Definition predta_produit_4_def (d0 : preDTA) : Prop :=
  forall (a a0 : ad) (s s0 : state),
  MapGet state (preDTA_produit_r a0 s0 d0) a = Some s ->
  exists a1 : ad,
    (exists a2 : ad,
       (exists s1 : state,
          (exists s2 : state,
             a = iad_conv a1 a2 /\
             MapGet state (M1 state a0 s0) a2 = Some s1 /\
             MapGet state d0 a1 = Some s2))).

Lemma pl_produit_rec_2 : forall tl : term_list, (forall u : term, term_list_occur u tl -> forall (d0 d1 : preDTA) (a0 a1 : ad), predta_compatible d0 d1 -> reconnaissance (preDTA_produit d0 d1) (iad_conv a0 a1) u -> reconnaissance d0 a0 u /\ reconnaissance d1 a1 u) -> forall (d0 d1 : preDTA) (plp : pl_path), predta_compatible d0 d1 -> pl_path_recon (preDTA_produit d0 d1) tl plp -> exists plp0 : pl_path, (exists plp1 : pl_path, plp = pl_path_product plp0 plp1 /\ pl_path_recon d0 tl plp0 /\ pl_path_recon d1 tl plp1).