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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                                                                          *)
(****************************************************************************)
(*                                 Words.v                                  *)
(****************************************************************************)
(* G. Huet - V5.8  Nov. 1994 *)
(*    ported V5.10 June 1995 *)
(*****************)
(* Boolean words *)
(*****************)
Inductive word : Set :=
| empty : word
| bit : bool -> word -> word.
(* Remark : word ~ bool list *)
(* word concatenation : logical definition *)
Inductive conc : word -> word -> word -> Prop :=
| conc_empty : forall v : word, conc empty v v
| conc_bit :
forall (u v w : word) (b : bool),
conc u v w -> conc (bit b u) v (bit b w).
(* word concatenation : functional definition *)
Fixpoint append (u : word) : word -> word :=
fun v : word =>
match u with
| empty => v
| bit b w => bit b (append w v)
end.

(* Specifications: *)
(* Relating the two definitions; unused below *)
Lemma conc_append : forall u v w : word, conc u v w -> w = append u v.
(* Associativity of append; unused below *)
Lemma assoc_append :
forall u v w : word, append u (append v w) = append (append u v) w.