Inductive structured_match_callstack mu (m: mem) (tm: mem):
callstack -> Values.block -> Values.block -> Prop :=
| st_mcs_nil:
forall hi bound tbound,
match_globalenvs prog (restrict (as_inj mu) (vis mu)) hi ->
Ple hi bound -> Ple hi tbound ->
structured_match_callstack mu m tm nil bound tbound
| st_mcs_cons:
forall cenv tf e le te sp lo hi cs bound tbound
(BOUND: Ple hi bound)
(TBOUND: Plt sp tbound)
(MTMP: match_temps (restrict (as_inj mu) (vis mu)) le te)
(MENV: match_env (restrict (as_inj mu) (vis mu)) cenv e sp lo hi)
(BOUND: match_bounds e m)
(PERM: padding_freeable (restrict (as_inj mu) (vis mu)) e tm sp tf.(fn_stackspace))
(SPlocal: locBlocksTgt mu sp = true) (*NEW*)
(*LENB: Here is the condition me_incr from match_env above, but we assert it on the entire as_inj mu*)
(ME_INCR: forall b tb delta,
(as_inj mu) b = Some(tb, delta) -> Plt b lo -> Plt tb sp)
(MCS: structured_match_callstack mu m tm cs lo sp),
structured_match_callstack mu m tm (Frame cenv tf e le te sp lo hi :: cs) bound tbound.

(* Specifications: *)
Lemma structured_match_callstack_incr_bound:
forall f m tm cs bound tbound bound' tbound',
structured_match_callstack f m tm cs bound tbound ->
Ple bound bound' -> Ple tbound tbound' ->
structured_match_callstack f m tm cs bound' tbound'.
Lemma structured_match_callstack_intern_invariant:
forall f1 m1 tm1 f2 m2 tm2 cs bound tbound (WD1: SM_wd f1) (WD2: SM_wd f2),
intern_incr f1 f2 ->
structured_match_callstack f1 m1 tm1 cs bound tbound ->
(forall b ofs p, Plt b bound -> Mem.perm m2 b ofs Max p -> Mem.perm m1 b ofs Max p) ->
(forall sp ofs, Plt sp tbound -> Mem.perm tm1 sp ofs Cur Freeable -> Mem.perm tm2 sp ofs Cur Freeable) ->
(forall b, Plt b bound -> (as_inj f2) b = (as_inj f1) b) ->
(forall b b' delta, (as_inj f2) b = Some(b', delta) -> Plt b' tbound -> (as_inj f1) b = Some(b', delta)) ->
structured_match_callstack f2 m2 tm2 cs bound tbound.
Lemma structured_match_callstack_match_globalenvs:
forall mu m tm cs bound tbound
(MCS: structured_match_callstack mu m tm cs bound tbound),
exists hi, Ple hi bound /\ Ple hi tbound /\
match_globalenvs prog (restrict (as_inj mu) (vis mu)) hi.
Lemma replace_locals_extern_incr_vis: forall mu nu pubS pubT
(Hnu: nu = replace_locals mu pubS pubT)
nu' (INC : extern_incr nu nu') (WDnu' : SM_wd nu') l m1'
b (VIS: vis mu b = true),
locBlocksSrc nu' b || DomSrc nu' b &&
(negb (locBlocksSrc nu' b) && REACH m1' (exportedSrc nu' l) b) = true.
Lemma structured_match_callstack_ext: forall mu m1 m2 cs bound
tbound
(MCS : structured_match_callstack mu m1 m2 cs bound tbound)
m1' (FwdSrc : mem_forward m1 m1')
nu vals1 vals2
(Hnu: nu = replace_locals mu
(fun b0 : Values.block =>
locBlocksSrc mu b0 &&
REACH m1 (exportedSrc mu vals1) b0)
(fun b0 : Values.block =>
locBlocksTgt mu b0 &&
REACH m2 (exportedTgt mu vals2) b0))
(SMvalMu : sm_valid mu m1 m2)
(PG : meminj_preserves_globals (Genv.globalenv prog) (as_inj mu))
nu' (INC : extern_incr nu nu')
(SEP : sm_inject_separated nu nu' m1 m2)
(UNMAPPED : Mem.unchanged_on
(fun (b : Values.block) (_ : Z) =>
locBlocksSrc nu b = true /\ pubBlocksSrc nu b = false) m1
m1')
m2' (FwdTgt : mem_forward m2 m2') (OUTOFREACH : Mem.unchanged_on (local_out_of_reach nu m1) m2 m2')
ret1 ret2 (WDnu' : SM_wd nu') (WDmu : SM_wd mu) (WDnu : SM_wd nu)
(BND: Ple bound (Mem.nextblock m1))
(TBND:Ple tbound (Mem.nextblock m2)),
structured_match_callstack
(replace_externs nu'
(fun b : Values.block =>
DomSrc nu' b &&
(negb (locBlocksSrc nu' b) && REACH m1' (exportedSrc nu' (ret1 :: nil)) b))
(fun b : Values.block =>
DomTgt nu' b &&
(negb (locBlocksTgt nu' b) && REACH m2' (exportedTgt nu' (ret2 :: nil)) b)))
m1' m2' cs bound tbound.