(* invariant de reconnaissance sur conv_0 : sens reciproque *)

Definition u_conv_rec_0_r (p0 : preDTA) (a0 : ad) (t : term)
  (pr0 : reconnaissance p0 a0 t) :=
  forall (p : preDTA) (a : ad),
  p0 = udta_conv_0 p -> a0 = uad_conv_0 a -> reconnaissance p a t.Definition u_conv_str_0_r (p0 : preDTA) (s0 : state) 
  (t : term) (pr : state_reconnait p0 s0 t) :=
  forall (p : preDTA) (s : state),
  p0 = udta_conv_0 p -> s0 = umpl_conv_0 s -> state_reconnait p s t.Definition u_conv_lr_0_r (p0 : preDTA) (pl0 : prec_list) 
  (t : term_list) (pr : liste_reconnait p0 pl0 t) :=
  forall (p : preDTA) (pl : prec_list),
  p0 = udta_conv_0 p -> pl0 = upl_conv_0 pl -> liste_reconnait p pl t.

(* Specifications: *)
Lemma u_conv0_0r : forall (d : preDTA) (a : ad) (t : term) (ladj : state) (e : MapGet state d a = Some ladj) (s : state_reconnait d ladj t), u_conv_str_0_r d ladj t s -> u_conv_rec_0_r d a t (rec_dta d a t ladj e s).
Lemma u_conv0_1r : forall (d : preDTA) (s : state) (c : ad) (tl : term_list) (l : prec_list) (e : MapGet prec_list s c = Some l) (l0 : liste_reconnait d l tl), u_conv_lr_0_r d l tl l0 -> u_conv_str_0_r d s (app c tl) (rec_st d s c tl l e l0).
Lemma u_conv0_2r : forall d : preDTA, u_conv_lr_0_r d prec_empty tnil (rec_empty d).
Lemma u_conv0_3r : forall (d : preDTA) (a : ad) (la ls : prec_list) (hd : term) (tl : term_list) (r : reconnaissance d a hd), u_conv_rec_0_r d a hd r -> forall l : liste_reconnait d la tl, u_conv_lr_0_r d la tl l -> u_conv_lr_0_r d (prec_cons a la ls) (tcons hd tl) (rec_consi d a la ls hd tl r l).
Lemma u_conv0_4r : forall (d : preDTA) (a : ad) (la ls : prec_list) (hd : term) (tl : term_list) (l : liste_reconnait d ls (tcons hd tl)), u_conv_lr_0_r d ls (tcons hd tl) l -> u_conv_lr_0_r d (prec_cons a la ls) (tcons hd tl) (rec_consn d a la ls hd tl l).
Lemma u_conv0_5r : forall (p : preDTA) (a : ad) (t : term) (r : reconnaissance p a t), u_conv_rec_0_r p a t r.
Lemma u_conv0_r : forall (p : preDTA) (a : ad) (t : term), reconnaissance (udta_conv_0 p) (uad_conv_0 a) t -> reconnaissance p a t.
Lemma u_conv_1_invar_0 : forall (d : preDTA) (a : ad) (ladj : state), MapGet state d a = Some ladj -> MapGet state (udta_conv_1 d) (uad_conv_1 a) = Some (umpl_conv_1 ladj).
Lemma u_conv_1_invar_1 : forall (s : state) (c : ad) (p : prec_list), MapGet prec_list s c = Some p -> MapGet prec_list (umpl_conv_1 s) c = Some (upl_conv_1 p).
Lemma u_conv_1_invar_2 : forall (d : preDTA) (a : ad) (ladj : state), MapGet state (udta_conv_1 d) (uad_conv_1 a) = Some (umpl_conv_1 ladj) -> MapGet state d a = Some ladj.
Lemma u_conv_1_invar_3 : forall (s : state) (c : ad) (p : prec_list), MapGet prec_list (umpl_conv_1 s) c = Some (upl_conv_1 p) -> MapGet prec_list s c = Some p.
Lemma u_conv_1_invar_4 : forall (d : preDTA) (a : ad) (ladj : state), MapGet state (udta_conv_1 d) (uad_conv_1 a) = Some ladj -> exists ladj0 : _, ladj = umpl_conv_1 ladj0.
Lemma u_conv_1_invar_5 : forall (d : preDTA) (a : ad) (ladj : state), MapGet state (udta_conv_1 d) (uad_conv_1 a) = Some ladj -> exists ladj0 : _, ladj = umpl_conv_1 ladj0 /\ MapGet state d a = Some ladj0.
Lemma u_conv_1_invar_6 : forall (s : state) (c : ad) (p : prec_list), MapGet prec_list (umpl_conv_1 s) c = Some p -> exists p0 : prec_list, p = upl_conv_1 p0.
Lemma u_conv_1_invar_7 : forall (s : state) (c : ad) (p : prec_list), MapGet prec_list (umpl_conv_1 s) c = Some p -> exists p0 : prec_list, p = upl_conv_1 p0 /\ MapGet prec_list s c = Some p0.
Lemma u_conv_1_invar_8 : forall (p0 : preDTA) (a0 : ad) (s0 : state), MapGet state (udta_conv_1 p0) a0 = Some s0 -> exists a1 : ad, a0 = uad_conv_1 a1.