(** ** 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.

(* Specifications: *)
Lemma exec_par_outside: forall m e r, ~In r (dests m) -> exec_par m e r = e r.
Lemma exec_par_lift: forall m1 s d m2 e, ~In d (dests m1) -> exec_par (m1 ++ (s, d) :: m2) e = exec_par ((s, d) :: m1 ++ m2) e.
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.
Lemma exec_seq_app: forall m1 m2 e, exec_seq (m1 ++ m2) e = exec_seq m2 (exec_seq m1 e).
Lemma exec_seq_rev_app: forall m1 m2 e, exec_seq_rev (m1 ++ m2) e = exec_seq_rev m1 (exec_seq_rev m2 e).
Lemma exec_seq_exec_seq_rev: forall m e, exec_seq_rev m e = exec_seq (List.rev m) e.
Lemma exec_seq_rev_exec_seq: forall m e, exec_seq m e = exec_seq_rev (List.rev m) e.
Lemma exec_par_update_no_read: forall s d m e, no_read m d -> ~In d (dests m) -> exec_par m (update d (e s) e) = update d (e s) (exec_par m e).
Lemma exec_par_append_eq: forall m1 m2 m3 e2 e3, exec_par m2 e2 = exec_par m3 e3 -> (forall r, In r (srcs m1) -> e2 r = e3 r) -> exec_par (m1 ++ m2) e2 = exec_par (m1 ++ m3) e3.
Lemma exec_par_combine: forall e sl dl, List.length sl = List.length dl -> list_norepet dl -> let e' := exec_par (combine sl dl) e in List.map e' dl = List.map e sl /\ (forall l, ~In l dl -> e' l = e l).
Lemma exec_par_outside: forall m e r, ~In r (dests m) -> exec_par m e r = e r.
Lemma exec_par_lift: forall m1 s d m2 e, ~In d (dests m1) -> exec_par (m1 ++ (s, d) :: m2) e = exec_par ((s, d) :: m1 ++ m2) e.
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.
Lemma exec_seq_app: forall m1 m2 e, exec_seq (m1 ++ m2) e = exec_seq m2 (exec_seq m1 e).
Lemma exec_seq_rev_app: forall m1 m2 e, exec_seq_rev (m1 ++ m2) e = exec_seq_rev m1 (exec_seq_rev m2 e).
Lemma exec_seq_exec_seq_rev: forall m e, exec_seq_rev m e = exec_seq (List.rev m) e.
Lemma exec_seq_rev_exec_seq: forall m e, exec_seq m e = exec_seq_rev (List.rev m) e.
Lemma exec_par_update_no_read: forall s d m e, no_read m d -> ~In d (dests m) -> exec_par m (update d (e s) e) = update d (e s) (exec_par m e).
Lemma exec_par_append_eq: forall m1 m2 m3 e2 e3, exec_par m2 e2 = exec_par m3 e3 -> (forall r, In r (srcs m1) -> e2 r = e3 r) -> exec_par (m1 ++ m2) e2 = exec_par (m1 ++ m3) e3.
Lemma exec_par_combine: forall e sl dl, List.length sl = List.length dl -> list_norepet dl -> let e' := exec_par (combine sl dl) e in List.map e' dl = List.map e sl /\ (forall l, ~In l dl -> e' l = e l).