(* calcul des automates modifiés *)
Fixpoint upl_conv_0 (p : prec_list) : prec_list :=
match p with
| prec_empty => prec_empty
| prec_cons a p0 p1 =>
prec_cons (uad_conv_0 a) (upl_conv_0 p0) (upl_conv_0 p1)
end.
Fixpoint upl_conv_1 (p : prec_list) : prec_list :=
match p with
| prec_empty => prec_empty
| prec_cons a p0 p1 =>
prec_cons (uad_conv_1 a) (upl_conv_1 p0) (upl_conv_1 p1)
end.
Fixpoint umpl_conv_0 (s : state) : state :=
match s with
| M0 => M0 prec_list
| M1 a p => M1 prec_list a (upl_conv_0 p)
| M2 p0 p1 => M2 prec_list (umpl_conv_0 p0) (umpl_conv_0 p1)
end.
Fixpoint umpl_conv_1 (s : state) : state :=
match s with
| M0 => M0 prec_list
| M1 a p => M1 prec_list a (upl_conv_1 p)
| M2 p0 p1 => M2 prec_list (umpl_conv_1 p0) (umpl_conv_1 p1)
end.
Fixpoint udta_conv_0_aux (d : preDTA) : preDTA :=
match d with
| M0 => M0 state
| M1 a s => M1 state a (umpl_conv_0 s)
| M2 s0 s1 => M2 state (udta_conv_0_aux s0) (udta_conv_0_aux s1)
end.
Fixpoint udta_conv_1_aux (d : preDTA) : preDTA :=
match d with
| M0 => M0 state
| M1 a s => M1 state a (umpl_conv_1 s)
| M2 s0 s1 => M2 state (udta_conv_1_aux s0) (udta_conv_1_aux s1)
end.
Definition udta_conv_0 (d : preDTA) : preDTA :=
M2 state (udta_conv_0_aux d) (M0 state).
Definition udta_conv_1 (d : preDTA) : preDTA :=
M2 state (M0 state) (udta_conv_1_aux d).
(* lemmes d'injectivité *)

Lemma umpl_conv_0_inj :
forall s0 s1 : state, umpl_conv_0 s0 = umpl_conv_0 s1 -> s0 = s1.