(** Truncation *)
Record shr_record := { shr_m : Z ; shr_r : bool ; shr_s : bool }.
Definition shr_1 mrs :=
let '(Build_shr_record m r s) := mrs in
let s := orb r s in
match m with
| Z0 => Build_shr_record Z0 false s
| Zpos xH => Build_shr_record Z0 true s
| Zpos (xO p) => Build_shr_record (Zpos p) false s
| Zpos (xI p) => Build_shr_record (Zpos p) true s
| Zneg xH => Build_shr_record Z0 true s
| Zneg (xO p) => Build_shr_record (Zneg p) false s
| Zneg (xI p) => Build_shr_record (Zneg p) true s
end.
Definition loc_of_shr_record mrs :=
match mrs with
| Build_shr_record _ false false => loc_Exact
| Build_shr_record _ false true => loc_Inexact Lt
| Build_shr_record _ true false => loc_Inexact Eq
| Build_shr_record _ true true => loc_Inexact Gt
end.
Definition shr_record_of_loc m l :=
match l with
| loc_Exact => Build_shr_record m false false
| loc_Inexact Lt => Build_shr_record m false true
| loc_Inexact Eq => Build_shr_record m true false
| loc_Inexact Gt => Build_shr_record m true true
end.

Theorem shr_m_shr_record_of_loc :
forall m l,
shr_m (shr_record_of_loc m l) = m.