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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
⊢ (γ₁ + γ₂).leftShift a n' hn' = γ₁.leftShift a n' hn' + γ₂.leftShift a n' hn'

Tactic:
ext p q hpq
  dsimp
  simp only [leftShift_v _ a n' hn' p q hpq (p + a) (by lia), add_v, comp_add, smul_add]

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

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' a : ℤ
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
x : R
⊢ (rightShiftLinearEquiv R K L n a n' hn).symm (x • γ) = x • γ.rightUnshift n hn

Tactic:
apply map_smul

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

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' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
x : R
⊢ (leftShiftLinearEquiv R K L n a n' hn).symm (x • γ) = x • γ.leftUnshift n hn

Tactic:
apply map_smul

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
B : Type u_5
inst✝³ : AddCommMonoid B
inst✝² : HasShift C B
inst✝¹ : HasShift D B
inst✝ : F.CommShift B
X : C
a b : B
eq :
  F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (commShiftIso F (a + b)).hom.app X =
    (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
      (shiftFunctor D b).map ((commShiftIso F a).hom.app X) ≫ (shiftFunctorAdd D a b).inv.app (F.obj X)
⊢ F.map ((shiftFunctorAdd C a b).inv.app X ≫ (shiftFunctorAdd' C b a (a + b) ⋯).hom.app X) =
    F.map ((shiftFunctorAdd C a b).inv.app X) ≫
      (commShiftIso F (a + b)).hom.app X ≫
        (shiftFunctorAdd' D b a (a + b) ⋯).hom.app (F.obj X) ≫
          (shiftFunctor D a).map ((commShiftIso F b).inv.app X) ≫ (commShiftIso F a).inv.app ((shiftFunctor C b).obj X)

Tactic:
Functor.map_comp

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
B : Type u_5
inst✝³ : AddCommMonoid B
inst✝² : HasShift C B
inst✝¹ : HasShift D B
inst✝ : F.CommShift B
X : C
a b : B
eq :
  F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (commShiftIso F (a + b)).hom.app X =
    (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
      (shiftFunctor D b).map ((commShiftIso F a).hom.app X) ≫ (shiftFunctorAdd D a b).inv.app (F.obj X)
⊢ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ F.map ((shiftFunctorAdd' C b a (a + b) ⋯).hom.app X) =
    F.map ((shiftFunctorAdd C a b).inv.app X) ≫
      (commShiftIso F (a + b)).hom.app X ≫
        (shiftFunctorAdd' D b a (a + b) ⋯).hom.app (F.obj X) ≫
          (shiftFunctor D a).map ((commShiftIso F b).inv.app X) ≫ (commShiftIso F a).inv.app ((shiftFunctor C b).obj X)
Module: Mathlib.CategoryTheory.Triangulated.Rotate

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (-(shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X✝.obj₁) ≫ f.hom₁ =
    -(shiftFunctor C (-1)).map (X✝.mor₃ ≫ (shiftFunctor C 1).map f.hom₁) ≫
        (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app Y✝.obj₁

Tactic:
Functor.map_comp

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (-(shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X✝.obj₁) ≫ f.hom₁ =
    -((shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctor C (-1)).map ((shiftFunctor C 1).map f.hom₁)) ≫
        (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app Y✝.obj₁
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mp
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h : T.mor₃ = 0
X : C
f : (CategoryTheory.shiftFunctor C 1).obj T.obj₁ ⟶ X
hg : T.mor₂ ≫ T.mor₃ ≫ f = 0
⊢ T.mor₃ ≫ f = 0

Tactic:
h,

Goal state after:
case mp
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h : T.mor₃ = 0
X : C
f : (CategoryTheory.shiftFunctor C 1).obj T.obj₁ ⟶ X
hg : T.mor₂ ≫ T.mor₃ ≫ f = 0
⊢ 0 ≫ f = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mp
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h : T.mor₁ = 0
X : C
f : X ⟶ T.obj₁
hg : (f ≫ T.mor₁) ≫ T.mor₂ = 0
⊢ f ≫ T.mor₁ = 0

Tactic:
h,

Goal state after:
case mp
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h : T.mor₁ = 0
X : C
f : X ⟶ T.obj₁
hg : (f ≫ T.mor₁) ≫ T.mor₂ = 0
⊢ f ≫ 0 = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
⊢ (y₂ - x₂ ≫ φ.hom₂) ≫ T'.mor₂ = 0

Tactic:
sub_comp,

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
⊢ y₂ ≫ T'.mor₂ - (x₂ ≫ φ.hom₂) ≫ T'.mor₂ = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ (Triangle.shiftFunctor (CochainComplex C ℤ) n).obj (triangle φ) ≅
    triangle ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ)

Tactic:
by
  refine Triangle.isoMk _ _ (Iso.refl _) (n.negOnePow • Iso.refl _) (shiftIso φ n) ?_ ?_ ?_
  · dsimp
    simp only [Linear.comp_units_smul, comp_id, id_comp, smul_smul,
      Int.units_mul_self, one_smul]
  · ext p
    dsimp
    simp only [Units.smul_def, shiftIso, Int.reduceNeg, Linear.smul_comp, id_comp,
      ext_to_iff _ _ (p + 1) rfl, shiftFunctor_obj_X', assoc, lift_f_fst_v, Cocycle.coe_smul,
      Cocycle.shift_coe, Cochain.smul_v, Cochain.shift_v', Linear.comp_smul, inr_f_fst_v,
      smul_zero, lift_f_snd_v, inr_f_snd_v, and_true]
  · ext p
    dsimp
    simp only [triangle, Triangle.mk_mor₃, Cocycle.homOf_f, Cocycle.rightShift_coe,
      Cocycle.coe_neg, Cochain.rightShift_neg, Cochain.neg_v, shiftFunctor_obj_X',
      (fst φ).1.rightShift_v 1 0 (zero_add 1) (p + n) (p + n) (add_zero (p + n)) (p + 1 + n)
        (by lia),
      shiftFunctor_obj_X, shiftFunctorObjXIso, shiftFunctorComm_hom_app_f, Preadditive.neg_comp,
      assoc, Iso.inv_hom_id, comp_id, smul_neg, Units.smul_def, shiftIso, Int.reduceNeg,
      (fst (φ⟦n⟧')).1.rightShift_v 1 0 (zero_add 1) p p (add_zero p) (p + 1) rfl,
      HomologicalComplex.XIsoOfEq_rfl, Iso.refl_inv, Preadditive.comp_neg, lift_f_fst_v,
      Cocycle.coe_smul, Cocycle.shift_coe, Cochain.smul_v, Cochain.shift_v']

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles C
⊢ T.rotate ∈ distinguishedTriangles C

Tactic:
by
  obtain ⟨K, L, φ, ⟨e⟩⟩ := hT
  exact ⟨_, _, _, ⟨(rotate _).mapIso e ≪≫ CochainComplex.mappingCone.rotateTrianglehIso φ⟩⟩

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v, u_1} C
inst✝ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
p : ℤ
this : Mono (S.f.f (p + 2))
r_f : ∀ (n : ℤ), (σ n).r ≫ S.f.f n = 𝟙 (S.X₂.X n) - S.g.f n ≫ (σ n).s
s_g : ∀ (n : ℤ), (σ n).s ≫ S.g.f n = 𝟙 (S.X₃.X n)
⊢ (σ p).s ≫ S.X₂.d p (p + 1) ≫ (𝟙 (S.X₂.X (p + 1)) - S.g.f (p + 1) ≫ (σ (p + 1)).s) ≫ S.X₂.d (p + 1) (p + 2) +
      S.X₃.d p (p + 1) ≫ (σ (p + 1)).s ≫ S.X₂.d (p + 1) (p + 2) ≫ (σ (p + 2)).r ≫ S.f.f (p + 2) =
    0

Tactic:
sub_comp,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v, u_1} C
inst✝ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
p : ℤ
this : Mono (S.f.f (p + 2))
r_f : ∀ (n : ℤ), (σ n).r ≫ S.f.f n = 𝟙 (S.X₂.X n) - S.g.f n ≫ (σ n).s
s_g : ∀ (n : ℤ), (σ n).s ≫ S.g.f n = 𝟙 (S.X₃.X n)
⊢ (σ p).s ≫
        S.X₂.d p (p + 1) ≫
          (𝟙 (S.X₂.X (p + 1)) ≫ S.X₂.d (p + 1) (p + 2) - (S.g.f (p + 1) ≫ (σ (p + 1)).s) ≫ S.X₂.d (p + 1) (p + 2)) +
      S.X₃.d p (p + 1) ≫ (σ (p + 1)).s ≫ S.X₂.d (p + 1) (p + 2) ≫ (σ (p + 2)).r ≫ S.f.f (p + 2) =
    0
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ m₁ ≫ v₁₃ = v₁₂

Tactic:
rw [← shiftFunctorCompIsoId_naturality_1 v₁₃ 1 (-1) (Int.add_right_neg 1)]

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ m₁ ≫
      (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₃ ≫
        (shiftFunctor C (-1)).map ((shiftFunctor C 1).map v₁₃) ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₁ =
    v₁₂
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
⊢ HomologicalComplex.homologyMap S.f i ≫ HomologicalComplex.homologyMap S.g i = 0

Tactic:
by rw [← HomologicalComplex.homologyMap_comp,
      S.zero, HomologicalComplex.homologyMap_zero]

Goal state after:
Module: Mathlib.CategoryTheory.CatCommSq

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ≌ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ≌ C₄
h : CatCommSq T.functor L R B.functor
X : C₁
⊢ (B.symm.unitIso.hom.app (R.obj (T.symm.inverse.obj X)) ≫
        B.symm.inverse.map ((iso T.inverse R L B.inverse).inv.app (T.symm.inverse.obj X)) ≫
          B.symm.inverse.map (L.map (T.symm.counitIso.hom.app X))) ≫
      (L ⋙ B.functor).map (T.unitIso.hom.app X) =
    (T.functor ⋙ R).map (T.unitIso.hom.app X) ≫ (iso T.functor L R B.functor).hom.app ((T.functor ⋙ T.inverse).obj X)

Tactic:
hInv_iso_inv_app

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ≌ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ≌ C₄
h : CatCommSq T.functor L R B.functor
X : C₁
⊢ (B.symm.unitIso.hom.app (R.obj (T.symm.inverse.obj X)) ≫
        B.symm.inverse.map
            (B.inverse.map (R.map (T.counitIso.inv.app (T.symm.inverse.obj X))) ≫
              B.inverse.map ((iso T.functor L R B.functor).hom.app (T.inverse.obj (T.symm.inverse.obj X))) ≫
                B.unitIso.inv.app (L.obj (T.inverse.obj (T.symm.inverse.obj X)))) ≫
          B.symm.inverse.map (L.map (T.symm.counitIso.hom.app X))) ≫
      (L ⋙ B.functor).map (T.unitIso.hom.app X) =
    (T.functor ⋙ R).map (T.unitIso.hom.app X) ≫ (iso T.functor L R B.functor).hom.app ((T.functor ⋙ T.inverse).obj X)
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
F₁ F₂ F₃ : C ⥤ E
F₁' F₂' F₃' : D ⥤ E
h₁ : Lifting L W F₁ F₁'
h₂ : Lifting L W F₂ F₂'
h₃ : Lifting L W F₃ F₃'
τ : F₁ ⟶ F₂
τ' : F₂ ⟶ F₃
X : C
⊢ (liftNatTrans L W F₁ F₂ F₁' F₂' τ ≫ liftNatTrans L W F₂ F₃ F₂' F₃' τ').app (L.obj X) =
    (liftNatTrans L W F₁ F₃ F₁' F₃' (τ ≫ τ')).app (L.obj X)

Tactic:
by
    simp only [NatTrans.comp_app, liftNatTrans_app, assoc, Iso.inv_hom_id_app_assoc]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
P : MorphismProperty D
inst✝¹ : P.RespectsIso
inst✝ : P.IsMultiplicative
h₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)
h₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), P (isoOfHom L W f hf).inv
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
X Y : C
f : X ⟶ Y
hf : W f
⊢ e.hom.app Y = (uniq W.Q L W).functor.map ((isoOfHom W.Q W f hf).inv ≫ W.Q.map f) ≫ e.hom.app Y

Tactic:
isoOfHom_inv_hom_id,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
P : MorphismProperty D
inst✝¹ : P.RespectsIso
inst✝ : P.IsMultiplicative
h₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)
h₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), P (isoOfHom L W f hf).inv
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
X Y : C
f : X ⟶ Y
hf : W f
⊢ e.hom.app Y = (uniq W.Q L W).functor.map (𝟙 (W.Q.obj Y)) ≫ e.hom.app Y
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁵ : L.IsLocalization W
inst✝⁴ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : Preadditive E
inst✝¹ : Preadditive D
inst✝ : L.Additive
G : D ⥤ E
h : (L ⋙ G).Additive
this : ∀ ⦃X Y : C⦄ (f g : L.obj X ⟶ L.obj Y), G.map (f + g) = G.map f + G.map g
X Y : D
f g : X ⟶ Y
hL : L.EssSurj
eq :
  G.map (L.objObjPreimageIso X).hom ≫ G.map ((f + g) ≫ (L.objObjPreimageIso Y).inv) =
    G.map (L.objObjPreimageIso X).hom ≫ (G.map f + G.map g) ≫ G.map (L.objObjPreimageIso Y).inv
⊢ G.map (f + g) = G.map f + G.map g

Tactic:
Functor.map_comp

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁵ : L.IsLocalization W
inst✝⁴ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : Preadditive E
inst✝¹ : Preadditive D
inst✝ : L.Additive
G : D ⥤ E
h : (L ⋙ G).Additive
this : ∀ ⦃X Y : C⦄ (f g : L.obj X ⟶ L.obj Y), G.map (f + g) = G.map f + G.map g
X Y : D
f g : X ⟶ Y
hL : L.EssSurj
eq :
  G.map (L.objObjPreimageIso X).hom ≫ G.map (f + g) ≫ G.map (L.objObjPreimageIso Y).inv =
    G.map (L.objObjPreimageIso X).hom ≫ (G.map f + G.map g) ≫ G.map (L.objObjPreimageIso Y).inv
⊢ G.map (f + g) = G.map f + G.map g
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: I : Type u
inst✝³ : SupSet I
φ : I → I
J : Type w
inst✝² : LinearOrder J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
i₀ : I
j : J
hj : Order.IsSuccLimit j
⊢ transfiniteIterate φ j i₀ = ⨆ x, transfiniteIterate φ (↑x) i₀

Tactic:
by
  dsimp [transfiniteIterate]
  rw [SuccOrder.limitRecOn_of_isSuccLimit _ _ _ hj]
  simp only [iSup_apply]

Goal state after:
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ : ∀ (i : I), i ≤ φ i
⊢ Monotone fun j ↦ transfiniteIterate φ j i₀

Tactic:
intro k

Goal state after:
I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ : ∀ (i : I), i ≤ φ i
k : J
⊢ ∀ ⦃b : J⦄, k ≤ b → (fun j ↦ transfiniteIterate φ j i₀) k ≤ (fun j ↦ transfiniteIterate φ j i₀) b
Module: Mathlib.CategoryTheory.ObjectProperty.FiniteProducts

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasInitial C
P Q : ObjectProperty C
inst✝¹ : Q.IsClosedUnderBinaryCoproducts
inst✝ : Q.IsClosedUnderColimitsOfShape (Discrete PEmpty.{1})
⊢ P.binaryCoproductsClosure ≤ Q ↔ P ≤ Q

Tactic:
refine ⟨fun h ↦ (P.le_colimitsClosure _).trans h, fun h ↦ ?_⟩
  let : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms Q
  exact colimitsClosure_le h

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

Goal state before:
pp: case mp
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P Q R : ObjectProperty C
inst✝ : IsTriangulated C
X : C
⊢ (P.extensionProduct Q).extensionProduct R X → P.extensionProduct (Q.extensionProduct R) X

Tactic:
intro ⟨Y, C, f, g, h, hT, ⟨A, B, f', g', h', hT', hP, hQ⟩, hR⟩
    obtain ⟨Y, g'', h'', hT''⟩ := distinguished_cocone_triangle (f' ≫ f)
    let o := someOctahedron rfl hT' hT hT''
    exact ⟨_, _, _, _, _, hT'', hP, ⟨_, _, _, _, _, o.mem, hQ, hR⟩⟩

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

Goal state before:
pp: case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n : ℕ
h : P.extensionProductIter (n + 1) = (P.extensionProductIter n).extensionProduct P
⊢ P.extensionProduct (P.extensionProductIter (n + 1)) = (P.extensionProductIter (n + 1)).extensionProduct P

Tactic:
h,

Goal state after:
case succ
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n : ℕ
h : P.extensionProductIter (n + 1) = (P.extensionProductIter n).extensionProduct P
⊢ P.extensionProduct ((P.extensionProductIter n).extensionProduct P) =
    ((P.extensionProductIter n).extensionProduct P).extensionProduct P
Module: Mathlib.CategoryTheory.Localization.Composition

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
⊢ (L₁ ⋙ L₂).IsLocalization W₃

Tactic:
let E₂ := Localization.uniq L₁ W₁.Q W₁

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
⊢ (L₁ ⋙ L₂).IsLocalization W₃
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ)
hφ' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map (φ ≫ φ'))

Tactic:
Functor.map_comp

Goal state after:
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
hφ : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ)
hφ' : ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ')
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c i).map φ ≫ (shortComplexFunctor C c i).map φ')
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case zero
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n n' : ℕ
h : n = n' + 1
⊢ P.extensionProductIter n = (P.extensionProductIter n').extensionProduct P

Tactic:
h,

Goal state after:
case zero
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} 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
P : ObjectProperty C
inst✝ : IsTriangulated C
n n' : ℕ
h : n = n' + 1
⊢ P.extensionProductIter (n' + 1) = (P.extensionProductIter n').extensionProduct P