(* Copyright (C) 2012  G. Gonthier, B. Ziliani, A. Nanevski, D. Dreyer  This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.  This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.  You should have received a copy of the GNU General Public License along with this program.  If not, see <http://www.gnu.org/licenses/>. *)
From mathcomp
From LemmaOverloading
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Delimit Scope stsep_scope with stsep.
Open Scope stsep_scope.
Definition lolli (p : _ -> Prop) q i m :=
forall i1 h, i = i1 :+ h -> def i -> p i1 ->
exists m1, m = m1 :+ h /\ def m /\ q i1 m1.
Notation "p '--o' q"   := (lolli p q) (at level 75) : stsep_scope.

Lemma fr_pre p i j : (p # top) i -> (p # top) (i :+ j).
Definition fr A (s : spec A) : spec A :=
(s.1 # top, fun x => s.1 --o s.2 x).