mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ mn + a = a''

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ F.shiftIso mn a a'' ⋯ =
    isoWhiskerRight (shiftFunctorAdd' C m n mn hnm) (F.shift a) ≪≫
      (shiftFunctor C m).associator (shiftFunctor C n) (F.shift a) ≪≫
        (shiftFunctor C m).isoWhiskerLeft (F.shiftIso n a a' ha') ≪≫ F.shiftIso m a' a'' ha''

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
b✝ c✝ : Paths V
g' : Quiver.Path Y✝ b✝
p : b✝ ⟶ c✝
ih :
  Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g'
⊢ f ≫ g'.cons p = Quiver.Path.cons (f ≫ g') p

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ mn + a = a''

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso mn a a'' ⋯).hom.app X =
    (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).hom.app X) ≫
      (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)
A : C
k : S.X₂.opcycles i ⟶ A
hk : opcyclesMap S.f i ≫ k = 0
⊢ (S.map (eval C c i)).f ≫ S.X₂.pOpcycles i ≫ k = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso mn a a'' ⋯).inv.app X =
    (F.shiftIso m a' a'' ha'').inv.app X ≫
      (F.shiftIso n a a' ha').inv.app ((shiftFunctor C m).obj X) ≫
        (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X =
    (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X) ≫ (F.shiftIso mn a a'' ⋯).hom.app X

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
X Y Z : C
n : M
f : X ⟶ (shiftFunctor C n).obj Y
g : Y ⟶ Z
a a' : M
ha' : n + a = a'
⊢ F.shiftMap (f ≫ (shiftFunctor C n).map g) a a' ha' = F.shiftMap f a a' ha' ≫ (F.shift a').map g

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
X Y Z : C
n : M
f : X ⟶ Y
g : Y ⟶ (shiftFunctor C n).obj Z
a a' : M
ha' : n + a = a'
⊢ F.shiftMap (f ≫ g) a a' ha' = (F.shift a).map f ≫ F.shiftMap g a a' ha'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
X Y : C
m : M
f : X ⟶ (shiftFunctor C m).obj Y
n mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ mn + a = a''

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g) =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
φ : V ⥤q C
X Y : V
f : X ⟶ Y
⊢ (lift φ).map f.toPath = φ.map f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
X Y : C
m : M
f : X ⟶ (shiftFunctor C m).obj Y
n mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ (F.shiftIso n a a' ha').hom.app X ≫ F.shiftMap f a' a'' ha'' =
    (F.shift a).map ((shiftFunctor C n).map f) ≫
      (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app Y) ≫ (F.shiftIso mn a a'' ⋯).hom.app Y

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
φ : V ⥤q C
⊢ of V ⋙q (lift φ).toPrefunctor = φ

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
G : Type u_5
inst✝³ : AddGroup G
inst✝² : HasShift C G
inst✝¹ : F.ShiftSequence M
inst✝ : F.ShiftSequence G
X Y : C
m : G
f : X ⟶ (shiftFunctor C m).obj Y
n : G
hnm : n + m = 0
a a' : G
ha' : m + a = a'
⊢ n + a' = a

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
Φ : Paths V ⥤ C
X Y b✝ c✝ : Paths V
p : Quiver.Path X b✝
f' : b✝ ⟶ c✝
ih : Φ.map p = Quiver.Path.rec (𝟙 (Φ.obj X)) (fun {b c} x f ihp ↦ ihp ≫ Φ.map f.toPath) p
⊢ Φ.map (p.cons f') = Φ.map p ≫ Φ.map f'.toPath

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
G : Type u_5
inst✝³ : AddGroup G
inst✝² : HasShift C G
inst✝¹ : F.ShiftSequence M
inst✝ : F.ShiftSequence G
X Y : C
m : G
f : X ⟶ (shiftFunctor C m).obj Y
n : G
hnm : n + m = 0
a a' : G
ha' : m + a = a'
⊢ m + n = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence G
X Y : C
m : G
f : X ⟶ (shiftFunctor C m).obj Y
n : G
hnm : n + m = 0
a a' : G
ha' : m + a = a'
⊢ m + n = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)
A : C
k :
  { X₁ := S.X₁.opcycles i, X₂ := S.X₂.opcycles i, X₃ := S.X₃.opcycles i, f := opcyclesMap S.f i, g := opcyclesMap S.g i,
        zero := ⋯ }.X₂ ⟶
    A
hk :
  { X₁ := S.X₁.opcycles i, X₂ := S.X₂.opcycles i, X₃ := S.X₃.opcycles i, f := opcyclesMap S.f i, g := opcyclesMap S.g i,
          zero := ⋯ }.f ≫
      k =
    0
⊢ { l //
    { X₁ := S.X₁.opcycles i, X₂ := S.X₂.opcycles i, X₃ := S.X₃.opcycles i, f := opcyclesMap S.f i,
            g := opcyclesMap S.g i, zero := ⋯ }.g ≫
        l =
      k }

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence G
X Y : C
m : G
f : X ⟶ (shiftFunctor C m).obj Y
n : G
hnm : n + m = 0
a a' : G
ha' : m + a = a'
⊢ (F.shiftIso n a' a ⋯).hom.app X ≫ F.shiftMap f a a' ha' =
    (F.shift a').map ((shiftFunctor C n).map f ≫ (shiftFunctorCompIsoId C m n ⋯).hom.app Y)

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
φ : V ⥤q C
Φ : Paths V ⥤ C
hΦ : of V ⋙q Φ.toPrefunctor = φ
⊢ Φ = lift φ

Tactic:
by

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : F.ShiftSequence M
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms A
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : ∀ (n : M), (shiftFunctor C n).PreservesZeroMorphisms
X Y : C
n a a' : M
ha' : n + a = a'
⊢ F.shiftMap 0 a a' ha' = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ { X₁ := S.X₁.opcycles i, X₂ := S.X₂.opcycles i, X₃ := S.X₃.opcycles i, f := opcyclesMap S.f i, g := opcyclesMap S.g i,
      zero := ⋯ }.Exact

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
G : Type u_5
inst✝³ : AddGroup G
inst✝² : HasShift C G
inst✝¹ : π.CommShift M
inst✝ : H.ShiftSequence M
a : M
⊢ ((shiftFunctor C 0).associator π (H.shift a)).symm ≪≫
      isoWhiskerRight (commShiftIso π 0) (H.shift a) ≪≫
        π.associator (shiftFunctor D 0) (H.shift a) ≪≫ π.isoWhiskerLeft (H.shiftIso 0 a a ⋯) =
    isoWhiskerRight (shiftFunctorZero C M) (π ⋙ H.shift a) ≪≫ (π ⋙ H.shift a).leftUnitor

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ cyclesMap S.f i ≫ cyclesMap S.g i = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ Mono (S.map (eval C c i)).f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
G : Type u_5
inst✝³ : AddGroup G
inst✝² : HasShift C G
inst✝¹ : π.CommShift M
inst✝ : H.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ ((shiftFunctor C (m + n)).associator π (H.shift a)).symm ≪≫
      isoWhiskerRight (commShiftIso π (m + n)) (H.shift a) ≪≫
        π.associator (shiftFunctor D (m + n)) (H.shift a) ≪≫ π.isoWhiskerLeft (H.shiftIso (m + n) a a'' ⋯) =
    isoWhiskerRight (shiftFunctorAdd C m n) (π ⋙ H.shift a) ≪≫
      (shiftFunctor C m).associator (shiftFunctor C n) (π ⋙ H.shift a) ≪≫
        (shiftFunctor C m).isoWhiskerLeft
            (((shiftFunctor C n).associator π (H.shift a)).symm ≪≫
              isoWhiskerRight (commShiftIso π n) (H.shift a) ≪≫
                π.associator (shiftFunctor D n) (H.shift a) ≪≫ π.isoWhiskerLeft (H.shiftIso n a a' ha')) ≪≫
          ((shiftFunctor C m).associator π (H.shift a')).symm ≪≫
            isoWhiskerRight (commShiftIso π m) (H.shift a') ≪≫
              π.associator (shiftFunctor D m) (H.shift a') ≪≫ π.isoWhiskerLeft (H.shiftIso m a' a'' ha'')

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
A : C
k : A ⟶ S.X₂.cycles i
hk : k ≫ cyclesMap S.g i = 0
⊢ (k ≫ S.X₂.iCycles i) ≫ (S.map (eval C c i)).g = 0

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : (of V).obj u ⟶ (of V).obj v) (q : v ⟶ w), P p → P (p ≫ (of V).map q)
⊢ P = ⊤

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P ((of V).map p ≫ q)
⊢ P = ⊤

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
V : Type u₁
inst✝ : Quiver V
F G : Paths V ⥤ C
α_app : (v : V) → F.obj v ⟶ G.obj v
α_nat : ∀ {X Y : V} (f : X ⟶ Y), F.map f.toPath ≫ α_app Y = α_app X ≫ G.map f.toPath
⊢ ∀ ⦃X Y : Paths V⦄ (f : X ⟶ Y), F.map f ≫ α_app Y = α_app X ≫ G.map f

Tactic:
by

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

Goal state before:
pp: V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ F = G

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y Z : C
f : Path X Y
g : Path Y Z
⊢ composePath (f.comp g) = composePath f ≫ composePath g

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
⊢ pathsHomRel C (𝟙 X).toPath (𝟙 { as := X }.as)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ pathsHomRel C (f ≫ g).toPath (f.toPath ≫ g.toPath)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : Quotient (pathsHomRel C)
⊢ (𝟭 (Quotient (pathsHomRel C))).obj X ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj X

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y Z : C
p : Path X Y
q : Path Y Z
⊢ W.paths (p.comp q) ↔ W.paths p ∧ W.paths q

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X✝ Y✝ : Quotient (pathsHomRel C)
f : X✝.as ⟶ Y✝.as
⊢ pathsHomRel C (f ≫ 𝟙 ((𝟭 (Quotient (pathsHomRel C))).obj { as := Y✝.as }).as)
    (𝟙 ((𝟭 (Quotient (pathsHomRel C))).obj { as := X✝.as }).as ≫
      ((quotientPathsTo C).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f)).toPath)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.IsStableUnderComposition
X Y : C
p : Path X Y
h : W (𝟙 X)
⊢ W.paths Path.nil → W (composePath Path.nil)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.IsStableUnderComposition
X Y : C
p : Path X Y
hp : W.paths p
h : W (𝟙 X)
⊢ W (composePath p)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.IsStableUnderComposition
X Y : C
p : Path X Y
⊢ W.paths Path.nil → 0 < Path.nil.length → W (composePath Path.nil)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X✝ Y✝ : Quotient (pathsHomRel C)
f : X✝.as ⟶ Y✝.as
⊢ (𝟭 (Quotient (pathsHomRel C))).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f) ≫
      (Quotient.casesOn (motive := fun t ↦
          Y✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj Y✝ ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj Y✝)) Y✝
          (fun as h ↦ ⋯ ▸ Iso.refl ((𝟭 (Quotient (pathsHomRel C))).obj { as := as })) ⋯).hom =
    (Quotient.casesOn (motive := fun t ↦
          X✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj X✝ ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj X✝)) X✝
          (fun as h ↦ ⋯ ▸ Iso.refl ((𝟭 (Quotient (pathsHomRel C))).obj { as := as })) ⋯).hom ≫
      (quotientPathsTo C ⋙ toQuotientPaths C).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f)

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
⊢ (toQuotientPaths C ⋙ quotientPathsTo C).map f ≫ (Iso.refl ((toQuotientPaths C ⋙ quotientPathsTo C).obj Y✝)).hom =
    (Iso.refl ((toQuotientPaths C ⋙ quotientPathsTo C).obj X✝)).hom ≫ (𝟭 C).map f

Tactic:
by

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : Quotient (pathsHomRel C)
⊢ (quotientPathsTo C).map
        ((NatIso.ofComponents
                (fun X ↦
                  Quotient.casesOn (motive := fun t ↦
                    X = t → ((𝟭 (Quotient (pathsHomRel C))).obj X ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj X)) X
                    (fun as h ↦ ⋯ ▸ Iso.refl ((𝟭 (Quotient (pathsHomRel C))).obj { as := as })) ⋯)
                ⋯).hom.app
          X) ≫
      (NatIso.ofComponents (fun x ↦ Iso.refl ((toQuotientPaths C ⋙ quotientPathsTo C).obj x)) ⋯).hom.app
        ((quotientPathsTo C).obj X) =
    𝟙 ((quotientPathsTo C).obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.IsStableUnderComposition
X Y : C
p : Path X Y
b✝¹ c✝ : C
f : b✝¹ ⟶ c✝
b✝ : C
a✝¹ : Path X b✝
a✝ : b✝ ⟶ b✝¹
hp : W.paths (a✝¹.cons a✝) → 0 < (a✝¹.cons a✝).length → W (composePath (a✝¹.cons a✝))
hp' : W.paths ((a✝¹.cons a✝).cons f)
hp'' : 0 < ((a✝¹.cons a✝).cons f).length
⊢ 0 < (a✝¹.cons a✝).length

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.IsStableUnderComposition
X Y : C
p : Path X Y
hp : W.paths p
h : 0 < p.length
⊢ W (composePath p)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X✝ Y✝ Z✝ : C
x✝³ : X✝ ⟶ Y✝
x✝² : Y✝ ⟶ Z✝
x✝¹ : W.paths.strictMap (pathComposition C) x✝³
x✝ : W.paths.strictMap (pathComposition C) x✝²
f✝¹ : X✝ ⟶ Y✝
hp : W.paths f✝¹
f✝ : Y✝ ⟶ Z✝
hq : W.paths f✝
⊢ W.paths.strictMap (pathComposition C) ((pathComposition C).map f✝¹ ≫ (pathComposition C).map f✝)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : W f
⊢ W.paths f.toPath

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
⊢ W.multiplicativeClosure = W.paths.strictMap (pathComposition C)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
A : C
k :
  A ⟶
    { X₁ := S.X₁.cycles i, X₂ := S.X₂.cycles i, X₃ := S.X₃.cycles i, f := cyclesMap S.f i, g := cyclesMap S.g i,
        zero := ⋯ }.X₂
hk :
  k ≫
      { X₁ := S.X₁.cycles i, X₂ := S.X₂.cycles i, X₃ := S.X₃.cycles i, f := cyclesMap S.f i, g := cyclesMap S.g i,
          zero := ⋯ }.g =
    0
⊢ { l //
    l ≫
        { X₁ := S.X₁.cycles i, X₂ := S.X₂.cycles i, X₃ := S.X₃.cycles i, f := cyclesMap S.f i, g := cyclesMap S.g i,
            zero := ⋯ }.f =
      k }

Tactic:
by

Goal state after: