLE d
inst✝ : ∀ (n : ℤ), Projective (K.X n)
⊢ K.IsKProjective

Tactic:
let L := ((opEquivalence C).functor.obj (op K))
  have (n : ℤ) : Injective (L.X n) := by
    dsimp [L]
    infer_instance
  have : L.IsStrictlyGE (-d) := by
    rw [isStrictlyGE_iff]
    intro i hi
    exact (K.isZero_of_isStrictlyLE d _ (by dsimp; lia)).op
  exact isKProjective_of_op (isKInjective_of_injective L (-d))

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives

Goal state before:
pp: C : Type u_1
H : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Abelian C
inst✝⁴ : Category.{v_2, u_2} H
inst✝³ : EnoughInjectives C
F : Plus C ⥤ H
inst✝² : HasDerivedCategory C
F' : DerivedCategory.Plus C ⥤ H
α : F ⟶ DerivedCategory.Plus.Qh ⋙ F'
inst✝¹ : F'.IsRightDerivedFunctor α (quasiIso C)
K : Plus C
inst✝ : ∀ (n : ℤ), Injective (K.obj.as.X n)
this : ∀ (Y : Plus (InjectiveObject C)), IsIso (α.app ((InjectiveObject.ι C).mapHomotopyCategoryPlus.obj Y))
⊢ (InjectiveObject.ι C).mapHomotopyCategoryPlus.essImage K

Tactic:
obtain ⟨X, hX⟩ := K
    obtain ⟨K, rfl⟩ := HomotopyCategory.quotient_obj_surjective X
    refine ⟨(quotient _).obj
      ((CochainComplex.Plus.fibrantObjectEquivalence C).inverse.obj
      ⟨⟨K, by simpa using hX⟩, ?_⟩), ⟨Iso.refl _⟩⟩
    dsimp [fibrantObjects]
    rwa [CochainComplex.Plus.modelCategoryQuillen.isFibrant_iff]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
ι : Type u_2
c : ComplexShape ι
inst✝ : c.EulerCharSigns
X : GradedObject ι (ModuleCat R)
⊢ (Function.support fun i ↦ ↑(c.χ i) * ↑(Module.finrank R ↑(X i))) = finrankSupport X

Tactic:
simp only [finrankSupport, Function.support_mul_of_ne_zero_left (fun i => Units.ne_zero (c.χ i))]
  ext i; simp [Function.mem_support]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
w✝⁵ w✝⁴ w✝³ : CochainComplex C ℤ
f : w✝⁵ ⟶ w✝⁴
g : w✝⁴ ⟶ w✝³
w✝² w✝¹ w✝ : CochainComplex C ℤ
f' : w✝² ⟶ w✝¹
g' : w✝¹ ⟶ w✝
φ : ComposableArrows.mk₂ f g ⟶ ComposableArrows.mk₂ f' g'
eq :
  map g g' (φ.app 1) (φ.app 2) ⋯ ≫ (CochainComplex.mappingConeCompTriangle f' g').mor₃ =
    (CochainComplex.mappingConeCompTriangle f g).mor₃ ≫
      (shiftFunctor (CochainComplex C ℤ) 1).map (map f f' (φ.app 0) (φ.app 1) ⋯)
⊢ (ComposableArrows.functorArrows (CochainComplex C ℤ) 1 2 2 SpectralObject._proof_2 SpectralObject._proof_4 ⋙
            composableArrowsFunctor C ⋙ quotient C (ComplexShape.up ℤ)).map
        φ ≫
      ((quotient C (ComplexShape.up ℤ)).mapTriangle.obj
          (CochainComplex.mappingConeCompTriangle
            ((ComposableArrows.mk₂ f' g').map' 0 1 composableArrowsFunctor._proof_1 spectralObjectMappingCone._proof_2)
            ((ComposableArrows.mk₂ f' g').map' 1 2 spectralObjectMappingCone._proof_2
              spectralObjectMappingCone._proof_3))).mor₃ =
    ((quotient C (ComplexShape.up ℤ)).mapTriangle.obj
          (CochainComplex.mappingConeCompTriangle
            ((ComposableArrows.mk₂ f g).map' 0 1 composableArrowsFunctor._proof_1 spectralObjectMappingCone._proof_2)
            ((ComposableArrows.mk₂ f g).map' 1 2 spectralObjectMappingCone._proof_2
              spectralObjectMappingCone._proof_3))).mor₃ ≫
      (ComposableArrows.functorArrows (CochainComplex C ℤ) 0 1 2 SpectralObject._proof_6 SpectralObject._proof_2 ⋙
            (composableArrowsFunctor C ⋙ quotient C (ComplexShape.up ℤ)) ⋙
              shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
        φ

Tactic:
dsimp [ComposableArrows.Precomp.map] at eq ⊢

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
w✝⁵ w✝⁴ w✝³ : CochainComplex C ℤ
f : w✝⁵ ⟶ w✝⁴
g : w✝⁴ ⟶ w✝³
w✝² w✝¹ w✝ : CochainComplex C ℤ
f' : w✝² ⟶ w✝¹
g' : w✝¹ ⟶ w✝
φ : ComposableArrows.mk₂ f g ⟶ ComposableArrows.mk₂ f' g'
eq :
  map g g' (φ.app 1) (φ.app 2) ⋯ ≫ (triangle g').mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map (inr f') =
    ((triangle g).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map (inr f)) ≫
      (shiftFunctor (CochainComplex C ℤ) 1).map (map f f' (φ.app 0) (φ.app 1) ⋯)
⊢ (quotient C (ComplexShape.up ℤ)).map (map g g' (φ.app 1) (φ.app 2) ⋯) ≫
      (quotient C (ComplexShape.up ℤ)).map ((triangle g').mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map (inr f')) ≫
        (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (CochainComplex.mappingCone f') =
    ((quotient C (ComplexShape.up ℤ)).map ((triangle g).mor₃ ≫ (shiftFunctor (CochainComplex C ℤ) 1).map (inr f)) ≫
        (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (CochainComplex.mappingCone f)) ≫
      (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
        ((quotient C (ComplexShape.up ℤ)).map (map f f' (φ.app 0) (φ.app 1) ⋯))