(* 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.
(* linked lists, storing a value and next pointer in consecutive locations *)
Definition llist (T : Type) := ptr.
Section LList.
Variable T : Type.
Notation llist := (llist T).
Fixpoint lseg (p q : ptr) (xs : seq T) {struct xs} :=
if xs is x::xt then
[Pred h | exists r, exists h',
h = p :-> x :+ (p .+ 1 :-> r :+ h') /\ h' \In lseg r q xt]
else [Pred h | p = q /\ h = empty].

(* Specifications: *)
Lemma lseg_add_last xs x p r h :
h \In lseg p r (rcons xs x) <->
exists q, exists h',
h = h' :+ (q :-> x :+ q .+ 1 :-> r) /\ h' \In lseg p q xs.
Lemma lseg_null xs q h :
def h -> h \In lseg null q xs ->
[/\ q = null, xs = [::] & h = empty].
Lemma lseg_neq xs p q h :
p != q -> h \In lseg p q xs ->
exists x, exists r, exists h',
[/\ xs = x :: behead xs,
p :-> x :+ (p .+ 1 :-> r :+ h') = h & h' \In lseg r q (behead xs)].
Lemma lseg_empty xs p q : empty \In lseg p q xs -> p = q /\ xs = [::].
Lemma lseg_case xs p q h :
h \In lseg p q xs ->
[/\ p = q, xs = [::] & h = empty] \/
exists x, exists r, exists h',
[/\ xs = x :: behead xs, h = p :-> x :+ (p .+ 1 :-> r :+ h') &
h' \In lseg r q (behead xs)].