mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
c₁₂ : ComplexShape I₁₂
inst✝³ : TotalComplexShape c₁ c₂ c₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : K.HasTotal c₁₂
inst✝ : L.HasTotal c₁₂
i₁₂ i₁₂' : I₁₂
⊢ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂ ≫ L.D₂ c₁₂ i₁₂ i₁₂' =
    K.D₂ c₁₂ i₁₂ i₁₂' ≫ GradedObject.mapMap (toGradedObjectMap φ) (c₁.π c₂ c₁₂) i₁₂'

Tactic:
cat_disch

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝¹³ : Category.{v_1, u_1} C₁
inst✝¹² : Category.{v_2, u_2} C₂
inst✝¹¹ : Category.{v_3, u_3} D
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
inst✝¹⁰ : HasZeroMorphisms C₁
inst✝⁹ : HasZeroMorphisms C₂
inst✝⁸ : Preadditive D
K₁ L₁ : HomologicalComplex C₁ c₁
φ₁ : K₁ ⟶ L₁
K₂ L₂ : HomologicalComplex C₂ c₂
φ₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
inst✝⁶ : ∀ (X₁ : C₁), (F.obj X₁).PreservesZeroMorphisms
c : ComplexShape J
inst✝⁵ : TotalComplexShape c₁ c₂ c
inst✝⁴ : TotalComplexShape c₂ c₁ c
inst✝³ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝² : DecidableEq J
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
⊢ mapBifunctorMap φ₂ φ₁ F.flip c ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom =
    (K₁.mapBifunctorFlipIso K₂ F c).hom ≫ mapBifunctorMap φ₁ φ₂ F c

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Twist

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : Type w
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
t : TwistShiftData C A
i j k : A
h : i + j = k
X : t.Category
⊢ (eqToHom ⋯).app X ≫ (shiftFunctorAdd t.Category i j).hom.app X =
    ↑(t.z i j) • (eqToHom ⋯).app X ≫ (shiftFunctorAdd C i j).hom.app X

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Twist

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : Type w
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
t : TwistShiftData C A
i j k : A
h : i + j = k
X : t.Category
⊢ (shiftFunctorAdd t.Category i j).inv.app X ≫ (eqToHom ⋯).app X =
    ↑(t.z i j)⁻¹ • (shiftFunctorAdd C i j).inv.app X ≫ (eqToHom ⋯).app X

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Shift.CommShiftTwo

Goal state before:
pp: C₁ : Type u_1
C₁' : Type u_2
C₂ : Type u_3
C₂' : Type u_4
D : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₁'
inst✝⁵ : Category.{v_3, u_3} C₂
inst✝⁴ : Category.{v_4, u_4} C₂'
inst✝³ : Category.{v_5, u_5} D
inst✝² : Preadditive D
inst✝¹ : HasShift D ℤ
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
x✝¹ x✝ : ℤ × ℤ
⊢ ∀ (a : ℤ × ℤ),
    (Functor.commShiftIso (𝟭 (PullbackShift D (AddMonoidHom.fst ℤ ℤ + AddMonoidHom.snd ℤ ℤ))) a).hom ≫
        Functor.whiskerRight (↑((-1) ^ (x✝¹.1 * x✝.2)))
          (shiftFunctor (PullbackShift D (AddMonoidHom.fst ℤ ℤ + AddMonoidHom.snd ℤ ℤ)) a) =
      (shiftFunctor (PullbackShift D (AddMonoidHom.fst ℤ ℤ + AddMonoidHom.snd ℤ ℤ)) a).whiskerLeft
          ↑((-1) ^ (x✝¹.1 * x✝.2)) ≫
        (Functor.commShiftIso (𝟭 (PullbackShift D (AddMonoidHom.fst ℤ ℤ + AddMonoidHom.snd ℤ ℤ))) a).hom

Tactic:
cat_disch

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

Goal state before:
pp: case e'_2
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
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
⊢ f ≫ biprod.inl - g ≫ biprod.inr = biprod.lift f (-g)

Tactic:
cat_disch

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

Goal state before:
pp: case e'_3
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
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
sq : CommSq f g inl inr
h : IsColimit (PushoutCocone.mk inl inr ⋯)
s : Cofork (biprod.lift f (-g)) 0
⊢ 0 = 0 ≫ s.π

Tactic:
cat_disch

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)
p q : ℤ
⊢ ((fun h ↦ homotopyOp h) ((fun h ↦ homotopyUnop h) h)).hom p q = h.hom p q

Tactic:
simp [homotopyOp_hom_eq _ p q (-p) (-q),
      homotopyUnop_hom_eq _ (-q) (-p) q p]

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

Goal state before:
pp: case e'_2
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
f : X₁ ⟶ X₂
g : X₁ ⟶ X₃
inl : X₂ ⟶ X₄
inr : X₃ ⟶ X₄
sq : CommSq f g inl inr
h : IsColimit sq.cokernelCofork
s : PushoutCocone f g
m : X₄ ⟶ s.pt
hm₁ : inl ≫ m = s.inl
hm₂ : inr ≫ m = s.inr
e_1✝ :
  (X₂ ⊞ X₃ ⟶ s.pt) =
    ((parallelPair (biprod.lift f (-g)) 0).obj WalkingParallelPair.one ⟶
      (CokernelCofork.ofπ (biprod.desc s.inl s.inr) ⋯).pt)
⊢ Cofork.π sq.cokernelCofork ≫ m = (CokernelCofork.ofπ (biprod.desc s.inl s.inr) ⋯).ι.app WalkingParallelPair.one

Tactic:
cat_disch

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

Goal state before:
pp: case e'_2
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
⊢ biprod.fst ≫ f - biprod.snd ≫ g = biprod.desc f (-g)

Tactic:
cat_disch

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

Goal state before:
pp: case e'_3
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
⊢ 0 = s.ι ≫ 0

Tactic:
cat_disch

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

Goal state before:
pp: case e'_2
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 sq.kernelFork
s : PullbackCone f g
m : s.pt ⟶ X₁
hm₁ : m ≫ fst = s.fst
hm₂ : m ≫ snd = s.snd
e_1✝ :
  (s.pt ⟶ X₂ ⊞ X₃) =
    ((KernelFork.ofι (biprod.lift s.fst s.snd) ⋯).pt ⟶
      (parallelPair (biprod.desc f (-g)) 0).obj WalkingParallelPair.zero)
⊢ m ≫ Fork.ι sq.kernelFork = (KernelFork.ofι (biprod.lift s.fst s.snd) ⋯).π.app WalkingParallelPair.zero

Tactic:
cat_disch

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n m p : ℤ
hm : n + 1 = m
hp : m + 1 = p
⊢ AddCommGrpCat.ofHom (Cocycle.toCochainAddMonoidHom K L m) ≫ (HomologicalComplex.sc' (K.HomComplex L) n m p).g = 0

Tactic:
cat_disch

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
hφ : ∀ (k : ℕ) (x : ↑(X k)), (↑(φ k x)).EqUpTo (↑x) (d + ↑k)
n : ℤ
⊢ (Cochain.ofHom f).v n n ⋯ = (δ (-1) 0 (limitSequence φ hφ x₀)).v n n ⋯

Tactic:
let k₀ := (n - d + 1).toNat

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
L : CochainComplex C ℤ
d : ℤ
inst✝¹ : L.IsStrictlyGE d
inst✝ : ∀ (n : ℤ), Injective (L.X n)
K : CochainComplex C ℤ
f : K ⟶ L
hK : HomologicalComplex.Acyclic K
X : ℕ → Set (Cochain K L (-1)) := fun n ↦ {α | (δ (-1) 0 α).EqUpTo (Cochain.ofHom f) (↑n + d - 1)}
x₀ : ↑(X 0) := ⟨0, ⋯⟩
φ : (n : ℕ) → ↑(X n) → ↑(X (n + 1)) := fun n α ↦ ⟨↑α + Cochain.single ⋯.choose (-1), ⋯⟩
hφ : ∀ (k : ℕ) (x : ↑(X k)), (↑(φ k x)).EqUpTo (↑x) (d + ↑k)
n : ℤ
k₀ : ℕ := (n - d + 1).toNat
⊢ (Cochain.ofHom f).v n n ⋯ = (δ (-1) 0 (limitSequence φ hφ x₀)).v n n ⋯
Module: Mathlib.AlgebraicTopology.ModelCategory.Cylinder

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
A : C
P : Precylinder A
inst✝ : HasBinaryCoproducts C
⊢ P.symm.i = (coprod.braiding A A).hom ≫ P.i

Tactic:
cat_disch

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
A : C
h : (cofibrations C).MapFactorizationData (trivialFibrations C) (codiag A)
⊢ (ofFactorizationData h).i = h.i

Tactic:
cat_disch

Goal state after:
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 := ⋯ }
⊢ P'.i = P.i ≫ fac.i

Tactic:
cat_disch

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{v_1, u_1} C₁
inst✝²² : Category.{v_2, u_2} C₂
inst✝²¹ : Category.{v_3, u_5} C₃
inst✝²⁰ : Category.{v_4, u_6} C₄
inst✝¹⁹ : Category.{v_6, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroMorphisms C₃
inst✝¹⁵ : Preadditive C₂₃
inst✝¹⁴ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹³ : G₂₃.PreservesZeroMorphisms
inst✝¹² : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹¹ : F.PreservesZeroMorphisms
inst✝¹⁰ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁹ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁸ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁷ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁶ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁵ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁴ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝³ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝² : DecidableEq ι₂₃
inst✝¹ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₂₃ : ι₂₃ := ⋯
h₁ : c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
h₂ : ¬c₁.π c₂₃ c₄ (i₁, c₂₃.next i₂₃) = j'
⊢ 0 + 0 = d₂ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' + d₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j'

Tactic:
congr 1
        · by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
          · rw [d₂_eq _ _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            intro h₄
            apply h₂
            rw [← h₄]
            dsimp [ComplexShape.r]
            rw [ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄, ComplexShape.next_π₁ c₃ c₂₃ h₃ i₃]
          · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₃]
        · by_cases h₃ : c₃.Rel i₃ (c₃.next i₃)
          · rw [d₃_eq _ _ _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            intro h₄
            apply h₂
            rw [← h₄]
            dsimp [ComplexShape.r]
            rw [ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄, ComplexShape.next_π₂ c₂ c₂₃ i₂ h₃]
          · rw [d₃_eq_zero _ _ _ _ _ _ _ _ _ _ _ _ h₃]

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

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✝ : IsCofibrant X
h✝ : RightHomotopyRel f g
P : PathObject Y
left✝ : P.IsGood
h : P.RightHomotopy f g
fac : (cofibrations C).MapFactorizationData (trivialFibrations C) P.ι :=
  (cofibrations C).factorizationData (trivialFibrations C) P.ι
P' : PathObject Y :=
  { P := fac.Z, p₀ := fac.p ≫ P.p₀, p₁ := fac.p ≫ P.p₁, ι := fac.i, ι_p₀ := ⋯, ι_p₁ := ⋯, weakEquivalence_ι := ⋯ }
⊢ P'.p = fac.p ≫ P.p

Tactic:
cat_disch

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K : CochainComplex C ℤ
⊢ plus C ((quotient C (ComplexShape.up ℤ)).obj K) ↔ CochainComplex.plus C K

Tactic:
refine ⟨?_, fun h ↦ ⟨_, h⟩⟩
  simp only [plus, ObjectProperty.strictMap_iff]
  rintro ⟨L, h, hL⟩
  obtain rfl : L = K := congr_arg Quotient.as hL
  exact h

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: 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
x✝ : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ (shiftFunctor C 1).obj W
hT : Triangle.mk a s b ∈ Pretriangulated.distinguishedTriangles
hW : B W
W' : C
c : W ⟶ W'
d : W' ⟶ X
h₁ : A W'
h₂ : B W'
fac : c ≫ d = a
U : C
hU : A U
e : X ⟶ U
f : U ⟶ (shiftFunctor C 1).obj W'
hT' : Triangle.mk d e f ∈ Pretriangulated.distinguishedTriangles
⊢ (Triangle.mk a s b).mor₁ ≫ 𝟙 (Triangle.mk a s b).obj₂ = c ≫ (Triangle.mk d e f).mor₁

Tactic:
cat_disch

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 ≤ b
⊢ t.natTransTruncLEOfLE a b h ≫ t.truncLEι b = t.truncLEι a

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: 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
x✝ : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ (shiftFunctor C 1).obj W
hT : Triangle.mk a s b ∈ Pretriangulated.distinguishedTriangles
hW : B W
W' : C
c : W ⟶ W'
d : W' ⟶ X
h₁ : A W'
h₂ : B W'
fac : c ≫ d = a
U : C
hU : A U
e : X ⟶ U
f : U ⟶ (shiftFunctor C 1).obj W'
hT' : Triangle.mk d e f ∈ Pretriangulated.distinguishedTriangles
g : (Triangle.mk a s b).obj₃ ⟶ (Triangle.mk d e f).obj₃
hg : (Triangle.mk a s b).mor₂ ≫ g = 𝟙 (Triangle.mk a s b).obj₂ ≫ (Triangle.mk d e f).mor₂
right✝ : (Triangle.mk a s b).mor₃ ≫ (shiftFunctor C 1).map c = g ≫ (Triangle.mk d e f).mor₃
⊢ s ≫ g = e

Tactic:
cat_disch

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.truncGEπ b ≫ (t.truncGTIsoTruncGE a b h).inv = t.truncGTπ a

Tactic:
subst h
  dsimp [truncGTIsoTruncGE, truncGTπ, truncGT]
  rw [Category.comp_id]

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

Goal state before:
pp: case refine_1
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.triangleLEGE a b h).whiskerLeft Triangle.π₁Toπ₂ ≫ (Iso.refl (t.triangleLEGE a b h ⋙ Triangle.π₂)).hom =
    (t.truncLEIsoTruncLT a b h).hom ≫ (t.triangleLTGE b).whiskerLeft Triangle.π₁Toπ₂

Tactic:
cat_disch

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

Goal state before:
pp: case refine_2
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.triangleLEGE a b h).whiskerLeft Triangle.π₂Toπ₃ ≫ (Iso.refl (t.triangleLEGE a b h ⋙ Triangle.π₃)).hom =
    (Iso.refl (t.triangleLEGE a b h ⋙ Triangle.π₂)).hom ≫ (t.triangleLTGE b).whiskerLeft Triangle.π₂Toπ₃

Tactic:
cat_disch

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.triangleLEGT a).whiskerLeft Triangle.π₁Toπ₂ ≫ (Iso.refl (t.triangleLEGT a ⋙ Triangle.π₂)).hom =
    (Iso.refl (t.triangleLEGT a ⋙ Triangle.π₁)).hom ≫ (t.triangleLEGE a b h).whiskerLeft Triangle.π₁Toπ₂

Tactic:
cat_disch

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.triangleLEGT a).whiskerLeft Triangle.π₂Toπ₃ ≫ (t.truncGTIsoTruncGE a b h).hom =
    (Iso.refl (t.triangleLEGT a ⋙ Triangle.π₂)).hom ≫ (t.triangleLEGE a b h).whiskerLeft Triangle.π₂Toπ₃

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: 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 X → B.trW s → ∃ Z s' b, A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : Y ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f s b ∈ Pretriangulated.distinguishedTriangles
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
U : C
hU : A U
a : U ⟶ Y
c : W ⟶ (shiftFunctor C 1).obj U
hT' : Triangle.mk a s' c ∈ Pretriangulated.distinguishedTriangles
⊢ (Triangle.mk f s b).mor₂ ≫ g = 𝟙 Y ≫ (Triangle.mk a s' c).mor₂

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: 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 X → B.trW s → ∃ Z s' b, A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : Y ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f s b ∈ Pretriangulated.distinguishedTriangles
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
U : C
hU : A U
a : U ⟶ Y
c : W ⟶ (shiftFunctor C 1).obj U
hT' : Triangle.mk a s' c ∈ Pretriangulated.distinguishedTriangles
t : (Triangle.mk f s b).obj₁ ⟶ (Triangle.mk a s' c).obj₁
ht : (Triangle.mk f s b).mor₁ ≫ 𝟙 Y = t ≫ (Triangle.mk a s' c).mor₁
ht' : (Triangle.mk f s b).mor₃ ≫ (shiftFunctor C 1).map t = g ≫ (Triangle.mk a s' c).mor₃
⊢ t ≫ a = f

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: 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 : Cᵒᵖ⦄ (s : X ⟶ Y), A.op X → B.op.trW s → ∃ Z s' b, A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'
X Y : C
s : X ⟶ Y
hY : A Y
hs : B.trW s
Z' : Cᵒᵖ
s' : op Y ⟶ Z'
b : op X ⟶ Z'
hZ' : A.op Z'
hs' : (A.op ⊓ B.op).trW s'
fac : s.op ≫ b = s'
⊢ b.unop ≫ s = s'.unop

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: 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' : C
s' : Z' ⟶ unop X
b : Z' ⟶ unop Y
hZ' : A Z'
hs' : (A ⊓ B).trW s'
fac : b ≫ s.unop = s'
⊢ s ≫ b.op = s'.op

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: case mpr.refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
Z : C
a : Y.obj ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X.obj
h : Triangle.mk f.hom a b ∈ Pretriangulated.distinguishedTriangles
hZ : (A ⊓ B) Z
⊢ (A.ι.mapTriangle.obj (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)))).mor₁ ≫
      (Iso.refl
          (A.ι.mapTriangle.obj
              (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)))).obj₂).hom =
    (Iso.refl
          (A.ι.mapTriangle.obj
              (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)))).obj₁).hom ≫
      (Triangle.mk f.hom a b).mor₁

Tactic:
cat_disch

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: case mpr.refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
A B : ObjectProperty C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
Z : C
a : Y.obj ⟶ Z
b : Z ⟶ (shiftFunctor C 1).obj X.obj
h : Triangle.mk f.hom a b ∈ Pretriangulated.distinguishedTriangles
hZ : (A ⊓ B) Z
⊢ (A.ι.mapTriangle.obj (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)))).mor₂ ≫
      (Iso.refl
          (A.ι.mapTriangle.obj
              (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)))).obj₃).hom =
    (Iso.refl
          (A.ι.mapTriangle.obj
              (Triangle.mk f (homMk a) (homMk (b ≫ (Functor.commShiftIso A.ι 1).inv.app X)))).obj₂).hom ≫
      (Triangle.mk f.hom a b).mor₂

Tactic:
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
T T' : Triangle C
f₁ f₂ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
a b : ℤ
h₀ : t.IsLE T.obj₁ a
h₁ : t.IsGE T'.obj₃ b
H : f₁.hom₂ = f₂.hom₂
hab : a ≤ b
this : ∀ (f : T ⟶ T'), f.hom₂ = 0 → f = 0
⊢ f₁ - f₂ = 0

Tactic:
cat_disch

Goal state after: