From HoTT.Basics Require Import Overture Equivalences Tactics.From HoTT.Basics Require Import Overture Equivalences Tactics.
From HoTT.WildCat Require Import Core SetoidRewrite AbEnriched EpiStable.
From HoTT.Spaces Require Import Nat.Core Finite.Fin.

Local Open Scope mc_add_scope.

(** * Homological algebra *)

(** Various "diagram lemmas" are proved, usually assuming the wild category satisfies [IsAbEpiStable].  The proofs use the diagram chasing method provided by WildCat.EpiStable. *)

(** We will often use the convention that a map [bc : B $-> C] is named as the juxtaposition of the domain and codomain.  When we want to highlight the importance of certain maps, they will be given different names. *)

(** ** The weak four lemmas *)

Section FourLemma.

  Context {A : Type} `{IsAbEpiStable A}.

  Context {B C D E B' C' D' E' : A}
    (bc : B $-> C) (cd : C $-> D) (de : D $-> E)
    (bc' : B' $-> C') (cd' : C' $-> D') (de' : D' $-> E')
    (g : B $-> B') (h : C $-> C') (i : D $-> D') (j : E $-> E')
    (sq2 : h $o bc $== bc' $o g) (sq3 : i $o cd $== cd' $o h)
    (sq4 : j $o de $== de' $o i)
    `{!CatIsAbExact cd de} `{!CatIsAbExact bc' cd'} `{!Epic g} `{!Monic j}.

  (** The diagram has the following shape:
<<
  B ---> C ---> D ---> E
  |      |      |      v
  |g     |h     |i     |j
  v      |      |      |
  v      v      v      v
  B' --> C' --> D' --> E'
>>
  The common assumptions are that the map [B $-> B'] is epi, the map [E $-> E'] is mono, and the sequences [C $-> D] + [D $-> E] and [B' $-> C'] + [C' $-> D'] are exact. *)

  (** If in addition the composite of [C' $-> D'] and [D' $-> C'] is zero and the map [D $-> D'] is epi, then the map [C $-> C'] is epi. *)
  Definition weak_four_epi {complex_cde' : de' $o cd' $== 0} `{!Epic i}
    : Epic h.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
Epic h
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
Epic h
    (* To show that [h : C $-> C'] is epi, we must show that an arbitrary generalized element [c' : elt P C'] lifts along [h : C $-> C']. *)
    apply epic_elt_lift; intros P c'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
c': elt P C'
Lift c' h

    (* Since the map [i : D $-> D'] is assumed epi, we can lift the element [cd' $o c' : elt P D'] to an element [d : elt _ D].  [ld] will be the name given to the proof that [d] is a lift. *)
    elt_lift_epic i (cd' $o c') d ld.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
Lift c' h

    (* Using exactness of the top row, we can further lift the element [d] to an element [c : elt _ C].  This step involves verifying that [d] is mapped to zero by [de : D $-> E], which requires the map [j] to be mono. *)
    (* We clear 2-cells and elements when they are no longer needed, to speed up the lifts.  The [using clear] clause is applied *after* the hypothesis is proved. *)
    elt_lift_exact cd de d c lc using clear sq4 complex_cde'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
de $o d $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
Lift c' h
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
de $o d $== 0 apply (ismonic' j).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
j $o (de $o d) $== 0
      rewrite_with_assoc sq4.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
de' $o (i $o d) $== 0
      rewrite ld.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
de' $o (cd' $o c') $== 0
      rewrite cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
de' $o cd' $o c' $== 0
      rewrite complex_cde'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_cde': de' $o cd' $== 0
Epic1: Epic i
P: A
d: elt P D
c': elt P C'
ld: i $o d $== cd' $o c'
0 $o c' $== 0
      apply precomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
Lift c' h

    (* The element [c] is not necessarily the correct lift of [c'].  We correct this by lifting the element [c' - h $o c] using exactness of the bottom row. This yields an element [b' : elt _ B']. *)
    elt_lift_exact bc' cd' (c' - h $o c) b' lb' using clear sq3 lc ld d.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
cd' $o (c' - h $o c) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
b': elt P B'
c: elt P C
c': elt P C'
lb': bc' $o b' $== c' - h $o c
Lift c' h
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
cd' $o (c' - h $o c) $== 0
      rewrite postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
cd' $o c' - cd' $o (h $o c) $== 0
      rewrite_with_assoc sq3^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
cd' $o c' - i $o (cd $o c) $== 0
      rewrite lc, ld.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
d: elt P D
ld: i $o d $== cd' $o c'
c: elt P C
lc: cd $o c $== d
cd' $o c' - cd' $o c' $== 0
      apply inverse_r_0gpd. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
b': elt P B'
c: elt P C
c': elt P C'
lb': bc' $o b' $== c' - h $o c
Lift c' h

    (* Since we know that the map [g : B $-> B'] is epi, we can lift [b'] to an element [b : elt _ B]. *)
    elt_lift_epic g b' b lb.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
Lift c' h

    (* We obtain the correct lift by adding together the incorrect lift and its correction. *)
    provide_lift (c + bc $o b).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
h $o (c + bc $o b) $== c'
    (* It remains to show that this is indeed a lift of [c']. *)
    lhs' napply postcomp_op.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
h $o c + h $o (bc $o b) $== c'
    rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
h $o c + bc' $o (g $o b) $== c'
    rewrite lb, lb'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
h $o c + (c' - h $o c) $== c'
    rewrite (comm_0gpd c').
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
h $o c + (- (h $o c) + c') $== c'
    lhs_V' napply assoc_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
h $o c - h $o c + c' $== c'
    rewrite inverse_r_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
Epic1: Epic i
P: A
c': elt P C'
c: elt P C
b': elt P B'
lb': bc' $o b' $== c' - h $o c
b: elt P B
lb: g $o b $== b'
0 + c' $== c'
    apply mon_unit_l_0gpd.
  Defined.

  (** If, in addition to the common assumptions, the composite of [B $-> C] and [C $-> D] is zero and the map [C $-> C'] is mono, then the map [D $-> D'] is mono. *)
  Definition weak_four_mono {complex_bcd : cd $o bc $== 0} `{!Monic h}
    : Monic i.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
Monic i
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
Monic i
    (* To show that a map is mono, it suffices to show that an arbitrary generalized element [d : elt P D] that goes to [0 : elt P D'] is homotopic to [0 : elt P D]. *)
    apply monic_via_elt_isemb_ab; intros P d p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
d $== 0

    (* Using exactness, we lift the element [d] to [c : elt _ C]. *)
    elt_lift_exact cd de d c lc using clear sq4.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
de $o d $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
c: elt P C
lc: cd $o c $== d
d $== 0
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
de $o d $== 0 apply (ismonic' j).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
j $o (de $o d) $== 0
      rewrite_with_assoc sq4.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
de' $o (i $o d) $== 0
      rewrite p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
sq4: j $o de $== de' $o i
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
de' $o 0 $== 0
      apply postcomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
c: elt P C
lc: cd $o c $== d
d $== 0

    (* Next we lift [h $o c] to [b' : elt _ B']. *)
    elt_lift_exact bc' cd' (h $o c) b' lb' using clear sq3 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
c: elt P C
lc: cd $o c $== d
cd' $o (h $o c) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
c: elt P C
d: elt P D
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
d $== 0
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
c: elt P C
lc: cd $o c $== d
cd' $o (h $o c) $== 0 rewrite_with_assoc sq3^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
c: elt P C
lc: cd $o c $== d
i $o (cd $o c) $== 0
      rewrite lc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
sq3: i $o cd $== cd' $o h
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
p: i $o d $== 0
c: elt P C
lc: cd $o c $== d
i $o d $== 0
      exact p. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
c: elt P C
d: elt P D
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
d $== 0

    (* We lift [b'] to [b : elt _ B]. *)
    elt_lift_epic g b' b lb.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
d $== 0

    (* Since [h] is monic, it follows that [b] is a lift of [c]. *)
    assert (p' : bc $o b $== c).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
bc $o b $== c
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
p': bc $o b $== c
d $== 0
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
bc $o b $== c apply (ismonic h).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
h $o (bc $o b) $== h $o c
      rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
bc' $o (g $o b) $== h $o c
      rewrite lb.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
bc' $o b' $== h $o c
      exact lb'. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
p': bc $o b $== c
d $== 0

    (* So [d] lifts all the way to [B].  Since [B -> C -> D] is a complex, [d] is zero. *)
    lhs_V' napply lc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
p': bc $o b $== c
cd $o c $== 0
    rewrite <- p'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
p': bc $o b $== c
cd $o (bc $o b) $== 0
    lhs' napply cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
p': bc $o b $== c
cd $o bc $o b $== 0
    rewrite complex_bcd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
B, C, D, E, B', C', D', E': A
bc: B $-> C
cd: C $-> D
de: D $-> E
bc': B' $-> C'
cd': C' $-> D'
de': D' $-> E'
g: B $-> B'
h: C $-> C'
i: D $-> D'
j: E $-> E'
sq2: h $o bc $== bc' $o g
CatIsAbExact0: CatIsAbExact cd de
CatIsAbExact1: CatIsAbExact bc' cd'
Epic0: Epic g
Monic0: Monic j
complex_bcd: cd $o bc $== 0
Monic1: Monic h
P: A
d: elt P D
c: elt P C
lc: cd $o c $== d
b': elt P B'
lb': bc' $o b' $== h $o c
b: elt P B
lb: g $o b $== b'
p': bc $o b $== c
0 $o b $== 0
    apply precomp_zero.
  Defined.

End FourLemma.

(** ** The 3x3-lemmas as presented in Mac Lane's book "Homology" *)

Section ThreeByThree.

  Context {A : Type} `{Is1Cat A}.

  Context {X00 X01 X02 X10 X11 X12 X20 X21 X22 : A}
    (f0i : X00 $-> X01) (f0j : X01 $-> X02)
    (fi0 : X00 $-> X10) (fi1 : X01 $-> X11) (fi2 : X02 $-> X12)
    (f1i : X10 $-> X11) (f1j : X11 $-> X12)
    (fj0 : X10 $-> X20) (fj1 : X11 $-> X21) (fj2 : X12 $-> X22)
    (f2i : X20 $-> X21) (f2j : X21 $-> X22)
    (sq1 : fi1 $o f0i $== f1i $o fi0) (sq2 : fi2 $o f0j $== f1j $o fi1)
    (sq3 : fj1 $o f1i $== f2i $o fj0) (sq4 : fj2 $o f1j $== f2j $o fj1).

  (** The diagram has the following shape:
<<
  X00 ----f0i----> X01 ----f0j----> X02
   |                |                |
   |                |                |
  fi0     sq1      fi1     sq2      fi2
   |                |                |
   |                |                |
   v                v                v
  X10 ----f1i----> X11 ----f1j----> X12
   |                |                |
   |                |                |
  fj0     sq3      fj1     sq4      fj2
   |                |                |
   |                |                |
   v                v                v
  X20 ----f2i----> X21 ----f2j----> X22
>>
  *)

  (** If all three columns and the last two rows are short exact, then the top row is short exact.  We prove the three parts of the conclusion as separate results, each requiring a different subset of the hypotheses (and in fact not all of the hypotheses are needed). *)

  Definition three_by_three_monic `{!Monic f1i} `{!Monic fi0} `{!Monic fi1}
    : Monic f0i.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
Monic0: Monic f1i
Monic1: Monic fi0
Monic2: Monic fi1
Monic f0i
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
Monic0: Monic f1i
Monic1: Monic fi0
Monic2: Monic fi1
Monic f0i
    napply (monic_cancel f0i fi1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
Monic0: Monic f1i
Monic1: Monic fi0
Monic2: Monic fi1
Monic (fi1 $o f0i)
    exact (monic_homotopic _ _ sq1^$).
  Defined.

  (** For the remaining results, we need this assumption. *)
  Context `{!IsAbEpiStable A}.

  (** The hypotheses aren't entirely minimal here.  Instead of the two exactness hypotheses, we really only need that the lifting property holds. *)
  Definition three_by_three_ex `{!Monic f2i} `{!Monic fi1} `{!Monic fi2}
    `{!CatIsAbExact fi0 fj0} `{!CatIsAbExact f1i f1j}
    {q : fj1 $o fi1 $== 0}
    : CatIsAbExact f0i f0j.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
CatIsAbExact f0i f0j
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
CatIsAbExact f0i f0j
    snapply Build_CatIsAbExact.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
f0j $o f0i $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
forall (P : A) (b : elt P X01), f0j $o b $== 0 -> Lift b f0i
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
f0j $o f0i $== 0 apply (ismonic' fi2).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
fi2 $o (f0j $o f0i) $== 0
      rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
f1j $o (fi1 $o f0i) $== 0
      lhs' napply (_ $@L sq1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
f1j $o (f1i $o fi0) $== 0
      lhs' napply cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
f1j $o f1i $o fi0 $== 0
      lhs' apply (isabcomplex $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
0 $o fi0 $== 0
      apply precomp_zero.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
forall (P : A) (b : elt P X01), f0j $o b $== 0 -> Lift b f0i intros P x01 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x01: elt P X01
p: f0j $o x01 $== 0
Lift x01 f0i
      clear sq4.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x01: elt P X01
p: f0j $o x01 $== 0
Lift x01 f0i

      elt_lift_exact f1i f1j (fi1 $o x01) x10 lx10 using clear sq2 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x01: elt P X01
p: f0j $o x01 $== 0
f1j $o (fi1 $o x01) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
Lift x01 f0i
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x01: elt P X01
p: f0j $o x01 $== 0
f1j $o (fi1 $o x01) $== 0 rewrite_with_assoc sq2^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x01: elt P X01
p: f0j $o x01 $== 0
fi2 $o (f0j $o x01) $== 0
        lhs' napply (_ $@L p).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x01: elt P X01
p: f0j $o x01 $== 0
fi2 $o 0 $== 0
        apply postcomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
Lift x01 f0i

      elt_lift_exact fi0 fj0 x10 x00 lx00 using clear sq3 q.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
fj0 $o x10 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
P: A
x01: elt P X01
x10: elt P X10
lx10: f1i $o x10 $== fi1 $o x01
x00: elt P X00
lx00: fi0 $o x00 $== x10
Lift x01 f0i
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
fj0 $o x10 $== 0 apply (ismonic' f2i).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
f2i $o (fj0 $o x10) $== 0
        rewrite_with_assoc sq3^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
fj1 $o (f1i $o x10) $== 0
        lhs' napply (_ $@L lx10).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
fj1 $o (fi1 $o x01) $== 0
        lhs' napply cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
fj1 $o fi1 $o x01 $== 0
        lhs' napply (q $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
q: fj1 $o fi1 $== 0
P: A
x10: elt P X10
x01: elt P X01
lx10: f1i $o x10 $== fi1 $o x01
0 $o x01 $== 0
        apply precomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
P: A
x01: elt P X01
x10: elt P X10
lx10: f1i $o x10 $== fi1 $o x01
x00: elt P X00
lx00: fi0 $o x00 $== x10
Lift x01 f0i

      provide_lift x00.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
P: A
x01: elt P X01
x10: elt P X10
lx10: f1i $o x10 $== fi1 $o x01
x00: elt P X00
lx00: fi0 $o x00 $== x10
f0i $o x00 $== x01
      apply (ismonic fi1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
P: A
x01: elt P X01
x10: elt P X10
lx10: f1i $o x10 $== fi1 $o x01
x00: elt P X00
lx00: fi0 $o x00 $== x10
fi1 $o (f0i $o x00) $== fi1 $o x01
      rewrite_with_assoc sq1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
P: A
x01: elt P X01
x10: elt P X10
lx10: f1i $o x10 $== fi1 $o x01
x00: elt P X00
lx00: fi0 $o x00 $== x10
f1i $o (fi0 $o x00) $== fi1 $o x01
      refine (_ $@ lx10).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f2i
Monic1: Monic fi1
Monic2: Monic fi2
CatIsAbExact0: CatIsAbExact fi0 fj0
CatIsAbExact1: CatIsAbExact f1i f1j
P: A
x01: elt P X01
x10: elt P X10
lx10: f1i $o x10 $== fi1 $o x01
x00: elt P X00
lx00: fi0 $o x00 $== x10
f1i $o (fi0 $o x00) $== f1i $o x10
      exact (_ $@L lx00).
  Defined.

  (** We could also do away with some hypothesis here. *)
  Definition three_by_three_epi `{!Monic fi2} `{!Epic f1j} `{!Epic fj0}
    {complex_f_2 : fj2 $o fi2 $== 0} {complex_f1_ : f1j $o f1i $== 0}
    `{!CatIsAbExact fi1 fj1} `{!CatIsAbExact f2i f2j}
    : Epic f0j.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
Epic f0j
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
Epic f0j
    clear sq1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
Epic f0j
    apply epic_elt_lift; intros P x02.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
Lift x02 f0j

    elt_lift_epic f1j (fi2 $o x02) x11 lx11.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
Lift x02 f0j

    elt_lift_exact f2i f2j (fj1 $o x11) x20 lx20 using clear sq4 complex_f_2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
f2j $o (fj1 $o x11) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
Lift x02 f0j
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
f2j $o (fj1 $o x11) $== 0 rewrite_with_assoc sq4^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
fj2 $o (f1j $o x11) $== 0
      lhs' napply (_ $@L lx11).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
fj2 $o (fi2 $o x02) $== 0
      lhs_V' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
fj2 $o fi2 $o x02 $== 0
      lhs' napply (complex_f_2 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f_2: fj2 $o fi2 $== 0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
0 $o x02 $== 0
      apply precomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
Lift x02 f0j

    elt_lift_epic fj0 x20 x10 lx10.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
Lift x02 f0j

    elt_lift_exact fi1 fj1 (x11 - f1i $o x10) x01 lx01 using clear sq3 lx10 lx20 x20.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o (x11 - f1i $o x10) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
Lift x02 f0j
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o (x11 - f1i $o x10) $== 0 lhs' rapply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 - fj1 $o (f1i $o x10) $== 0
      rewrite_with_assoc sq3.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 - f2i $o (fj0 $o x10) $== 0
      rewrite lx10.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x11: elt P X11
x02: elt P X02
lx11: f1j $o x11 $== fi2 $o x02
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 - f2i $o x20 $== 0
      apply (inverse_r_homotopy_0gpd lx20^$). }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
Lift x02 f0j

    provide_lift x01.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f0j $o x01 $== x02
    apply (ismonic fi2).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
fi2 $o (f0j $o x01) $== fi2 $o x02
    rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o (fi1 $o x01) $== fi2 $o x02
    rewrite lx01.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o (x11 - f1i $o x10) $== fi2 $o x02
    lhs' napply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o x11 - f1j $o (f1i $o x10) $== fi2 $o x02
    rewrite <- cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o x11 - f1j $o f1i $o x10 $== fi2 $o x02
    rewrite complex_f1_.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o x11 - 0 $o x10 $== fi2 $o x02
    rewrite precomp_zero.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o x11 - 0 $== fi2 $o x02
    lhs' napply mon_unit_l_inv_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic fi2
Epic0: Epic f1j
Epic1: Epic fj0
complex_f1_: f1j $o f1i $== 0
CatIsAbExact0: CatIsAbExact fi1 fj1
CatIsAbExact1: CatIsAbExact f2i f2j
P: A
x02: elt P X02
x11: elt P X11
lx11: f1j $o x11 $== fi2 $o x02
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o x11 $== fi2 $o x02
    exact lx11.
  Defined.

  (** If every column is exact, the top and bottom rows are short exact, and the middle row is a complex, then the middle row is short exact.  Like above, we prove the three parts of this separately, with fewer hypotheses. *)

  Definition middle_three_by_three_mono
    `{!Monic f0i} `{!Monic f2i} `{!Monic fi1}
    `{!CatIsAbExact fi0 fj0}
    : Monic f1i.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
Monic f1i
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
Monic f1i
    clear sq2 sq4.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
Monic f1i
    apply monic_via_elt_isemb_ab; intros P x10 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x10 $== 0

    elt_lift_exact fi0 fj0 x10 x00 lx00 using clear sq3.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
fj0 $o x10 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
x10 $== 0
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
fj0 $o x10 $== 0 apply (ismonic' f2i).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
f2i $o (fj0 $o x10) $== 0
      rewrite_with_assoc sq3^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
fj1 $o (f1i $o x10) $== 0
      lhs' napply (_ $@L p).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
fj1 $o 0 $== 0
      apply postcomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
x10 $== 0

    lhs_V' napply lx00.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
fi0 $o x00 $== 0
    rhs_V' napply (postcomp_zero fi0).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
fi0 $o x00 $== fi0 $o 0
    apply cat_postwhisker.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
x00 $== 0
    rapply (ismonic' (fi1 $o f0i)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
fi1 $o f0i $o x00 $== 0
    lhs' napply (sq1 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
f1i $o fi0 $o x00 $== 0
    lhs' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
f1i $o (fi0 $o x00) $== 0
    lhs' napply (_ $@L lx00).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Monic0: Monic f0i
Monic1: Monic f2i
Monic2: Monic fi1
CatIsAbExact0: CatIsAbExact fi0 fj0
P: A
x10: elt P X10
p: f1i $o x10 $== 0
x00: elt P X00
lx00: fi0 $o x00 $== x10
f1i $o x10 $== 0
    exact p.
  Defined.

  Definition middle_three_by_three_ex {cpx : f1j $o f1i $== 0}
    `{!Epic fj0} `{!Monic fi2} `{!CatIsAbExact f0i f0j}
    `{!CatIsAbExact f2i f2j} `{!CatIsAbExact fi1 fj1}
    : CatIsAbExact f1i f1j.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
CatIsAbExact f1i f1j
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
CatIsAbExact f1i f1j
    snapply (Build_CatIsAbExact _ _ _ _ _ cpx).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
forall (P : A) (b : elt P X11), f1j $o b $== 0 -> Lift b f1i
    intros P x11 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
Lift x11 f1i

    elt_lift_exact f2i f2j (fj1 $o x11) x20 lx20 using clear sq4.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
f2j $o (fj1 $o x11) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
Lift x11 f1i
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
f2j $o (fj1 $o x11) $== 0 rewrite_with_assoc sq4^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
fj2 $o (f1j $o x11) $== 0
      lhs' napply (_ $@L p).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
fj2 $o 0 $== 0
      apply postcomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
Lift x11 f1i

    elt_lift_epic fj0 x20 x10 lx10.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
Lift x11 f1i

    elt_lift_exact fi1 fj1 (x11 - (f1i $o x10)) x01 lx01 using clear sq3 lx10 lx20 x20.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o (x11 - f1i $o x10) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
Lift x11 f1i
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o (x11 - f1i $o x10) $== 0 lhs' napply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 - fj1 $o (f1i $o x10) $== 0
      apply inverse_r_homotopy_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 $== fj1 $o (f1i $o x10)
      rewrite_with_assoc sq3.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 $== f2i $o (fj0 $o x10)
      rhs' napply (_ $@L lx10).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x20: elt P X20
lx20: f2i $o x20 $== fj1 $o x11
x10: elt P X10
lx10: fj0 $o x10 $== x20
fj1 $o x11 $== f2i $o x20
      exact lx20^$. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
Lift x11 f1i

    elt_lift_exact f0i f0j x01 x00 lx00 using clear sq2 p cpx.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f0j $o x01 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
Lift x11 f1i
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f0j $o x01 $== 0 apply (ismonic' fi2).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
fi2 $o (f0j $o x01) $== 0
      rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o (fi1 $o x01) $== 0
      lhs' napply (_ $@L lx01).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o (x11 - f1i $o x10) $== 0
      lhs' napply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
f1j $o x11 - f1j $o (f1i $o x10) $== 0
      rewrite p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
0 - f1j $o (f1i $o x10) $== 0
      rewrite cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
0 - f1j $o f1i $o x10 $== 0
      rewrite cpx.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
0 - 0 $o x10 $== 0
      rewrite precomp_zero.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
cpx: f1j $o f1i $== 0
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x11: elt P X11
p: f1j $o x11 $== 0
x01: elt P X01
x10: elt P X10
lx01: fi1 $o x01 $== x11 - f1i $o x10
0 - 0 $== 0
      apply inverse_r_0gpd. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
Lift x11 f1i

    provide_lift ((fi0 $o x00) + x10).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
f1i $o (fi0 $o x00 + x10) $== x11
    lhs' napply postcomp_op.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
f1i $o (fi0 $o x00) + f1i $o x10 $== x11
    rewrite_with_assoc sq1^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
fi1 $o (f0i $o x00) + f1i $o x10 $== x11
    rewrite lx00.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
fi1 $o x01 + f1i $o x10 $== x11
    rewrite lx01.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
x11 - f1i $o x10 + f1i $o x10 $== x11
    lhs' napply assoc_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
x11 + (- (f1i $o x10) + f1i $o x10) $== x11
    rewrite inverse_l_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic fj0
Monic0: Monic fi2
CatIsAbExact0: CatIsAbExact f0i f0j
CatIsAbExact1: CatIsAbExact f2i f2j
CatIsAbExact2: CatIsAbExact fi1 fj1
P: A
x10: elt P X10
x01: elt P X01
x11: elt P X11
lx01: fi1 $o x01 $== x11 - f1i $o x10
x00: elt P X00
lx00: f0i $o x00 $== x01
x11 + 0 $== x11
    napply mon_unit_r_0gpd.
  Defined.

  Definition middle_three_by_three_epi `{!Epic f0j} `{!Epic f2j}
    `{!Epic fj1} `{!CatIsAbExact fi2 fj2}
    : Epic f1j.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
Epic f1j
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq1: fi1 $o f0i $== f1i $o fi0
sq2: fi2 $o f0j $== f1j $o fi1
sq3: fj1 $o f1i $== f2i $o fj0
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
Epic f1j
    clear sq1 sq3.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
Epic f1j
    apply epic_elt_lift; intros P x12.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x12: elt P X12
Lift x12 f1j

    elt_lift_epic (f2j $o fj1) (fj2 $o x12) x11 lx11.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
Lift x12 f1j

    elt_lift_exact fi2 fj2 (x12 - (f1j $o x11)) x02 lx02 using clear sq4 lx11.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
fj2 $o (x12 - f1j $o x11) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x02: elt P X02
x12: elt P X12
x11: elt P X11
lx02: fi2 $o x02 $== x12 - f1j $o x11
Lift x12 f1j
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
fj2 $o (x12 - f1j $o x11) $== 0 lhs' napply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
fj2 $o x12 - fj2 $o (f1j $o x11) $== 0
      apply inverse_r_homotopy_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
fj2 $o x12 $== fj2 $o (f1j $o x11)
      rhs_V' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
fj2 $o x12 $== fj2 $o f1j $o x11
      rhs' napply (sq4 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
sq4: fj2 $o f1j $== f2j $o fj1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
lx11: f2j $o fj1 $o x11 $== fj2 $o x12
fj2 $o x12 $== f2j $o fj1 $o x11
      exact lx11^$. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x02: elt P X02
x12: elt P X12
x11: elt P X11
lx02: fi2 $o x02 $== x12 - f1j $o x11
Lift x12 f1j

    elt_lift_epic f0j x02 x01 lx01.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
Lift x12 f1j

    provide_lift ((fi1 $o x01) + x11).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
f1j $o (fi1 $o x01 + x11) $== x12
    lhs' napply postcomp_op.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
f1j $o (fi1 $o x01) + f1j $o x11 $== x12
    rewrite_with_assoc sq2^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
fi2 $o (f0j $o x01) + f1j $o x11 $== x12
    rewrite lx01.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
fi2 $o x02 + f1j $o x11 $== x12
    rewrite lx02.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
x12 - f1j $o x11 + f1j $o x11 $== x12
    lhs' napply assoc_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
x12 + (- (f1j $o x11) + f1j $o x11) $== x12
    rewrite inverse_l_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
X00, X01, X02, X10, X11, X12, X20, X21, X22: A
f0i: X00 $-> X01
f0j: X01 $-> X02
fi0: X00 $-> X10
fi1: X01 $-> X11
fi2: X02 $-> X12
f1i: X10 $-> X11
f1j: X11 $-> X12
fj0: X10 $-> X20
fj1: X11 $-> X21
fj2: X12 $-> X22
f2i: X20 $-> X21
f2j: X21 $-> X22
sq2: fi2 $o f0j $== f1j $o fi1
IsAbEpiStable0: IsAbEpiStable A
Epic0: Epic f0j
Epic1: Epic f2j
Epic2: Epic fj1
CatIsAbExact0: CatIsAbExact fi2 fj2
P: A
x11: elt P X11
x12: elt P X12
x02: elt P X02
lx02: fi2 $o x02 $== x12 - f1j $o x11
x01: elt P X01
lx01: f0j $o x01 $== x02
x12 + 0 $== x12
    apply mon_unit_r_0gpd.
  Defined.

End ThreeByThree.

(** ** The spider lemma *)

Section Spider.

  Context {A : Type} `{IsAbEpiStable A}.

  (** Given a diagram
<<
    A1 -> A2
      \   ^
       v /
        X
       ^ \
      /   v
    B1 -> B2
>>
  where the diagonals are exact, if the map [B1 $-> B2] is mono/epi, so is the map [A1 $-> A2]. *)

  Context {A1 A2 X B1 B2 : A} (a : A1 $-> A2) (ax : A1 $-> X)
    (xa : X $-> A2) (b : B1 $-> B2) (bx : B1 $-> X) (xb : X $-> B2)
    {tr1 : xa $o ax $== a} {tr2 : xb $o bx $== b}
    `{!Monic ax} `{!CatIsAbExact ax xb} `{!Epic xb}
    `{!Monic bx} `{!CatIsAbExact bx xa} `{!Epic xa}.

  Definition spider_mono `{!Monic b} : Monic a.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
Monic a
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
Monic a
    apply monic_via_elt_isemb_ab; intros P a1 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
a1: elt P A1
p: a $o a1 $== 0
a1 $== 0
    apply (ismonic' ax a1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
a1: elt P A1
p: a $o a1 $== 0
ax $o a1 $== 0

    elt_lift_exact bx xa (ax $o a1) b1 lb1 using clear tr1 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
a1: elt P A1
p: a $o a1 $== 0
xa $o (ax $o a1) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
ax $o a1 $== 0
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
a1: elt P A1
p: a $o a1 $== 0
xa $o (ax $o a1) $== 0 lhs' napply cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
a1: elt P A1
p: a $o a1 $== 0
xa $o ax $o a1 $== 0
      lhs' napply (tr1 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
a1: elt P A1
p: a $o a1 $== 0
a $o a1 $== 0
      exact p. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
ax $o a1 $== 0

    assert (b1_is_zero : b1 $== 0).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b1 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b1_is_zero: b1 $== 0
ax $o a1 $== 0
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b1 $== 0 apply (ismonic' b).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b $o b1 $== 0
      lhs_V' napply (tr2 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
xb $o bx $o b1 $== 0
      rewrite_with_assoc_opp lb1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
xb $o ax $o a1 $== 0
      lhs' apply (isabcomplex $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
0 $o a1 $== 0
      apply precomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b1_is_zero: b1 $== 0
ax $o a1 $== 0

    lhs_V' napply lb1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b1_is_zero: b1 $== 0
bx $o b1 $== 0
    lhs' napply (_ $@L b1_is_zero).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Monic2: Monic b
P: A
b1: elt P B1
a1: elt P A1
lb1: bx $o b1 $== ax $o a1
b1_is_zero: b1 $== 0
bx $o 0 $== 0
    apply postcomp_zero.
  Defined.

  Definition spider_epi `{!Epic b} : Epic a.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
Epic a
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
Epic a
    apply epic_elt_lift; intros P a2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
Lift a2 a

    elt_lift_epic xa a2 x lx.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
Lift a2 a

    elt_lift_epic b (xb $o x) b1 lb1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
x: elt P X
lx: xa $o x $== a2
b1: elt P B1
lb1: b $o b1 $== xb $o x
Lift a2 a

    elt_lift_exact ax xb (x - (bx $o b1)) a1 la1 using clear tr2 lb1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
x: elt P X
lx: xa $o x $== a2
b1: elt P B1
lb1: b $o b1 $== xb $o x
xb $o (x - bx $o b1) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
Lift a2 a
    {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
x: elt P X
lx: xa $o x $== a2
b1: elt P B1
lb1: b $o b1 $== xb $o x
xb $o (x - bx $o b1) $== 0 rewrite postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
x: elt P X
lx: xa $o x $== a2
b1: elt P B1
lb1: b $o b1 $== xb $o x
xb $o x - xb $o (bx $o b1) $== 0
      rewrite <- cat_assoc, tr2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
x: elt P X
lx: xa $o x $== a2
b1: elt P B1
lb1: b $o b1 $== xb $o x
xb $o x - b $o b1 $== 0
      rewrite <- lb1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
tr2: xb $o bx $== b
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
a2: elt P A2
x: elt P X
lx: xa $o x $== a2
b1: elt P B1
lb1: b $o b1 $== xb $o x
b $o b1 - b $o b1 $== 0
      apply inverse_r_0gpd. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
Lift a2 a

    provide_lift a1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
a $o a1 $== a2
    lhs_V' napply (tr1 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o ax $o a1 $== a2
    lhs' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o (ax $o a1) $== a2
    lhs' napply (_ $@L la1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o (x - bx $o b1) $== a2
    rewrite postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o x - xa $o (bx $o b1) $== a2
    rewrite <- cat_assoc, isabcomplex.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o x - 0 $o b1 $== a2
    rewrite precomp_zero.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o x - 0 $== a2
    rewrite mon_unit_l_inv_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, X, B1, B2: A
a: A1 $-> A2
ax: A1 $-> X
xa: X $-> A2
b: B1 $-> B2
bx: B1 $-> X
xb: X $-> B2
tr1: xa $o ax $== a
Monic0: Monic ax
CatIsAbExact0: CatIsAbExact ax xb
Epic0: Epic xb
Monic1: Monic bx
CatIsAbExact1: CatIsAbExact bx xa
Epic1: Epic xa
Epic2: Epic b
P: A
x: elt P X
a2: elt P A2
lx: xa $o x $== a2
a1: elt P A1
b1: elt P B1
la1: ax $o a1 $== x - bx $o b1
xa $o x $== a2
    exact lx.
  Defined.

End Spider.

(** ** The incomplete snail lemma *)

Section Snail.

  Context {A : Type} `{IsAbEpiStable A}.

  (** Here we will prove the incomplete snail lemma.  The complete snail lemma was developed as a more general analogue of the snake lemma that holds in certain non-abelian contexts.  The incomplete snail lemma is a lemma that one can use to prove the complete snail lemma.  In fact, if the incomplete snail lemma is true and if every kernel map in the category has a cokernel, then the complete snail lemma is true.  See Janelidze, Vitale; The snail lemma in a pointed regular category. *)

  Context {W1 W2 X Y1 Y2 Z : A} (w : W1 $-> W2) (wx : W1 $-> X)
    (wz : W2 $-> Z) (xw : X $-> W2) (xy : X $-> Y2)
    (y : Y1 $-> Y2) (yx : Y1 $-> X) (yz : Y2 $-> Z)
    (tr1 :  w $== xw $o wx) (tr2 : y $== xy $o yx)
    (sq1 : wz $o xw $== yz $o xy).

  (** We assume that we have a diagram of shape:
<<
        W1 = = W1
        |      |
        |      |
        v      v
  Y1 -> X ---> W2 ---> 0
  ||    |      |
  ||    |      |
  ||    v      v
  Y1 -> Y2 --> Z
>>
  Each row and the first column is assumed to be exact.  We will show that the last column is exact as well. *)

  Definition incomplete_snail `{!Epic xw} `{!CatIsAbExact wx xy}
    `{!CatIsAbExact yx xw} `{!CatIsAbExact y yz}
    : CatIsAbExact w wz.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
CatIsAbExact w wz
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
CatIsAbExact w wz
    snapply Build_CatIsAbExact.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
wz $o w $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
forall (P : A) (b : elt P W2), wz $o b $== 0 -> Lift b w
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
wz $o w $== 0 lhs' napply (_ $@L tr1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
wz $o (xw $o wx) $== 0
      rewrite_with_assoc sq1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
yz $o (xy $o wx) $== 0
      lhs' apply (_ $@L isabcomplex).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
yz $o 0 $== 0
      apply postcomp_zero.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
forall (P : A) (b : elt P W2), wz $o b $== 0 -> Lift b w intros P w2 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
p: wz $o w2 $== 0
Lift w2 w
      elt_lift_epic xw w2 x lx.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
p: wz $o w2 $== 0
x: elt P X
lx: xw $o x $== w2
Lift w2 w

      elt_lift_exact y yz (xy $o x) y1 ly1 using clear sq1 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
p: wz $o w2 $== 0
x: elt P X
lx: xw $o x $== w2
yz $o (xy $o x) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
Lift w2 w
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
p: wz $o w2 $== 0
x: elt P X
lx: xw $o x $== w2
yz $o (xy $o x) $== 0 rewrite_with_assoc sq1^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
p: wz $o w2 $== 0
x: elt P X
lx: xw $o x $== w2
wz $o (xw $o x) $== 0
        lhs' napply (_ $@L lx).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
sq1: wz $o xw $== yz $o xy
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
p: wz $o w2 $== 0
x: elt P X
lx: xw $o x $== w2
wz $o w2 $== 0
        exact p. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
Lift w2 w

      elt_lift_exact wx xy (x - yx $o y1) w1 lw1 using clear tr2 ly1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
xy $o (x - yx $o y1) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
Lift w2 w
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
xy $o (x - yx $o y1) $== 0 lhs' rapply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
xy $o x - xy $o (yx $o y1) $== 0
        rewrite <- cat_assoc, <- tr2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
xy $o x - y $o y1 $== 0
        rewrite ly1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
tr2: y $== xy $o yx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
x: elt P X
w2: elt P W2
lx: xw $o x $== w2
y1: elt P Y1
ly1: y $o y1 $== xy $o x
xy $o x - xy $o x $== 0
        apply inverse_r_0gpd. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
Lift w2 w

      provide_lift w1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
w $o w1 $== w2
      lhs' napply (tr1 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o wx $o w1 $== w2
      lhs' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o (wx $o w1) $== w2
      lhs' napply (_ $@L lw1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o (x - yx $o y1) $== w2
      lhs' rapply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o x - xw $o (yx $o y1) $== w2
      rewrite <- cat_assoc, isabcomplex.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o x - 0 $o y1 $== w2
      rewrite precomp_zero.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o x - 0 $== w2
      lhs' napply mon_unit_l_inv_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
W1, W2, X, Y1, Y2, Z: A
w: W1 $-> W2
wx: W1 $-> X
wz: W2 $-> Z
xw: X $-> W2
xy: X $-> Y2
y: Y1 $-> Y2
yx: Y1 $-> X
yz: Y2 $-> Z
tr1: w $== xw $o wx
Epic0: Epic xw
CatIsAbExact0: CatIsAbExact wx xy
CatIsAbExact1: CatIsAbExact yx xw
CatIsAbExact2: CatIsAbExact y yz
P: A
w2: elt P W2
x: elt P X
lx: xw $o x $== w2
w1: elt P W1
y1: elt P Y1
lw1: wx $o w1 $== x - yx $o y1
xw $o x $== w2
      exact lx.
  Defined.

End Snail.

(** ** Two-by-three lemmas *)

Section TwoByThree.

  Context {A : Type} `{IsAbEpiStable A}.

  (** Here we will show how to transport the structure of an exact sequence across a morphism of composable pairs satisfying some properties. *)

  Context {A1 A2 B1 B2 C1 C2 : A}
    (f1 : A1 $-> B1) (g1 : B1 $-> C1)
    (f2 : A2 $-> B2) (g2 : B2 $-> C2)
    (a : A1 $-> A2) (b : B1 $-> B2) (c : C1 $-> C2)
    (sq1 : b $o f1 $== f2 $o a) (sq2 : c $o g1 $== g2 $o b)
    `{!Epic a} `{!Monic c}.

  (** In general, the diagrams will be of the shape:
<<
  A1 ---> B1 ---> C1
  |        |       v
  |        |       |
  v        |       |
  v        v       v
  A2 ---> B2 ---> C2
>>
  In particular the map [A1 $-> A2] is epi and the map [C1 $-> C2] is mono. *)

  (** If the middle map [B1 $-> B2] is also epi and the upper morphisms [A1 $-> B1] and [B1 $-> C1] form an exact sequence, then the lower morphisms [A2 $-> B2] and [B2 $-> C2] form an exact sequence. *)
  Definition two_by_three_epi `{!Epic b} `{!CatIsAbExact f1 g1}
    : CatIsAbExact f2 g2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
CatIsAbExact f2 g2
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
CatIsAbExact f2 g2
    napply Build_CatIsAbExact.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
g2 $o f2 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
forall (P : A) (b0 : elt P B2), g2 $o b0 $== 0 -> Lift b0 f2
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
g2 $o f2 $== 0 apply (isepic a).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
g2 $o f2 $o a $== 0 $o a
      rhs' napply precomp_zero.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
g2 $o f2 $o a $== 0
      rewrite_with_assoc_opp sq1^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
g2 $o b $o f1 $== 0
      lhs_V' napply (sq2 $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
c $o g1 $o f1 $== 0
      lhs' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
c $o (g1 $o f1) $== 0
      lhs' apply (_ $@L isabcomplex).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
c $o 0 $== 0
      apply postcomp_zero.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
forall (P : A) (b0 : elt P B2), g2 $o b0 $== 0 -> Lift b0 f2 intros P b2 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
Lift b2 f2

      elt_lift_epic b b2 b1 lb1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
Lift b2 f2

      elt_lift_exact f1 g1 b1 a1 la1 using clear sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
g1 $o b1 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
a1: elt P A1
la1: f1 $o a1 $== b1
Lift b2 f2
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
g1 $o b1 $== 0 apply (ismonic' c).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
c $o (g1 $o b1) $== 0
        rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
g2 $o (b $o b1) $== 0
        by lhs' napply (_ $@L lb1). }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
a1: elt P A1
la1: f1 $o a1 $== b1
Lift b2 f2

      provide_lift (a $o a1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
a1: elt P A1
la1: f1 $o a1 $== b1
f2 $o (a $o a1) $== b2
      rewrite_with_assoc sq1^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Epic1: Epic b
CatIsAbExact0: CatIsAbExact f1 g1
P: A
b2: elt P B2
p: g2 $o b2 $== 0
b1: elt P B1
lb1: b $o b1 $== b2
a1: elt P A1
la1: f1 $o a1 $== b1
b $o (f1 $o a1) $== b2
      by lhs' napply (_ $@L la1).
  Defined.

  (** If the middle map [B1 $-> B2] is now mono and the lower morphisms [A2 $-> B2] and [B2 $-> C2] form an exact sequence, then the upper morphisms [A1 $-> B1] and [B1 $-> C1] form an exact sequence. *)
  Definition two_by_three_mono `{!Monic b} `{!CatIsAbExact f2 g2}
    : CatIsAbExact f1 g1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
CatIsAbExact f1 g1
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
CatIsAbExact f1 g1
    napply Build_CatIsAbExact.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
g1 $o f1 $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
forall (P : A) (b0 : elt P B1), g1 $o b0 $== 0 -> Lift b0 f1
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
g1 $o f1 $== 0 apply (ismonic' c).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
c $o (g1 $o f1) $== 0
      rewrite_with_assoc sq2.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
g2 $o (b $o f1) $== 0
      lhs' napply (_ $@L sq1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
g2 $o (f2 $o a) $== 0
      lhs_V' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
g2 $o f2 $o a $== 0
      lhs' apply (isabcomplex $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
0 $o a $== 0
      apply precomp_zero.
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
forall (P : A) (b0 : elt P B1), g1 $o b0 $== 0 -> Lift b0 f1 intros P b1 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
p: g1 $o b1 $== 0
Lift b1 f1

      elt_lift_exact f2 g2 (b $o b1) a2 la2 using clear sq2 p.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
p: g1 $o b1 $== 0
g2 $o (b $o b1) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
a2: elt P A2
b1: elt P B1
la2: f2 $o a2 $== b $o b1
Lift b1 f1
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
p: g1 $o b1 $== 0
g2 $o (b $o b1) $== 0 rewrite_with_assoc sq2^$.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
p: g1 $o b1 $== 0
c $o (g1 $o b1) $== 0
        lhs' napply (_ $@L p).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
sq2: c $o g1 $== g2 $o b
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
p: g1 $o b1 $== 0
c $o 0 $== 0
        apply postcomp_zero. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
a2: elt P A2
b1: elt P B1
la2: f2 $o a2 $== b $o b1
Lift b1 f1

      elt_lift_epic a a2 a1 la1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
a2: elt P A2
la2: f2 $o a2 $== b $o b1
a1: elt P A1
la1: a $o a1 $== a2
Lift b1 f1

      provide_lift a1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
a2: elt P A2
la2: f2 $o a2 $== b $o b1
a1: elt P A1
la1: a $o a1 $== a2
f1 $o a1 $== b1
      apply (ismonic b).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
a2: elt P A2
la2: f2 $o a2 $== b $o b1
a1: elt P A1
la1: a $o a1 $== a2
b $o (f1 $o a1) $== b $o b1
      rewrite_with_assoc sq1.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
a2: elt P A2
la2: f2 $o a2 $== b $o b1
a1: elt P A1
la1: a $o a1 $== a2
f2 $o (a $o a1) $== b $o b1
      lhs' napply (_ $@L la1).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
A1, A2, B1, B2, C1, C2: A
f1: A1 $-> B1
g1: B1 $-> C1
f2: A2 $-> B2
g2: B2 $-> C2
a: A1 $-> A2
b: B1 $-> B2
c: C1 $-> C2
sq1: b $o f1 $== f2 $o a
Epic0: Epic a
Monic0: Monic c
Monic1: Monic b
CatIsAbExact0: CatIsAbExact f2 g2
P: A
b1: elt P B1
a2: elt P A2
la2: f2 $o a2 $== b $o b1
a1: elt P A1
la1: a $o a1 $== a2
f2 $o a2 $== b $o b1
      exact la2.
  Defined.

End TwoByThree.

(** ** The diamond lemma *)

Section Diamond.

  Context {Ae : Type} `{H0 : Is1Cat Ae} `{!IsAbEpiStable Ae}.

  Context {A B C D E F G H : Ae}
    (ab : A $-> B) (bc : B $-> C) (cd : C $-> D) (de : D $-> E)
    (ah : A $-> H) (hg : H $-> G) (gf : G $-> F) (fe : F $-> E)
    (hb : H $-> B) (bd : B $-> D) (hf : H $-> F) (fd : F $-> D)
    (tru : ab $== hb $o ah) (trr : bd $== cd $o bc)
    (trd : fe $== de $o fd) (trl : hf $== gf $o hg)
    (sq : bd $o hb $== fd $o hf)
    `{!CatIsAbExact ab bc} `{!CatIsAbExact cd de}
    `{!CatIsAbExact ah hg} `{!CatIsAbExact gf fe}
    `{!Epic hg} `{!Monic cd}.

  (** The diagram has the shape:
<<
          A
         / \
        /   \
       v     v
      H ----> B
     /|       |\
    / |       | \
   v  |       |  v
  G   |       |   C
   \  |       |  /
    \ |       | /
     vv       vv
      F ----> D
       \     /
        \   /
         v v
          E
>>
  where each diagonal is assumed to be exact. *)

  (** If the map [H $-> B] is mono, then so is [G $-> F]. *)
  Definition diamond_lemma_mono `{!Monic hb} : Monic gf.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
Monic gf
  Proof.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
Monic gf
    clear trd.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
Monic gf
    apply monic_via_elt_isemb_ab.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
forall (P : Ae) (b : elt P G), gf $o b $== 0 -> b $== 0
    intros P g p.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
g $== 0

    elt_lift_epic hg g h lh.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
g $== 0

    elt_lift_exact ab bc (hb $o h) a la using clear trr trl p sq.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
bc $o (hb $o h) $== 0
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
g $== 0
    {
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
bc $o (hb $o h) $== 0 apply (ismonic' cd).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
cd $o (bc $o (hb $o h)) $== 0
      lhs_V' napply cat_assoc.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
cd $o bc $o (hb $o h) $== 0
      lhs_V' napply (trr $@R _).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
bd $o (hb $o h) $== 0
      rewrite_with_assoc sq.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
fd $o (hf $o h) $== 0
      rewrite trl.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
fd $o (gf $o hg $o h) $== 0
      rewrite cat_assoc, lh, p.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
g: elt P G
p: gf $o g $== 0
h: elt P H
lh: hg $o h $== g
fd $o 0 $== 0
      apply postcomp_zero. }
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
g $== 0

    assert (la' : ah $o a $== h).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
ah $o a $== h
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
la': ah $o a $== h
g $== 0
    {
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
ah $o a $== h apply (ismonic hb).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
hb $o (ah $o a) $== hb $o h
      lhs_V' napply cat_assoc.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
hb $o ah $o a $== hb $o h
      rewrite <- tru.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
ab $o a $== hb $o h
      exact la. }
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
la': ah $o a $== h
g $== 0

    lhs_V' napply lh.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
la': ah $o a $== h
hg $o h $== 0
    lhs_V' napply (_ $@L la').
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
la': ah $o a $== h
hg $o (ah $o a) $== 0
    lhs' napply cat_assoc_opp.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
la': ah $o a $== h
hg $o ah $o a $== 0
    lhs' rapply (isabcomplex $@R _).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Monic1: Monic hb
P: Ae
h: elt P H
g: elt P G
lh: hg $o h $== g
a: elt P A
la: ab $o a $== hb $o h
la': ah $o a $== h
0 $o a $== 0
    apply precomp_zero.
  Defined.

  (** If the map [F $-> D] is epi, then so is the map [B $-> C]. *)
  Definition diamond_lemma_epi `{!Epic fd} : Epic bc.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
Epic bc
  Proof.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
tru: ab $== hb $o ah
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
Epic bc
    clear tru.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
Epic bc
    apply epic_elt_lift; intros P c.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
c: elt P C
Lift c bc

    elt_lift_epic fd (cd $o c) f lf.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
Lift c bc

    elt_lift_exact gf fe f g lg using clear trd.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
fe $o f $== 0
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
c: elt P C
f: elt P F
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
Lift c bc
    {
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
fe $o f $== 0 lhs' napply (trd $@R f).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
de $o fd $o f $== 0
      lhs' napply cat_assoc.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
de $o (fd $o f) $== 0
      lhs' napply (de $@L lf).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
de $o (cd $o c) $== 0
      lhs_V' napply cat_assoc.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
de $o cd $o c $== 0
      lhs' rapply (isabcomplex $@R c).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trd: fe $== de $o fd
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
0 $o c $== 0
      napply precomp_zero. }
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
c: elt P C
f: elt P F
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
Lift c bc

    elt_lift_epic hg g h lh.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
Lift c bc

    provide_lift (hb $o h).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
bc $o (hb $o h) $== c
    apply (ismonic cd).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
cd $o (bc $o (hb $o h)) $== cd $o c
    lhs' napply cat_assoc_opp.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
cd $o bc $o (hb $o h) $== cd $o c
    lhs_V' napply (trr $@R _).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
bd $o (hb $o h) $== cd $o c
    lhs' napply cat_assoc_opp.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
bd $o hb $o h $== cd $o c
    lhs' napply (sq $@R _).
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
fd $o hf $o h $== cd $o c
    lhs' napply cat_assoc.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
fd $o (hf $o h) $== cd $o c
    rewrite trl.
Ae: Type
IsGraph0: IsGraph Ae
Is2Graph0: Is2Graph Ae
Is01Cat0: Is01Cat Ae
H0: Is1Cat Ae
IsAbEpiStable0: IsAbEpiStable Ae
A, B, C, D, E, F, G, H: Ae
ab: A $-> B
bc: B $-> C
cd: C $-> D
de: D $-> E
ah: A $-> H
hg: H $-> G
gf: G $-> F
fe: F $-> E
hb: H $-> B
bd: B $-> D
hf: H $-> F
fd: F $-> D
trr: bd $== cd $o bc
trl: hf $== gf $o hg
sq: bd $o hb $== fd $o hf
CatIsAbExact0: CatIsAbExact ab bc
CatIsAbExact1: CatIsAbExact cd de
CatIsAbExact2: CatIsAbExact ah hg
CatIsAbExact3: CatIsAbExact gf fe
Epic0: Epic hg
Monic0: Monic cd
Epic1: Epic fd
P: Ae
f: elt P F
c: elt P C
lf: fd $o f $== cd $o c
g: elt P G
lg: gf $o g $== f
h: elt P H
lh: hg $o h $== g
fd $o (gf $o hg $o h) $== cd $o c
    by rewrite cat_assoc, lh, lg, lf.
  Defined.

End Diamond.

(** ** The baby dragon lemmas *)

Section Baby_Dragon.

  (** The dragon lemmas involve diagrams which in general have the same shape, but can be elongated or shortened.  For example, the case [n = 2] below has shape:
<<
          top 0     top 1    top 2
          /   \     /  \     /  \
         /     \   /    \   /    \
        v       v v      v v      v
       mid 0   mid 1    mid 2    mid 3
         \     /   \     /  \     /
          \   /     \   /    \   /
           v v       v v      v v
          bot 0     bot 1    bot 2
>>
  where all the diagonal sequences are exact, the maps [top 0 $-> middle 0], [middle 1 $-> bot 1], and [middle 2 $-> bot 2] are epic, and the maps [top 0 $-> middle 1], [top 1 $-> middle 2], and [middle 3 $-> bot 2] are mono.

  For general [n], the diagram will have a similar shape, with [n + 1] objects in the top and bottom rows. *)

  (** The baby dragon lemma has an epi statement and a mono statement.  We shall first prove the epi part, then the mono part.  To do the epi part, we will show that if the map [middle 0 $-> bot 0] is epi, then so is [top 2 $-> middle 3].  Due to the lack of symmetry at the edges of the diagram, we need to prove a more general statement first to make the induction go through. *)

  Context {A : Type} `{IsAbEpiStable A}.

  (** We make a datatype for the above diagram with [n+1] squares. *)
  Record BabyDragon {n : nat} := {
    top : Fin n.+1 -> A;
    mid : Fin n.+2 -> A;
    bot : Fin n.+1 -> A;
    top_l (k : Fin n.+1) : top k $-> mid (fin_incl k);
    top_r (k : Fin n.+1) : top k $-> mid (fsucc k);
    bot_l (k : Fin n.+1) : mid (fin_incl k) $-> bot k;
    bot_r (k : Fin n.+1) : mid (fsucc k) $-> bot k;
    sq (k : Fin n.+1) : (bot_l k) $o (top_l k) $== (bot_r k) $o (top_r k);
    exact_r (k : Fin n) :: CatIsAbExact (top_r (fin_incl k)) (bot_l (fsucc k));
    exact_l (k : Fin n) :: CatIsAbExact (top_l (fsucc k)) (bot_r (fin_incl k))
  }.
  (** Note that [bot_l k] refers to the bottom-left map in the [k]th square, which points down and to the *right*. *)

  Arguments BabyDragon : clear implicits.

  (** In order to make an inductive argument, we need two ways of truncating a diagram.  This one drops the right-hand square, keeping the left [n+1] squares. *)
  Definition baby_dragon_trunc_left (n : nat) (D : BabyDragon n.+1)
    : BabyDragon n.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
BabyDragon n
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
BabyDragon n
    snapply Build_BabyDragon; intro k; cbn.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+2
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal $-> ?Goal0@{k:=fin_incl k}
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal $->
?Goal0@{k:=match k with
           | inl i' => inl (fsucc i')
           | inr tt => inr tt
           end}
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal0@{k:=fin_incl k} $-> ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal0@{k:=match k with
           | inl i' => inl (fsucc i')
           | inr tt => inr tt
           end} $->
?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal4 $o ?Goal2 $== ?Goal5 $o ?Goal3
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact ?Goal3@{k:=fin_incl k} ?Goal4@{k:=fsucc k}
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact ?Goal2@{k:=fsucc k} ?Goal5@{k:=fin_incl k}
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A exact (top D (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+2
A exact (mid D (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A exact (bot D (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
top D (fin_incl k) $-> mid D (fin_incl (fin_incl k)) exact (top_l _ (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
top D (fin_incl k) $->
mid D
  (fin_incl match k with
            | inl i' => inl (fsucc i')
            | inr tt => inr tt
            end) exact (top_r _ (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
mid D (fin_incl (fin_incl k)) $-> bot D (fin_incl k) exact (bot_l _ (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
mid D
  (fin_incl match k with
            | inl i' => inl (fsucc i')
            | inr tt => inr tt
            end) $->
bot D (fin_incl k) exact (bot_r _ (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
bot_l D (fin_incl k) $o top_l D (fin_incl k) $==
bot_r D (fin_incl k) $o top_r D (fin_incl k) exact (sq _ (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact (top_r D (fin_incl (fin_incl k))) (bot_l D (fin_incl (fsucc k))) exact (exact_r _ (fin_incl k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact (top_l D (fin_incl (fsucc k))) (bot_r D (fin_incl (fin_incl k))) exact (exact_l _ (fin_incl k)).
  Defined.

  (** This one drops the left-hand square, keeping the right [n+1] squares. *)
  Definition baby_dragon_trunc_right (n : nat) (D : BabyDragon n.+1)
    : BabyDragon n.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
BabyDragon n
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
BabyDragon n
    snapply Build_BabyDragon; intro k; cbn.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+2
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal $-> ?Goal0@{k:=fin_incl k}
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal $->
?Goal0@{k:=match k with
           | inl i' => inl (fsucc i')
           | inr tt => inr tt
           end}
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal0@{k:=fin_incl k} $-> ?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal0@{k:=match k with
           | inl i' => inl (fsucc i')
           | inr tt => inr tt
           end} $->
?Goal1
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
?Goal4 $o ?Goal2 $== ?Goal5 $o ?Goal3
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact ?Goal3@{k:=fin_incl k} ?Goal4@{k:=fsucc k}
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact ?Goal2@{k:=fsucc k} ?Goal5@{k:=fin_incl k}
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A exact (top D (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+2
A exact (mid D (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
A exact (bot D (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
top D (fsucc k) $-> mid D (fsucc (fin_incl k)) exact (top_l _ (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
top D (fsucc k) $->
mid D (fsucc match k with
             | inl i' => inl (fsucc i')
             | inr tt => inr tt
             end) exact (top_r _ (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
mid D (fsucc (fin_incl k)) $-> bot D (fsucc k) exact (bot_l _ (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
mid D (fsucc match k with
             | inl i' => inl (fsucc i')
             | inr tt => inr tt
             end) $->
bot D (fsucc k) exact (bot_r _ (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n.+1
bot_l D (fsucc k) $o top_l D (fsucc k) $==
bot_r D (fsucc k) $o top_r D (fsucc k) exact (sq _ (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact (top_r D (fsucc (fin_incl k))) (bot_l D (fsucc (fsucc k))) exact (exact_r _ (fsucc k)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
k: Fin n
CatIsAbExact (top_l D (fsucc (fsucc k))) (bot_r D (fsucc (fin_incl k))) exact (exact_l _ (fsucc k)).
  Defined.

  (** This is a helper lemma that can be proved by induction.  The epi part of the dragon lemma follows immediately. *)
  Local Instance baby_dragon_aux_left (n : nat) (D : BabyDragon n)
    `{!forall k : Fin (n.+1), Epic (bot_l D k)} `{!Epic (top_l D fin_zero)}
    : Epic (bot_r D fin_last $o top_r D fin_last).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n
H1: forall k : Fin n.+1, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
Epic (bot_r D fin_last $o top_r D fin_last)
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n
H1: forall k : Fin n.+1, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
Epic (bot_r D fin_last $o top_r D fin_last)
    induction n.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
D: BabyDragon 0
H1: forall k : Fin 1, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
Epic (bot_r D fin_last $o top_r D fin_last)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
Epic (bot_r D fin_last $o top_r D fin_last)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
D: BabyDragon 0
H1: forall k : Fin 1, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
Epic (bot_r D fin_last $o top_r D fin_last) rapply (epic_homotopic _ _ (sq _ fin_zero)).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
Epic (bot_r D fin_last $o top_r D fin_last) apply epic_elt_lift; intros P b_last.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
Lift b_last (bot_r D fin_last $o top_r D fin_last)

      elt_lift_epic (bot_l D fin_last) b_last m_last lm_last.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
Lift b_last (bot_r D fin_last $o top_r D fin_last)

      pose (D' := baby_dragon_trunc_left n D).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
D':= baby_dragon_trunc_left n D: BabyDragon n
Lift b_last (bot_r D fin_last $o top_r D fin_last)
      elt_lift_epic (bot_r D' fin_last $o top_r D' fin_last)
                    (bot_r D' fin_last $o m_last) t_penult lt_penult.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
D':= baby_dragon_trunc_left n D: BabyDragon n
t_penult: elt P (top D' fin_last)
lt_penult: bot_r D' fin_last $o top_r D' fin_last $o t_penult $==
bot_r D' fin_last $o m_last
Lift b_last (bot_r D fin_last $o top_r D fin_last)
      unfold D' in t_penult, lt_penult; clear D'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_penult: bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult $==
bot_r (baby_dragon_trunc_left n D) fin_last $o m_last
Lift b_last (bot_r D fin_last $o top_r D fin_last)

      elt_lift_exact (top_l D (fsucc fin_last)) (bot_r D (fin_incl fin_last))
        (m_last - top_r D (fin_incl fin_last) $o t_penult) t_last lt_last
        using clear IHn lt_penult.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_penult: bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult $==
bot_r (baby_dragon_trunc_left n D) fin_last $o m_last
bot_r D (fin_incl fin_last) $o
(m_last - top_r D (fin_incl fin_last) $o t_penult) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
Lift b_last (bot_r D fin_last $o top_r D fin_last)
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_penult: bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult $==
bot_r (baby_dragon_trunc_left n D) fin_last $o m_last
bot_r D (fin_incl fin_last) $o
(m_last - top_r D (fin_incl fin_last) $o t_penult) $== 0 lhs' napply postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_penult: bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult $==
bot_r (baby_dragon_trunc_left n D) fin_last $o m_last
bot_r D (fin_incl fin_last) $o m_last -
bot_r D (fin_incl fin_last) $o (top_r D (fin_incl fin_last) $o t_penult) $==
0
        rewrite <- cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_penult: bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult $==
bot_r (baby_dragon_trunc_left n D) fin_last $o m_last
bot_r D (fin_incl fin_last) $o m_last -
bot_r D (fin_incl fin_last) $o top_r D (fin_incl fin_last) $o t_penult $== 0
        rewrite <- lt_penult.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Epic (bot_l D0 k)) ->
Epic (top_l D0 fin_zero) -> Epic (bot_r D0 fin_last $o top_r D0 fin_last)
P: A
b_last: elt P (bot D fin_last)
m_last: elt P (mid D (fin_incl fin_last))
lm_last: bot_l D fin_last $o m_last $== b_last
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_penult: bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult $==
bot_r (baby_dragon_trunc_left n D) fin_last $o m_last
bot_r (baby_dragon_trunc_left n D) fin_last $o
top_r (baby_dragon_trunc_left n D) fin_last $o t_penult -
bot_r D (fin_incl fin_last) $o top_r D (fin_incl fin_last) $o t_penult $== 0
        apply inverse_r_0gpd. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
Lift b_last (bot_r D fin_last $o top_r D fin_last)

      provide_lift t_last.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_r D fin_last $o top_r D fin_last $o t_last $== b_last
      lhs_V' napply (sq D fin_last $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o top_l D fin_last $o t_last $== b_last
      lhs' napply cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o (top_l D fin_last $o t_last) $== b_last
      lhs' napply (_ $@L lt_last).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o (m_last - top_r D (fin_incl fin_last) $o t_penult) $==
b_last
      rewrite postcomp_op_inv.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o m_last -
bot_l D fin_last $o (top_r D (fin_incl fin_last) $o t_penult) $== b_last
      rewrite <- cat_assoc.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o m_last -
bot_l D fin_last $o top_r D (fin_incl fin_last) $o t_penult $== b_last
      rewrite isabcomplex.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o m_last - 0 $o t_penult $== b_last
      rewrite precomp_zero.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o m_last - 0 $== b_last
      rewrite mon_unit_l_inv_0gpd.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Epic (bot_l D k)
Epic0: Epic (top_l D fin_zero)
P: A
m_last: elt P (mid D (fin_incl fin_last))
b_last: elt P (bot D fin_last)
lm_last: bot_l D fin_last $o m_last $== b_last
t_last: elt P (top D (fsucc fin_last))
t_penult: elt P (top (baby_dragon_trunc_left n D) fin_last)
lt_last: top_l D (fsucc fin_last) $o t_last $==
m_last - top_r D (fin_incl fin_last) $o t_penult
bot_l D fin_last $o m_last $== b_last
      exact lm_last.
  Defined.

  (** The epi form of the baby dragon lemma states that the map [top fin_last $-> middle fin_last] is epic. *)
  Definition baby_dragon_epi (n : nat) (D : BabyDragon n)
    `{!forall k : Fin (n.+1), Epic (bot_l D k)}
    `{!Epic (top_l D fin_zero)} `{!Monic (bot_r D fin_last)}
    : Epic (top_r D fin_last)
    := epic_cancel_monic _ _.

  (** This is a helper lemma that can be proved by induction.  The mono part of the dragon lemma follows immediately. *)
  Local Instance baby_dragon_aux_right (n : nat) (D : BabyDragon n)
    `{!forall k : Fin (n.+1), Monic (top_r D k)}
    `{!Monic (bot_r D fin_last)}
    : Monic (bot_l D fin_zero $o top_l D fin_zero).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n
H1: forall k : Fin n.+1, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
Monic (bot_l D fin_zero $o top_l D fin_zero)
  Proof.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n
H1: forall k : Fin n.+1, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
Monic (bot_l D fin_zero $o top_l D fin_zero)
    induction n.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
D: BabyDragon 0
H1: forall k : Fin 1, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
Monic (bot_l D fin_zero $o top_l D fin_zero)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
Monic (bot_l D fin_zero $o top_l D fin_zero)
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
D: BabyDragon 0
H1: forall k : Fin 1, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
Monic (bot_l D fin_zero $o top_l D fin_zero) rapply (monic_homotopic _ _ (sq _ _)^$).
    -
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
Monic (bot_l D fin_zero $o top_l D fin_zero) apply monic_via_elt_isemb_ab; intros P t zero_t.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t: elt P (top D fin_zero)
zero_t: bot_l D fin_zero $o top_l D fin_zero $o t $== 0
t $== 0

      elt_lift_exact (top_l D (fsucc fin_zero)) (bot_r D fin_zero)
        (top_r D fin_zero $o t) t' lift_t'
        using clear zero_t.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t: elt P (top D fin_zero)
zero_t: bot_l D fin_zero $o top_l D fin_zero $o t $== 0
bot_r D fin_zero $o (top_r D fin_zero $o t) $== 0
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
t $== 0
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t: elt P (top D fin_zero)
zero_t: bot_l D fin_zero $o top_l D fin_zero $o t $== 0
bot_r D fin_zero $o (top_r D fin_zero $o t) $== 0 lhs' napply cat_assoc_opp.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t: elt P (top D fin_zero)
zero_t: bot_l D fin_zero $o top_l D fin_zero $o t $== 0
bot_r D fin_zero $o top_r D fin_zero $o t $== 0
        lhs_V' napply (sq D fin_zero $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t: elt P (top D fin_zero)
zero_t: bot_l D fin_zero $o top_l D fin_zero $o t $== 0
bot_l D fin_zero $o top_l D fin_zero $o t $== 0
        exact zero_t. }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
t $== 0

      rapply (ismonic' (top_r D fin_zero)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
top_r D fin_zero $o t $== 0
      lhs_V' napply lift_t'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
top_l D (fsucc fin_zero) $o t' $== 0
      rhs_V' napply (postcomp_zero (top_l D (fsucc fin_zero))).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
top_l D (fsucc fin_zero) $o t' $== top_l D (fsucc fin_zero) $o 0
      apply cat_postwhisker.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
t' $== 0
      set (D' := baby_dragon_trunc_right n D).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
D':= baby_dragon_trunc_right n D: BabyDragon n
t' $== 0
      rapply (ismonic' (bot_r D' fin_zero $o top_r D' fin_zero)).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
D':= baby_dragon_trunc_right n D: BabyDragon n
Monic (bot_r D' fin_zero $o top_r D' fin_zero)
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
D':= baby_dragon_trunc_right n D: BabyDragon n
bot_r D' fin_zero $o top_r D' fin_zero $o t' $== 0
      {
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
D':= baby_dragon_trunc_right n D: BabyDragon n
Monic (bot_r D' fin_zero $o top_r D' fin_zero) rapply (monic_homotopic _ _ (sq _ _)). }
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
D':= baby_dragon_trunc_right n D: BabyDragon n
bot_r D' fin_zero $o top_r D' fin_zero $o t' $== 0
      lhs_V' napply (sq D' fin_zero $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
D':= baby_dragon_trunc_right n D: BabyDragon n
bot_l D' fin_zero $o top_l D' fin_zero $o t' $== 0
      unfold D'; clear D'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
bot_l (baby_dragon_trunc_right n D) fin_zero $o
top_l (baby_dragon_trunc_right n D) fin_zero $o t' $== 0
      rewrite_with_assoc_opp lift_t'.
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
bot_l (baby_dragon_trunc_right n D) fin_zero $o top_r D fin_zero $o t $== 0
      lhs' apply (isabcomplex $@R _).
A: Type
IsGraph0: IsGraph A
Is2Graph0: Is2Graph A
Is01Cat0: Is01Cat A
H: Is1Cat A
H0: IsAbEpiStable A
n: nat
D: BabyDragon n.+1
H1: forall k : Fin n.+2, Monic (top_r D k)
Monic0: Monic (bot_r D fin_last)
IHn: forall D0 : BabyDragon n,
(forall k : Fin n.+1, Monic (top_r D0 k)) ->
Monic (bot_r D0 fin_last) -> Monic (bot_l D0 fin_zero $o top_l D0 fin_zero)
P: A
t': elt P (top D (fsucc fin_zero))
t: elt P (top D fin_zero)
lift_t': top_l D (fsucc fin_zero) $o t' $== top_r D fin_zero $o t
0 $o t $== 0
      apply precomp_zero.
  Defined.

  (** The mono form of the baby dragon lemma states that the map [middle fin_zero $-> bot fin_zero] is monic. *)
  Definition baby_dragon_mono (n : nat) (D : BabyDragon n)
    `{!forall k : Fin (n.+1), Monic (top_r D k)}
    `{!Monic (bot_r D fin_last)} `{!Epic (top_l D fin_zero)}
    : Monic (bot_l D fin_zero)
    := monic_cancel_epic _ _.

End Baby_Dragon.