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).