(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* 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 Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* ----                          machine.v                        ----- *)
(*                                                                          *)
(* Author: Pierre Casteran.                                                 *)
(*    LABRI, URA CNRS 1304,                                                 *)
(*    Departement d'Informatique, Universite Bordeaux I,                    *)
(*    33405 Talence CEDEX,                                                  *)
(*    e-mail:  casteran@labri.u-bordeaux.fr                                 *)
(* This file describes the implementation of our powering algorithm: *)
(*  Let us imagine an abstract machine for computing on the monoid (M,o,u) (see "monoid.v"). This machine has a register X and a stack S. *)
Inductive Instr : Set :=
| MUL : Instr
| SQR : Instr
| PUSH : Instr
| SWAP : Instr.
(* sequences of instructions *)
Inductive Code : Set :=
| End : Code
| seq : Instr -> Code -> Code.
(* code appending *)
(*Recursive Definition app:Code->Code->Code:= End c' => c' | (seq i  c) c' => (seq i ( app c  c')). *)
Fixpoint app (c : Code) : Code -> Code :=
fun c' : Code => match c with
| End => c'
| seq i c => seq i (app c c')
end.
(* semantics *)
(*************)
Section Monoid.
Variable M : Set.
Variable MO : monoid M.
Let uM := u _ MO.
Let oM := o _ MO.
Inductive Stack : Set :=
| emptystack : Stack
| push : M -> Stack -> Stack.
Definition top (s : Stack) :=
match s return M with
| emptystack => uM
| push m _ => m
end.
Definition pop (s : Stack) :=
match s return Stack with
| emptystack => emptystack
| push _ r => r
end.
(* configurations of the abstract machine *)
(******************************************)
Record Config : Set := config {config_X : M; config_S : Stack}.

(* Specifications: *)
Lemma Config_inv :
forall (a a' : M) (s s' : Stack),
a = a' -> s = s' -> config a s = config a' s'.