(** ** Transitions preserve semantics *)
(** We define the semantics of states as follows. For a triple [mu, sigma, tau], we perform the moves [tau] in reverse sequential order, then the moves [mu ++ sigma] in parallel. *)
Definition statemove (st: state) (e: env) :=
match st with
| State mu sigma tau => exec_par (mu ++ sigma) (exec_seq_rev tau e)
end.
(** In preparation to showing that transitions preserve semantics, we prove various properties of the parallel and sequential interpretations of moves. *)

Lemma exec_par_ident:
forall m1 r m2 e,
is_mill (m1 ++ (r, r) :: m2) ->
exec_par (m1 ++ (r, r) :: m2) e = exec_par (m1 ++ m2) e.