End pulb_setoid_def.
(* composition *)
Definition Comp_PULB_mor (a b c : PULB_ob) (f : PULB_mor_setoid a b) :=
match
f in (PULB_mor p p0) return (PULB_mor_setoid p0 c -> PULB_mor_setoid p c)
with
| IP1 =>
(* IP1 *)
match c as p return (PULB_mor P1 p -> PULB_mor P1 p) with
| P1 =>
(* P1 *)  fun _ => IP1 (* P2 *)
| P2 => fun _ => F12
(* P3 *)
| P3 =>
fun g => PULB_13_rect (fun _ : PULB_mor P1 P3 => PULB_mor P1 P3) g
end
(* IP1 *)
| IP2 =>
match c as p return (PULB_mor P2 p -> PULB_mor P2 p) with
| P1 =>
(* P1 *)
fun g =>
PULB_21_rect (fun _ : PULB_mor P2 P1 => PULB_mor P2 P1) g
(* P2 *)
| P2 => fun _ => IP2
(* P3 *)
| P3 =>
fun g => PULB_23_rect (fun _ : PULB_mor P2 P3 => PULB_mor P2 P3) g
end
(* IP3 *)
| IP3 =>
match c as p return (PULB_mor P3 p -> PULB_mor P3 p) with
| P1 =>
(* P1 *)
fun g =>
PULB_31_rect (fun _ : PULB_mor P3 P1 => PULB_mor P3 P1) g
(* P2 *)
| P2 => fun _ => F32
(* P3 *)
| P3 => fun _ => IP3
end
(* F12  *)
| F12 =>
match c as p return (PULB_mor P2 p -> PULB_mor P1 p) with
| P1 =>
(* P1 *)
fun g =>
PULB_21_rect (fun _ : PULB_mor P2 P1 => PULB_mor P1 P1) g
(* P2 *)
| P2 => fun _ => F12
(* P3 *)
| P3 =>
fun g => PULB_23_rect (fun _ : PULB_mor P2 P3 => PULB_mor P1 P3) g
end
(* F32  *)
| F32 =>
match c as p return (PULB_mor P2 p -> PULB_mor P3 p) with
| P1 =>
(* P1 *)
fun g =>
PULB_21_rect (fun _ : PULB_mor P2 P1 => PULB_mor P3 P1) g
(* P2 *)
| P2 => fun _ => F32
(* P3 *)
| P3 =>
fun g => PULB_23_rect (fun _ : PULB_mor P2 P3 => PULB_mor P3 P3) g
end
end.

Lemma Assoc_PULB : Assoc_law Comp_PULB.
Definition Id_PULB (a : PULB_ob) :=
match a as a' return (PULB_mor a' a') with
| P1 => (* P1 *)  IP1
| P2 => IP2
| P3 => IP3
end.