Definition MATCH (d:CL_core) mu c1 m1 c2 m2:Prop :=
  match_states (restrict (as_inj mu) (vis mu)) c1 m1 c2 m2 /\
  REACH_closed m1 (vis mu) /\
  meminj_preserves_globals ge (as_inj mu) /\
  globalfunction_ptr_inject (as_inj mu) /\
  (forall b, isGlobalBlock ge b = true -> frgnBlocksSrc mu b = true) /\
  sm_valid mu m1 m2 /\ SM_wd mu /\ Mem.inject (as_inj mu) m1 m2.

Lemma MATCH_restrict: forall d mu c1 m1 c2 m2 X (MC: MATCH d mu c1 m1 c2 m2) (HX: forall b : block, vis mu b = true -> X b = true) (RX: REACH_closed m1 X), MATCH d (restrict_sm mu X) c1 m1 c2 m2.