(*LENB: We omit the clauses INJ and PG, since they are required for the enitre mu, not just restrict mu (vis mu), and hence better enforced uniformly in definition MatchCore below*)
Inductive structured_match_cores: core_data -> SM_Injection -> CSharpMin_core -> mem -> CMin_core -> mem -> Prop :=
| SMC_states:
forall d fn s k e le m tfn ts tk sp te tm cenv xenv mu lo hi cs sz
(TRF: transl_funbody cenv sz fn = OK tfn)
(TR: transl_stmt cenv xenv s = OK ts)
(*(MINJ: Mem.inject (as_inj mu) m tm)*)
(MCS: structured_match_callstack mu m tm
(Frame cenv tfn e le te sp lo hi :: cs)
(Mem.nextblock m) (Mem.nextblock tm))
(MK: match_cont k tk  cenv xenv cs)
(*(PG: meminj_preserves_globals ge (as_inj mu))*), structured_match_cores d mu (CSharpMin_State fn s k e le) m (CMin_State tfn ts tk (Vptr sp Int.zero) te) tm | SMC_state_seq: forall d fn s1 s2 k e le m tfn ts1 tk sp te tm cenv xenv mu lo hi cs sz (TRF: transl_funbody cenv sz fn = OK tfn) (TR: transl_stmt cenv xenv s1 = OK ts1) (*(MINJ: Mem.inject (as_inj mu) m tm)*)
(MCS: structured_match_callstack mu m tm
(Frame cenv tfn e le te sp lo hi :: cs)
(Mem.nextblock m) (Mem.nextblock tm))
(MK: match_cont (Csharpminor.Kseq s2 k) tk cenv xenv cs)
(*(PG: meminj_preserves_globals ge (as_inj mu))*), structured_match_cores d mu (CSharpMin_State fn (Csharpminor.Sseq s1 s2) k e le) m (CMin_State tfn ts1 tk (Vptr sp Int.zero) te) tm | SMC_callstate: forall d fd args k m tfd targs tk tm mu cs cenv (TR: transl_fundef fd = OK tfd) (*(MINJ: Mem.inject (as_inj mu) m tm)*)
(MCS: structured_match_callstack mu m tm cs (Mem.nextblock m) (Mem.nextblock tm))
(MK: match_cont k tk cenv nil cs)
(ISCC: Csharpminor.is_call_cont k)
(ARGSINJ: val_list_inject (restrict (as_inj mu) (vis mu)) args targs)
(*(PG: meminj_preserves_globals ge (as_inj mu))*),  structured_match_cores d mu (CSharpMin_Callstate fd args k) m (CMin_Callstate tfd targs tk) tm | SMC_returnstate: forall d v k m tv tk tm mu cs cenv (*(MINJ: Mem.inject (as_inj mu) m tm)*)
(MCS: structured_match_callstack mu m tm cs (Mem.nextblock m) (Mem.nextblock tm))
(MK: match_cont k tk cenv nil cs)
(RESINJ: val_inject (restrict (as_inj mu) (vis mu)) v tv)
(*(PG: meminj_preserves_globals ge (as_inj mu))*), structured_match_cores d mu (CSharpMin_Returnstate v k) m (CMin_Returnstate tv tk) tm. (* Definition protected m mu:= forall b, REACH m (extBlocksSrc mu) b = true -> locBlocksSrc mu b = true -> REACH m (frgnBlocksSrc mu) b = true.  Lemma protected_restr: forall m mu X (PROT : protected m mu), protected m (restrict_sm mu X). *)
Definition Match_cores d mu c1 m1 c2 m2:Prop :=
structured_match_cores d mu c1 m1 c2 m2 (*(restrict_sm mu (vis mu)) doesn't work here, since
some of the conditions of match_env are "global"*) /\
REACH_closed m1 (vis mu) /\
meminj_preserves_globals ge (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(* /\ protected m1 mu*).

Lemma transl_expr_correct:
forall mu m tm cenv tf e lenv te sp lo hi cs
(MINJ: Mem.inject (as_inj mu) m tm)
(RC: REACH_closed m (vis mu))
(MATCH: structured_match_callstack mu m tm
(Frame cenv tf e lenv te sp lo hi :: cs)
(Mem.nextblock m) (Mem.nextblock tm)),
forall a v,
Csharpminor.eval_expr ge e lenv m a v ->
forall ta app
(TR: transl_expr cenv a = OK (ta, app)),
exists tv,
eval_expr tge (Vptr sp Int.zero) te tm ta tv
/\ val_inject (restrict (as_inj mu) (vis mu)) v tv
/\ val_match_approx app v.