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Definition coord_msg tag (from to : nid) : bool :=
[&& from == cn, to \in pts & tagFromCoordinator tag].
Definition participant_msg tag (from to : nid) : bool :=
[&& from \in pts, to == cn & tagFromParticipant tag].
Definition internal_msg (tag : nat) (from to : nid) : bool :=
coord_msg tag from to || participant_msg tag from to.

Lemma internal_msg_tagFromCoordinator_to_pts tag from to :
internal_msg tag from to ->
tagFromCoordinator tag ->
to \in pts.