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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                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                                 gram_g.v                                 *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)

Require Import Ensf.
Require Import Words.
Require Import fonctions.
Require Import Relations.
Require Import gram.

Inductive Deriveg (X R : Ensf) : Word -> Word -> Prop :=
  | Deriveg1 :
      forall (u v : Word) (A : Elt),
      dans (couple A (word u)) R -> Deriveg X R (cons A v) (Append u v)
  | Deriveg2 :
      forall (u v : Word) (x : Elt),
      dans x X -> Deriveg X R u v -> Deriveg X R (cons x u) (cons x v).

Hint Resolve Deriveg1.
Hint Resolve Deriveg2.


Definition Derivegstar (X R : Ensf) := Rstar Word (Deriveg X R).

Lemma Deriveg_Derive :
 forall (X R : Ensf) (u v : Word), Deriveg X R u v -> Derive R u v.
(* Goal: forall (X R : Ensf) (u v : Word) (_ : Deriveg X R u v), Derive R u v *)
intros X R u v Der_g.
(* Goal: Derive R u v *)
elim Der_g.
(* Goal: forall (u v : Word) (x : Elt) (_ : dans x X) (_ : Deriveg X R u v) (_ : Derive R u v), Derive R (cons x u) (cons x v) *)
intros.
(* Goal: Derive R (cons A v0) (Append u0 v0) *)
(* Goal: forall (u v : Word) (x : Elt) (_ : dans x X) (_ : Deriveg X R u v) (_ : Derive R u v), Derive R (cons x u) (cons x v) *)
apply Derive1; auto.
(* Goal: forall (u v : Word) (x : Elt) (_ : dans x X) (_ : Deriveg X R u v) (_ : Derive R u v), Derive R (cons x u) (cons x v) *)
intros.
(* Goal: Derive R (cons x u0) (cons x v0) *)
apply Derive2; auto.
Qed.

Lemma Derivegstar_Derivestar :
 forall (X R : Ensf) (u v : Word), Derivegstar X R u v -> Derivestar R u v.
(* Goal: forall (X R : Ensf) (u v : Word) (_ : Derivegstar X R u v), Derivestar R u v *)
unfold Derivegstar, Rstar, Derivestar in |- *.
(* Goal: forall (X R : Ensf) (u v : Word) (_ : forall (P : forall (_ : Word) (_ : Word), Prop) (_ : forall u0 : Word, P u0 u0) (_ : forall (u0 v0 w : Word) (_ : Deriveg X R u0 v0) (_ : P v0 w), P u0 w), P u v), Rstar Word (Derive R) u v *)
intros X R x y Derivegstar_x_y.
(* Goal: Rstar Word (Derive R) x y *)
pattern x, y in |- *.
(* Goal: (fun w w0 : Word => Rstar Word (Derive R) w w0) x y *)
apply Derivegstar_x_y.
(* Goal: forall u : Word, Rstar Word (Derive R) u u *)
(* Goal: forall (u v w : Word) (_ : Deriveg X R u v) (_ : Rstar Word (Derive R) v w), Rstar Word (Derive R) u w *)
intro. apply Rstar_reflexive.
(* Goal: forall (u v w : Word) (_ : Deriveg X R u v) (_ : Rstar Word (Derive R) v w), Rstar Word (Derive R) u w *)
intros u v w Der Der_star.
(* Goal: Rstar Word (Derive R) u w *)
apply Rstar_R with v.
(* Goal: Derive R u v *)
(* Goal: Rstar Word (Derive R) v w *)
apply Deriveg_Derive with X; assumption.
(* Goal: Rstar Word (Derive R) v w *)
assumption.
Qed.

 


Axiom
  Derivestar_Derivegstar :
    forall (X R : Ensf) (u v : Word), Derivestar R u v -> Derivegstar X R u v.

Hint Resolve Derivestar_Derivegstar.