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Words.v
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Words.v
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(* 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) *)
(****************************************************************************)
(* 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 *)
Require Import Bool.
(*****************)
(* 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.
(* Relating the two definitions; unused below *)
Lemma conc_append : forall u v w : word, conc u v w -> w = append u v.
Proof.
simple induction 1; simpl in |- *; trivial.
simple induction 2; trivial.
Qed.
(* Associativity of append; unused below *)
Lemma assoc_append :
forall u v w : word, append u (append v w) = append (append u v) w.
Proof.
simple induction u; simpl in |- *; intros; auto.
rewrite H; trivial.
Qed.
(**************)
(* Singletons *)
(**************)
Definition single (b : bool) := bit b empty.
(*********************)
(* Parities of words *)
(*********************)
Inductive odd : word -> Prop :=
even_odd : forall w : word, even w -> forall b : bool, odd (bit b w)
with even : word -> Prop :=
| even_empty : even empty
| odd_even : forall w : word, odd w -> forall b : bool, even (bit b w).
Hint Resolve odd_even even_empty even_odd.
Lemma not_odd_empty : ~ odd empty.
Proof.
unfold not in |- *; intro od.
inversion od.
Qed.
Hint Resolve not_odd_empty.
Lemma inv_odd : forall (w : word) (b : bool), odd (bit b w) -> even w.
Proof.
intros w b od.
inversion od; trivial.
Qed.
Lemma inv_even : forall (w : word) (b : bool), even (bit b w) -> odd w.
Proof.
intros w b ev.
inversion ev; trivial.
Qed.
(**********************)
(* (odd w) + (even w) *)
(**********************)
Lemma odd_or_even : forall w : word, odd w \/ even w.
Proof.
simple induction w; auto.
simple induction 1; intros.
right; auto.
left; auto.
Qed.
Lemma not_odd_and_even : forall w : word, odd w -> even w -> False.
Proof.
simple induction w; intros.
elim not_odd_empty; trivial.
apply H.
apply inv_even with b; trivial.
apply inv_odd with b; trivial.
Qed.
(************************)
(* Parities of subwords *)
(************************)
Lemma odd_even_conc :
forall u v w : word,
conc u v w ->
odd w /\ (odd u /\ even v \/ even u /\ odd v) \/
even w /\ (odd u /\ odd v \/ even u /\ even v).
Proof.
simple induction 1; intros.
elim (odd_or_even v0); auto.
elim H1; [ right | left ]; intuition.
Qed.
Lemma even_conc :
forall u v w : word,
conc u v w -> even w -> odd u /\ odd v \/ even u /\ even v.
Proof.
intros u v w c e; elim odd_even_conc with u v w; intros.
elim H; intro o; elim not_odd_and_even with w; auto.
elim H; intros; trivial.
trivial.
Qed.