(** * Determinisation of the transition relation *)
(** We now define a deterministic variant [dtransition] of the transition relation [transition].  [dtransition] is deterministic in the sense that at most one transition applies to a given state. *)
Inductive dtransition: state -> state -> Prop :=
| dtr_nop: forall r mu tau,
dtransition (State ((r, r) :: mu) nil tau)
(State mu nil tau)
| dtr_start: forall s d mu tau,
s <> d ->
dtransition (State ((s, d) :: mu) nil tau)
(State mu ((s, d) :: nil) tau)
| dtr_push: forall mu1 d r mu2 s sigma tau,
no_read mu1 d ->
dtransition (State (mu1 ++ (d, r) :: mu2) ((s, d) :: sigma) tau)
(State (mu1 ++ mu2) ((d, r) :: (s, d) :: sigma) tau)
| dtr_loop_pop: forall mu s r0 d  sigma tau,
no_read mu r0 ->
dtransition (State mu ((s, r0) :: sigma ++ (r0, d) :: nil) tau)
(State mu (sigma ++ (temp r0, d) :: nil) ((s, r0) :: (r0, temp r0) :: tau))
| dtr_pop: forall mu s0 d0 s1 d1 sigma tau,
no_read mu d1 -> d1 <> s0 ->
dtransition (State mu ((s1, d1) :: sigma ++ (s0, d0) :: nil) tau)
(State mu (sigma ++ (s0, d0) :: nil) ((s1, d1) :: tau))
| dtr_last: forall mu s d tau,
no_read mu d ->
dtransition (State mu ((s, d) :: nil) tau)
(State mu nil ((s, d) :: tau)).
Definition dtransitions: state -> state -> Prop :=
clos_refl_trans state dtransition.
(** Every deterministic transition corresponds to a sequence of non-deterministic transitions. *)

(* Specifications: *)
Lemma transition_determ:
forall st st',
dtransition st st' ->
state_wf st ->
transitions st st'.
Lemma transitions_determ:
forall st st',
dtransitions st st' ->
state_wf st ->
transitions st st'.