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tactic
stringlengths 1
5.59k
| name
stringlengths 1
85
| haveDraft
stringlengths 1
44.5k
| goal
stringlengths 7
64.3k
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|---|---|---|---|
dsimp [dNext]
|
dsimp
|
C.d i ((ComplexShape.down ℕ).next i) ≫ f ((ComplexShape.down ℕ).next i) i = C.d i (i - 1) ≫ f (i - 1) i
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
⊢ (dNext i : ((i j : ℕ) → C.X i ⟶ D.X j) → (C.X i ⟶ D.X i)) f = C.d i (i - 1) ≫ f (i - 1) i
|
cases i
|
zero
|
C.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = C.d 0 (0 - 1) ≫ f (0 - 1) 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
⊢ C.d i ((ComplexShape.down ℕ).next i) ≫ f ((ComplexShape.down ℕ).next i) i = C.d i (i - 1) ≫ f (i - 1) i
|
cases i
|
succ
|
C.d (n✝ + 1) ((ComplexShape.down ℕ).next (n✝ + 1)) ≫ f ((ComplexShape.down ℕ).next (n✝ + 1)) (n✝ + 1) =
C.d (n✝ + 1) (n✝ + 1 - 1) ≫ f (n✝ + 1 - 1) (n✝ + 1)
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
zero : C.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = C.d 0 (0 - 1) ≫ f (0 - 1) 0
⊢ C.d i ((ComplexShape.down ℕ).next i) ≫ f ((ComplexShape.down ℕ).next i) i = C.d i (i - 1) ≫ f (i - 1) i
|
congr
|
succ
|
(ComplexShape.down ℕ).next (n✝ + 1) = n✝ + 1 - 1
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
n✝ : ℕ
⊢ C.d (n✝ + 1) ((ComplexShape.down ℕ).next (n✝ + 1)) ≫ f ((ComplexShape.down ℕ).next (n✝ + 1)) (n✝ + 1) =
C.d (n✝ + 1) (n✝ + 1 - 1) ≫ f (n✝ + 1 - 1) (n✝ + 1)
|
congr
|
succ
|
(ComplexShape.down ℕ).next (n✝ + 1) = n✝ + 1 - 1
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
n✝ : ℕ
succ : (ComplexShape.down ℕ).next (n✝ + 1) = n✝ + 1 - 1
⊢ C.d (n✝ + 1) ((ComplexShape.down ℕ).next (n✝ + 1)) ≫ f ((ComplexShape.down ℕ).next (n✝ + 1)) (n✝ + 1) =
C.d (n✝ + 1) (n✝ + 1 - 1) ≫ f (n✝ + 1 - 1) (n✝ + 1)
|
congr
|
succ
|
(ComplexShape.down ℕ).next (n✝ + 1) = n✝ + 1 - 1
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : ChainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
n✝ : ℕ
succ : (ComplexShape.down ℕ).next (n✝ + 1) = n✝ + 1 - 1
succ₁ : (ComplexShape.down ℕ).next (n✝ + 1) = n✝ + 1 - 1
⊢ C.d (n✝ + 1) ((ComplexShape.down ℕ).next (n✝ + 1)) ≫ f ((ComplexShape.down ℕ).next (n✝ + 1)) (n✝ + 1) =
C.d (n✝ + 1) (n✝ + 1 - 1) ≫ f (n✝ + 1 - 1) (n✝ + 1)
|
dsimp [prevD]
|
dsimp
|
f i ((ComplexShape.up ℕ).prev i) ≫ D.d ((ComplexShape.up ℕ).prev i) i = f i (i - 1) ≫ D.d (i - 1) i
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : CochainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
⊢ (prevD i : ((i j : ℕ) → C.X i ⟶ D.X j) → (C.X i ⟶ D.X i)) f = f i (i - 1) ≫ D.d (i - 1) i
|
cases i
|
zero
|
f 0 ((ComplexShape.up ℕ).prev 0) ≫ D.d ((ComplexShape.up ℕ).prev 0) 0 = f 0 (0 - 1) ≫ D.d (0 - 1) 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : CochainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
⊢ f i ((ComplexShape.up ℕ).prev i) ≫ D.d ((ComplexShape.up ℕ).prev i) i = f i (i - 1) ≫ D.d (i - 1) i
|
cases i
|
succ
|
f (n✝ + 1) ((ComplexShape.up ℕ).prev (n✝ + 1)) ≫ D.d ((ComplexShape.up ℕ).prev (n✝ + 1)) (n✝ + 1) =
f (n✝ + 1) (n✝ + 1 - 1) ≫ D.d (n✝ + 1 - 1) (n✝ + 1)
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : CochainComplex V ℕ
i : ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
zero : f 0 ((ComplexShape.up ℕ).prev 0) ≫ D.d ((ComplexShape.up ℕ).prev 0) 0 = f 0 (0 - 1) ≫ D.d (0 - 1) 0
⊢ f i ((ComplexShape.up ℕ).prev i) ≫ D.d ((ComplexShape.up ℕ).prev i) i = f i (i - 1) ≫ D.d (i - 1) i
|
congr
|
succ
|
(ComplexShape.up ℕ).prev (n✝ + 1) = n✝ + 1 - 1
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : CochainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
n✝ : ℕ
⊢ f (n✝ + 1) ((ComplexShape.up ℕ).prev (n✝ + 1)) ≫ D.d ((ComplexShape.up ℕ).prev (n✝ + 1)) (n✝ + 1) =
f (n✝ + 1) (n✝ + 1 - 1) ≫ D.d (n✝ + 1 - 1) (n✝ + 1)
|
congr
|
succ
|
(ComplexShape.up ℕ).prev (n✝ + 1) = n✝ + 1 - 1
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : CochainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
n✝ : ℕ
succ : (ComplexShape.up ℕ).prev (n✝ + 1) = n✝ + 1 - 1
⊢ f (n✝ + 1) ((ComplexShape.up ℕ).prev (n✝ + 1)) ≫ D.d ((ComplexShape.up ℕ).prev (n✝ + 1)) (n✝ + 1) =
f (n✝ + 1) (n✝ + 1 - 1) ≫ D.d (n✝ + 1 - 1) (n✝ + 1)
|
congr
|
succ
|
(ComplexShape.up ℕ).prev (n✝ + 1) = n✝ + 1 - 1
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
C D : CochainComplex V ℕ
f : (i j : ℕ) → C.X i ⟶ D.X j
n✝ : ℕ
succ : (ComplexShape.up ℕ).prev (n✝ + 1) = n✝ + 1 - 1
succ₁ : (ComplexShape.up ℕ).prev (n✝ + 1) = n✝ + 1 - 1
⊢ f (n✝ + 1) ((ComplexShape.up ℕ).prev (n✝ + 1)) ≫ D.d ((ComplexShape.up ℕ).prev (n✝ + 1)) (n✝ + 1) =
f (n✝ + 1) (n✝ + 1 - 1) ≫ D.d (n✝ + 1 - 1) (n✝ + 1)
|
ext n
|
h
|
(nullHomotopicMap hom ≫ g).f n = (nullHomotopicMap fun i j ↦ hom i j ≫ g.f j).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → C.X i ⟶ D.X j
g : D ⟶ E
⊢ nullHomotopicMap hom ≫ g = nullHomotopicMap fun i j ↦ hom i j ≫ g.f j
|
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
|
h
|
(C.dFrom n ≫ hom (c.next n) n + hom n (c.prev n) ≫ D.dTo n) ≫ g.f n =
C.dFrom n ≫ hom (c.next n) n ≫ g.f n + (hom n (c.prev n) ≫ g.f (c.prev n)) ≫ E.dTo n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → C.X i ⟶ D.X j
g : D ⟶ E
n : ι
⊢ (nullHomotopicMap hom ≫ g).f n = (nullHomotopicMap fun i j ↦ hom i j ≫ g.f j).f n
|
ext n
|
h
|
(nullHomotopicMap' hom ≫ g).f n = (nullHomotopicMap' fun i j hij ↦ hom i j sorry ≫ g.f j).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
⊢ nullHomotopicMap' hom ≫ g = nullHomotopicMap' fun i j hij ↦ hom i j hij ≫ g.f j
|
rw [nullHomotopicMap', nullHomotopicMap_comp]
|
h
|
(nullHomotopicMap fun i j ↦ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j).f n =
(nullHomotopicMap' fun i j hij ↦ hom i j sorry ≫ g.f j).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n : ι
⊢ (nullHomotopicMap' hom ≫ g).f n = (nullHomotopicMap' fun i j hij ↦ hom i j hij ≫ g.f j).f n
|
nullHomotopicMap',
|
h
|
((nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g).f n =
(nullHomotopicMap' fun i j hij ↦ hom i j sorry ≫ g.f j).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n : ι
⊢ (nullHomotopicMap' hom ≫ g).f n = (nullHomotopicMap' fun i j hij ↦ hom i j hij ≫ g.f j).f n
|
nullHomotopicMap_comp
|
h
|
(nullHomotopicMap fun i j ↦ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j).f n =
(nullHomotopicMap' fun i j hij ↦ hom i j sorry ≫ g.f j).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n : ι
⊢ ((nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g).f n =
(nullHomotopicMap' fun i j hij ↦ hom i j hij ≫ g.f j).f n
|
congr
|
h
|
(fun i j ↦ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j) = fun i j ↦
dite (c.Rel j i) ((fun i j hij ↦ hom i j sorry ≫ g.f j) i j) fun x ↦ 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n : ι
⊢ (nullHomotopicMap fun i j ↦ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j).f n =
(nullHomotopicMap' fun i j hij ↦ hom i j hij ≫ g.f j).f n
|
ext i j
|
h
|
(dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j = dite (c.Rel j i) ((fun i j hij ↦ hom i j sorry ≫ g.f j) i j) fun x ↦ 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n : ι
⊢ (fun i j ↦ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j) = fun i j ↦
dite (c.Rel j i) ((fun i j hij ↦ hom i j hij ≫ g.f j) i j) fun x ↦ 0
|
split_ifs
|
pos
|
hom i j sorry ≫ g.f j = (fun i j hij ↦ hom i j sorry ≫ g.f j) i j sorry
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n i j : ι
⊢ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j = dite (c.Rel j i) ((fun i j hij ↦ hom i j hij ≫ g.f j) i j) fun x ↦ 0
|
split_ifs
|
neg
|
0 ≫ g.f j = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n i j : ι
pos : hom i j _fvar.102933 ≫ g.f j = (fun i j hij ↦ hom i j hij ≫ g.f j) i j _fvar.102933
⊢ (dite (c.Rel j i) (hom i j) fun x ↦ 0) ≫ g.f j = dite (c.Rel j i) ((fun i j hij ↦ hom i j hij ≫ g.f j) i j) fun x ↦ 0
|
zero_comp
|
neg
|
0 = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
n i j : ι
h✝ : ¬c.Rel j i
⊢ 0 ≫ g.f j = 0
|
ext n
|
h
|
(f ≫ nullHomotopicMap hom).f n = (nullHomotopicMap fun i j ↦ f.f i ≫ hom i j).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → D.X i ⟶ E.X j
⊢ f ≫ nullHomotopicMap hom = nullHomotopicMap fun i j ↦ f.f i ≫ hom i j
|
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
|
h
|
f.f n ≫ (D.dFrom n ≫ hom (c.next n) n + hom n (c.prev n) ≫ E.dTo n) =
C.dFrom n ≫ f.f (c.next n) ≫ hom (c.next n) n + (f.f n ≫ hom n (c.prev n)) ≫ E.dTo n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → D.X i ⟶ E.X j
n : ι
⊢ (f ≫ nullHomotopicMap hom).f n = (nullHomotopicMap fun i j ↦ f.f i ≫ hom i j).f n
|
ext n
|
h
|
(f ≫ nullHomotopicMap' hom).f n = (nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j sorry).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
⊢ f ≫ nullHomotopicMap' hom = nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij
|
rw [nullHomotopicMap', comp_nullHomotopicMap]
|
h
|
(nullHomotopicMap fun i j ↦ f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0).f n =
(nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j sorry).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n : ι
⊢ (f ≫ nullHomotopicMap' hom).f n = (nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij).f n
|
nullHomotopicMap',
|
h
|
(f ≫ nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0).f n =
(nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j sorry).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n : ι
⊢ (f ≫ nullHomotopicMap' hom).f n = (nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij).f n
|
comp_nullHomotopicMap
|
h
|
(nullHomotopicMap fun i j ↦ f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0).f n =
(nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j sorry).f n
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n : ι
⊢ (f ≫ nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0).f n =
(nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij).f n
|
congr
|
h
|
(fun i j ↦ f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0) = fun i j ↦
dite (c.Rel j i) ((fun i j hij ↦ f.f i ≫ hom i j sorry) i j) fun x ↦ 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n : ι
⊢ (nullHomotopicMap fun i j ↦ f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0).f n =
(nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij).f n
|
ext i j
|
h
|
(f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0) = dite (c.Rel j i) ((fun i j hij ↦ f.f i ≫ hom i j sorry) i j) fun x ↦ 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n : ι
⊢ (fun i j ↦ f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0) = fun i j ↦
dite (c.Rel j i) ((fun i j hij ↦ f.f i ≫ hom i j hij) i j) fun x ↦ 0
|
split_ifs
|
pos
|
f.f i ≫ hom i j sorry = (fun i j hij ↦ f.f i ≫ hom i j sorry) i j sorry
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n i j : ι
⊢ (f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0) = dite (c.Rel j i) ((fun i j hij ↦ f.f i ≫ hom i j hij) i j) fun x ↦ 0
|
split_ifs
|
neg
|
f.f i ≫ 0 = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n i j : ι
pos : f.f i ≫ hom i j _fvar.109603 = (fun i j hij ↦ f.f i ≫ hom i j hij) i j _fvar.109603
⊢ (f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0) = dite (c.Rel j i) ((fun i j hij ↦ f.f i ≫ hom i j hij) i j) fun x ↦ 0
|
comp_zero
|
neg
|
0 = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
n i j : ι
h✝ : ¬c.Rel j i
⊢ f.f i ≫ 0 = 0
|
ext i
|
h
|
((G.mapHomologicalComplex c).map (nullHomotopicMap hom)).f i = (nullHomotopicMap fun i j ↦ G.map (hom i j)).f i
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (G.mapHomologicalComplex c).map (nullHomotopicMap hom) = nullHomotopicMap fun i j ↦ G.map (hom i j)
|
dsimp [nullHomotopicMap, dNext, prevD]
|
h
|
G.map (C.d i (c.next i) ≫ hom (c.next i) i + hom i (c.prev i) ≫ D.d (c.prev i) i) =
G.map (C.d i (c.next i)) ≫ G.map (hom (c.next i) i) + G.map (hom i (c.prev i)) ≫ G.map (D.d (c.prev i) i)
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → C.X i ⟶ D.X j
i : ι
⊢ ((G.mapHomologicalComplex c).map (nullHomotopicMap hom)).f i = (nullHomotopicMap fun i j ↦ G.map (hom i j)).f i
|
ext n
|
h
|
((G.mapHomologicalComplex c).map (nullHomotopicMap' hom)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j sorry)).f n
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (G.mapHomologicalComplex c).map (nullHomotopicMap' hom) = nullHomotopicMap' fun i j hij ↦ G.map (hom i j hij)
|
rw [nullHomotopicMap', map_nullHomotopicMap]
|
h
|
(nullHomotopicMap fun i j ↦ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j sorry)).f n
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n : ι
⊢ ((G.mapHomologicalComplex c).map (nullHomotopicMap' hom)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j hij)).f n
|
nullHomotopicMap',
|
h
|
((G.mapHomologicalComplex c).map (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j sorry)).f n
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n : ι
⊢ ((G.mapHomologicalComplex c).map (nullHomotopicMap' hom)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j hij)).f n
|
map_nullHomotopicMap
|
h
|
(nullHomotopicMap fun i j ↦ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j sorry)).f n
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n : ι
⊢ ((G.mapHomologicalComplex c).map (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j hij)).f n
|
congr
|
h
|
(fun i j ↦ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0)) = fun i j ↦
dite (c.Rel j i) ((fun i j hij ↦ G.map (hom i j sorry)) i j) fun x ↦ 0
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n : ι
⊢ (nullHomotopicMap fun i j ↦ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0)).f n =
(nullHomotopicMap' fun i j hij ↦ G.map (hom i j hij)).f n
|
ext i j
|
h
|
G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0) = dite (c.Rel j i) ((fun i j hij ↦ G.map (hom i j sorry)) i j) fun x ↦ 0
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n : ι
⊢ (fun i j ↦ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0)) = fun i j ↦
dite (c.Rel j i) ((fun i j hij ↦ G.map (hom i j hij)) i j) fun x ↦ 0
|
split_ifs
|
pos
|
G.map (hom i j sorry) = (fun i j hij ↦ G.map (hom i j sorry)) i j sorry
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n i j : ι
⊢ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0) = dite (c.Rel j i) ((fun i j hij ↦ G.map (hom i j hij)) i j) fun x ↦ 0
|
split_ifs
|
neg
|
G.map 0 = 0
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n i j : ι
pos : G.map (hom i j _fvar.118355) = (fun i j hij ↦ G.map (hom i j hij)) i j _fvar.118355
⊢ G.map (dite (c.Rel j i) (hom i j) fun x ↦ 0) = dite (c.Rel j i) ((fun i j hij ↦ G.map (hom i j hij)) i j) fun x ↦ 0
|
G.map_zero
|
neg
|
0 = 0
|
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
W : Type u_2
inst✝² : Category.{u_3, u_2} W
inst✝¹ : Preadditive W
G : V ⥤ W
inst✝ : G.Additive
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
n i j : ι
h✝ : ¬c.Rel j i
⊢ G.map 0 = 0
|
apply nullHomotopy fun i j => dite (c.Rel j i) (h i j) fun _ => 0
|
apply
|
∀ (i j : ι), ¬c.Rel j i → (dite (c.Rel j i) (h i j) fun x ↦ 0) = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h✝ k : D ⟶ E
i : ι
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ Homotopy (nullHomotopicMap' h) 0
|
dsimp only [nullHomotopicMap]
|
dsimp
|
(dNext k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom +
(prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁
|
dNext_eq hom r₁₀,
|
dNext_eq
|
C.d k₁ k₀ ≫ hom k₀ k₁ + (prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (dNext k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom +
(prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁
|
prevD_eq hom r₂₁
|
prevD_eq
|
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₁ k₀ ≫ hom k₀ k₁ + (prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁
|
simp only [nullHomotopicMap']
|
simp
|
(nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₁ =
C.d k₁ k₀ ≫ h k₀ k₁ sorry + h k₁ k₂ sorry ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁
|
rw [nullHomotopicMap_f r₂₁ r₁₀]
|
rw
|
(C.d k₁ k₀ ≫ dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) + (dite (c.Rel k₂ k₁) (h k₁ k₂) fun x ↦ 0) ≫ D.d k₂ k₁ =
C.d k₁ k₀ ≫ h k₀ k₁ sorry + h k₁ k₂ sorry ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₁ =
C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁
|
nullHomotopicMap_f r₂₁ r₁₀
|
nullHomotopicMap_f
|
(C.d k₁ k₀ ≫ dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) + (dite (c.Rel k₂ k₁) (h k₁ k₂) fun x ↦ 0) ≫ D.d k₂ k₁ =
C.d k₁ k₀ ≫ h k₀ k₁ sorry + h k₁ k₂ sorry ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₁ =
C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁
|
split_ifs
|
split_ifs
|
C.d k₁ k₀ ≫ h k₀ k₁ sorry + h k₁ k₂ sorry ≫ D.d k₂ k₁ = C.d k₁ k₀ ≫ h k₀ k₁ sorry + h k₁ k₂ sorry ≫ D.d k₂ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₂ k₁ k₀ : ι
r₂₁ : c.Rel k₂ k₁
r₁₀ : c.Rel k₁ k₀
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (C.d k₁ k₀ ≫ dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) + (dite (c.Rel k₂ k₁) (h k₁ k₂) fun x ↦ 0) ≫ D.d k₂ k₁ =
C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁
|
dsimp only [nullHomotopicMap]
|
dsimp
|
(dNext k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom +
(prevD k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom =
hom k₀ k₁ ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (nullHomotopicMap hom).f k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
rw [prevD_eq hom r₁₀, dNext, AddMonoidHom.mk'_apply, C.shape, zero_comp, zero_add]
|
a
|
¬c.Rel k₀ (c.next k₀)
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (dNext k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom +
(prevD k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom =
hom k₀ k₁ ≫ D.d k₁ k₀
|
prevD_eq hom r₁₀,
|
prevD_eq
|
(dNext k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (dNext k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom +
(prevD k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom =
hom k₀ k₁ ≫ D.d k₁ k₀
|
dNext,
|
this
|
(AddMonoidHom.mk' (fun f ↦ C.d k₀ (c.next k₀) ≫ f (c.next k₀) k₀) sorry :
((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀))
hom +
hom k₀ k₁ ≫ D.d k₁ k₀ =
hom k₀ k₁ ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (dNext k₀ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀)) hom + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
AddMonoidHom.mk'_apply,
|
this
|
C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (AddMonoidHom.mk' (fun f ↦ C.d k₀ (c.next k₀) ≫ f (c.next k₀) k₀) ⋯ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₀ ⟶ D.X k₀))
hom +
hom k₀ k₁ ≫ D.d k₁ k₀ =
hom k₀ k₁ ≫ D.d k₁ k₀
|
C.shape,
|
this
|
0 ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
C.shape,
|
a
|
¬c.Rel k₀ (c.next k₀)
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hom : (i j : ι) → C.X i ⟶ D.X j
this : 0 ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ k₁ ≫ D.d k₁ k₀ = hom k₀ k₁ ≫ D.d k₁ k₀
|
simp only [nullHomotopicMap']
|
simp
|
(nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₀ = h k₀ k₁ sorry ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap' h).f k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀
|
rw [nullHomotopicMap_f_of_not_rel_left r₁₀ hk₀]
|
rw
|
(dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) ≫ D.d k₁ k₀ = h k₀ k₁ sorry ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀
|
nullHomotopicMap_f_of_not_rel_left r₁₀ hk₀
|
nullHomotopicMap_f_of_not_rel_left
|
(dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) ≫ D.d k₁ k₀ = h k₀ k₁ sorry ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀
|
split_ifs
|
split_ifs
|
h k₀ k₁ sorry ≫ D.d k₁ k₀ = h k₀ k₁ sorry ≫ D.d k₁ k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) ≫ D.d k₁ k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀
|
dsimp only [nullHomotopicMap]
|
dsimp
|
(dNext k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom +
(prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁
|
rw [dNext_eq hom r₁₀, prevD, AddMonoidHom.mk'_apply, D.shape, comp_zero, add_zero]
|
a
|
¬c.Rel (c.prev k₁) k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (dNext k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom +
(prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁
|
dNext_eq hom r₁₀,
|
dNext_eq
|
C.d k₁ k₀ ≫ hom k₀ k₁ + (prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom = C.d k₁ k₀ ≫ hom k₀ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (dNext k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom +
(prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom =
C.d k₁ k₀ ≫ hom k₀ k₁
|
prevD,
|
this
|
C.d k₁ k₀ ≫ hom k₀ k₁ +
(AddMonoidHom.mk' (fun f ↦ f k₁ (c.prev k₁) ≫ D.d (c.prev k₁) k₁) sorry :
((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁))
hom =
C.d k₁ k₀ ≫ hom k₀ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₁ k₀ ≫ hom k₀ k₁ + (prevD k₁ : ((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁)) hom = C.d k₁ k₀ ≫ hom k₀ k₁
|
AddMonoidHom.mk'_apply,
|
this
|
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ (c.prev k₁) ≫ D.d (c.prev k₁) k₁ = C.d k₁ k₀ ≫ hom k₀ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₁ k₀ ≫ hom k₀ k₁ +
(AddMonoidHom.mk' (fun f ↦ f k₁ (c.prev k₁) ≫ D.d (c.prev k₁) k₁) ⋯ :
((i j : ι) → C.X i ⟶ D.X j) → (C.X k₁ ⟶ D.X k₁))
hom =
C.d k₁ k₀ ≫ hom k₀ k₁
|
D.shape,
|
this
|
C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ (c.prev k₁) ≫ 0 = C.d k₁ k₀ ≫ hom k₀ k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ (c.prev k₁) ≫ D.d (c.prev k₁) k₁ = C.d k₁ k₀ ≫ hom k₀ k₁
|
D.shape,
|
a
|
¬c.Rel (c.prev k₁) k₁
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
hom : (i j : ι) → C.X i ⟶ D.X j
this : C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ (c.prev k₁) ≫ 0 = C.d k₁ k₀ ≫ hom k₀ k₁
⊢ C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ (c.prev k₁) ≫ D.d (c.prev k₁) k₁ = C.d k₁ k₀ ≫ hom k₀ k₁
|
simp only [nullHomotopicMap']
|
simp
|
(nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ sorry
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀
|
rw [nullHomotopicMap_f_of_not_rel_right r₁₀ hk₁]
|
rw
|
(C.d k₁ k₀ ≫ dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) = C.d k₁ k₀ ≫ h k₀ k₁ sorry
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀
|
nullHomotopicMap_f_of_not_rel_right r₁₀ hk₁
|
nullHomotopicMap_f_of_not_rel_right
|
(C.d k₁ k₀ ≫ dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) = C.d k₁ k₀ ≫ h k₀ k₁ sorry
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀
|
split_ifs
|
split_ifs
|
C.d k₁ k₀ ≫ h k₀ k₁ sorry = C.d k₁ k₀ ≫ h k₀ k₁ sorry
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₁ k₀ : ι
r₁₀ : c.Rel k₁ k₀
hk₁ : ∀ (l : ι), ¬c.Rel l k₁
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (C.d k₁ k₀ ≫ dite (c.Rel k₁ k₀) (h k₀ k₁) fun x ↦ 0) = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀
|
dsimp [nullHomotopicMap, dNext, prevD]
|
dsimp
|
C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ (nullHomotopicMap hom).f k₀ = 0
|
rw [C.shape, D.shape, zero_comp, comp_zero, add_zero]
|
a
|
¬c.Rel (c.prev k₀) k₀
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
|
rw [C.shape, D.shape, zero_comp, comp_zero, add_zero]
|
a
|
¬c.Rel k₀ (c.next k₀)
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j
a : ¬c.Rel (c.prev k₀) k₀
⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
|
C.shape,
|
this
|
0 ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j
⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
|
C.shape,
|
a
|
¬c.Rel k₀ (c.next k₀)
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
hom : (i j : ι) → C.X i ⟶ D.X j
this : 0 ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
⊢ C.d k₀ (c.next k₀) ≫ hom (c.next k₀) k₀ + hom k₀ (c.prev k₀) ≫ D.d (c.prev k₀) k₀ = 0
|
simp only [nullHomotopicMap']
|
simp
|
(nullHomotopicMap fun i j ↦ dite (c.Rel j i) (h i j) fun x ↦ 0).f k₀ = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D : HomologicalComplex V c
k₀ : ι
hk₀ : ∀ (l : ι), ¬c.Rel k₀ l
hk₀' : ∀ (l : ι), ¬c.Rel l k₀
h : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
⊢ (nullHomotopicMap' h).f k₀ = 0
|
dsimp [prevD]
|
dsimp
|
f j ((ComplexShape.down ℕ).prev j) ≫ Q.d ((ComplexShape.down ℕ).prev j) j = f j (j + 1) ≫ Q.d (j + 1) j
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
j : ℕ
⊢ (prevD j : ((i j : ℕ) → P.X i ⟶ Q.X j) → (P.X j ⟶ Q.X j)) f = f j (j + 1) ≫ Q.d (j + 1) j
|
have : (ComplexShape.down ℕ).prev j = j + 1 := ChainComplex.prev ℕ j
|
have
|
f j ((ComplexShape.down ℕ).prev j) ≫ Q.d ((ComplexShape.down ℕ).prev j) j = f j (j + 1) ≫ Q.d (j + 1) j
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
j : ℕ
⊢ f j ((ComplexShape.down ℕ).prev j) ≫ Q.d ((ComplexShape.down ℕ).prev j) j = f j (j + 1) ≫ Q.d (j + 1) j
|
dsimp [dNext]
|
dsimp
|
P.d (i + 1) ((ComplexShape.down ℕ).next (i + 1)) ≫ f ((ComplexShape.down ℕ).next (i + 1)) (i + 1) =
P.d (i + 1) i ≫ f i (i + 1)
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
i : ℕ
⊢ (dNext (i + 1) : ((i j : ℕ) → P.X i ⟶ Q.X j) → (P.X (i + 1) ⟶ Q.X (i + 1))) f = P.d (i + 1) i ≫ f i (i + 1)
|
have : (ComplexShape.down ℕ).next (i + 1) = i := ChainComplex.next_nat_succ _
|
have
|
P.d (i + 1) ((ComplexShape.down ℕ).next (i + 1)) ≫ f ((ComplexShape.down ℕ).next (i + 1)) (i + 1) =
P.d (i + 1) i ≫ f i (i + 1)
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
i : ℕ
⊢ P.d (i + 1) ((ComplexShape.down ℕ).next (i + 1)) ≫ f ((ComplexShape.down ℕ).next (i + 1)) (i + 1) =
P.d (i + 1) i ≫ f i (i + 1)
|
dsimp [dNext]
|
dsimp
|
P.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ (dNext 0 : ((i j : ℕ) → P.X i ⟶ Q.X j) → (P.X 0 ⟶ Q.X 0)) f = 0
|
rw [P.shape, zero_comp]
|
a
|
¬(ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ P.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0
|
P.shape,
|
this
|
0 ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ P.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0
|
P.shape,
|
a
|
¬(ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
this : 0 ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0
⊢ P.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0
|
rw [ChainComplex.next_nat_zero]
|
a
|
¬(ComplexShape.down ℕ).Rel 0 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ ¬(ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)
|
ChainComplex.next_nat_zero
|
a
|
¬(ComplexShape.down ℕ).Rel 0 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ ¬(ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)
|
dsimp
|
a
|
¬1 = 0
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ ¬(ComplexShape.down ℕ).Rel 0 0
|
subst j
|
subst
|
(mkInductiveAux₂ e zero sorry one sorry succ i).snd.fst ≫ (xPrevIso Q sorry).hom =
(xNextIso P sorry).inv ≫ (mkInductiveAux₂ e zero sorry one sorry succ (i + 1)).fst
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i j : ℕ
h : i + 1 = j
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom =
(xNextIso P h).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ j).fst
|
rcases i with (_ | _ | i)
|
zero
|
(mkInductiveAux₂ e zero sorry one sorry succ 0).snd.fst ≫ (xPrevIso Q sorry).hom =
(xNextIso P sorry).inv ≫ (mkInductiveAux₂ e zero sorry one sorry succ (0 + 1)).fst
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q ⋯).hom =
(xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst
|
rcases i with (_ | _ | i)
|
succ
|
(mkInductiveAux₂ e zero sorry one sorry succ (0 + 1)).snd.fst ≫ (xPrevIso Q sorry).hom =
(xNextIso P sorry).inv ≫ (mkInductiveAux₂ e zero sorry one sorry succ (0 + 1 + 1)).fst
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
zero : (mkInductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst ≫ (xPrevIso Q ⋯).hom =
(xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q ⋯).hom =
(xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst
|
rcases i with (_ | _ | i)
|
succ
|
(mkInductiveAux₂ e zero sorry one sorry succ (i + 1 + 1)).snd.fst ≫ (xPrevIso Q sorry).hom =
(xNextIso P sorry).inv ≫ (mkInductiveAux₂ e zero sorry one sorry succ (i + 1 + 1 + 1)).fst
|
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
zero : (mkInductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst ≫ (xPrevIso Q ⋯).hom =
(xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst
succ : (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
(xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1 + 1)).fst
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q ⋯).hom =
(xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst
|
rw [dif_neg]
|
hnc
|
¬i + 1 = j
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i j : ℕ
w : ¬(ComplexShape.down ℕ).Rel j i
⊢ (if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom else 0) = 0
|
dif_neg
|
dif_neg
|
0 = 0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i j : ℕ
w : ¬(ComplexShape.down ℕ).Rel j i
⊢ (if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom else 0) = 0
|
dif_neg
|
hnc
|
¬i + 1 = j
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i j : ℕ
w : ¬(ComplexShape.down ℕ).Rel j i
dif_neg : 0 = 0
⊢ (if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom else 0) = 0
|
dsimp
|
dsimp
|
e.f i =
(dFrom P i ≫
(fromNext i : ((i j : ℕ) → P.X i ⟶ Q.X j) → (xNext P i ⟶ Q.X i)) fun i j ↦
if h : i + 1 = j then (mkInductiveAux₂ e zero sorry one sorry succ i).snd.fst ≫ (xPrevIso Q sorry).hom
else 0) +
((toPrev i : ((i j : ℕ) → P.X i ⟶ Q.X j) → (P.X i ⟶ xPrev Q i)) fun i j ↦
if h : i + 1 = j then (mkInductiveAux₂ e zero sorry one sorry succ i).snd.fst ≫ (xPrevIso Q sorry).hom
else 0) ≫
dTo Q i +
0
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
(n : ℕ) →
(p :
(f : P.X n ⟶ Q.X (n + 1)) ×'
(f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
(f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
⊢ e.f i =
(((dNext i : ((i j : ℕ) → P.X i ⟶ Q.X j) → (P.X i ⟶ Q.X i)) fun i j ↦
if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom
else 0) +
(prevD i : ((i j : ℕ) → P.X i ⟶ Q.X j) → (P.X i ⟶ Q.X i)) fun i j ↦
if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom
else 0) +
Hom.f 0 i
|
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