(*The blocks explicitly exported via call arguments, plus the already shared blocks*)
Definition exportedSrc mu vals b := orb (getBlocks vals b) (sharedSrc mu b).Definition exportedTgt mu vals b := orb (getBlocks vals b) (sharedTgt mu b).

(* Specifications: *)
Lemma exported_inject: forall mu (WD: SM_wd mu) vals1 vals2 (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b (SRC: exportedSrc mu vals1 b = true ), exists jb d, as_inj mu b = Some (jb, d) /\ exportedTgt mu vals2 jb = true.
Lemma val_inject_sub_on j k: forall v1 v2 (V: val_inject j v1 v2) (HK: forall b, getBlocks (v1::nil) b = true -> j b = k b), val_inject k v1 v2.
Lemma val_inject_sub_on' j k: forall v1 v2 (V: val_inject j v1 v2) (HK: forall b b2 d, getBlocks (v1::nil) b = true -> j b = Some(b2,d) -> k b = Some(b2,d)), val_inject k v1 v2.
Lemma val_list_inject_sub_on j k: forall vals1 vals2 (V: Val.inject_list j vals1 vals2) (HK: forall b, getBlocks vals1 b = true -> j b = k b), Val.inject_list k vals1 vals2.
Lemma val_list_inject_sub_on' j k: forall vals1 vals2 (V: Val.inject_list j vals1 vals2) (HK: forall b b2 d, getBlocks vals1 b = true -> j b = Some(b2,d) -> k b = Some(b2,d)), Val.inject_list k vals1 vals2.
Lemma REACH_shared_of: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (shared_of mu) m1 m2) (ValInjMu : Forall2 (val_inject (shared_of mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) B (HB: forall b b2 d, shared_of mu b = Some(b2,d) -> B b2 = true), exists b2 d, shared_of mu b1 = Some (b2, d) /\ REACH m2 (fun b => orb (getBlocks vals2 b) (B b)) b2 = true.
Lemma REACH_as_inj: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) B (HB: forall b b2 d, shared_of mu b = Some(b2,d) -> B b2 = true), exists b2 d, as_inj mu b1 = Some (b2, d) /\ REACH m2 (fun b => orb (getBlocks vals2 b) (B b)) b2 = true.
Lemma REACH_local: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = true), exists b2 d, local_of mu b1 = Some (b2, d).
Lemma REACH_extern: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = false), exists b2 d, extern_of mu b1 = Some (b2, d).
Lemma REACH_as_inj_REACH: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true), exists b2 d, as_inj mu b1 = Some (b2, d) /\ REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma REACH_local_REACH: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = true), exists b2 d, local_of mu b1 = Some (b2, d) /\ REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma REACH_local_REACH': forall mu m1 vals1 b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (WD: SM_wd mu) m2 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) (locBSrc : locBlocksSrc mu b1 = true) b2 d (LOC: local_of mu b1 = Some (b2, d)), REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma REACH_extern_REACH: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = false), exists b2 d, extern_of mu b1 = Some (b2, d) /\ REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma pubBlocksSrc_REACH: forall m1 mu (WD: SM_wd mu) vals b, pubBlocksSrc mu b = true -> REACH m1 (exportedSrc mu vals) b = true.
Lemma getBlocks_REACH_exportedSrc m mu vals b: forall (GB: getBlocks vals b = true), REACH m (exportedSrc mu vals) b = true.
Lemma exported_inject: forall mu (WD: SM_wd mu) vals1 vals2 (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b (SRC: exportedSrc mu vals1 b = true ), exists jb d, as_inj mu b = Some (jb, d) /\ exportedTgt mu vals2 jb = true.
Lemma REACH_shared_of: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (shared_of mu) m1 m2) (ValInjMu : Forall2 (val_inject (shared_of mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) B (HB: forall b b2 d, shared_of mu b = Some(b2,d) -> B b2 = true), exists b2 d, shared_of mu b1 = Some (b2, d) /\ REACH m2 (fun b => orb (getBlocks vals2 b) (B b)) b2 = true.
Lemma REACH_as_inj: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) B (HB: forall b b2 d, shared_of mu b = Some(b2,d) -> B b2 = true), exists b2 d, as_inj mu b1 = Some (b2, d) /\ REACH m2 (fun b => orb (getBlocks vals2 b) (B b)) b2 = true.
Lemma REACH_local: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = true), exists b2 d, local_of mu b1 = Some (b2, d).
Lemma REACH_extern: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = false), exists b2 d, extern_of mu b1 = Some (b2, d).
Lemma REACH_as_inj_REACH: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true), exists b2 d, as_inj mu b1 = Some (b2, d) /\ REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma REACH_local_REACH: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = true), exists b2 d, local_of mu b1 = Some (b2, d) /\ REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma REACH_local_REACH': forall mu m1 vals1 b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (WD: SM_wd mu) m2 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) (locBSrc : locBlocksSrc mu b1 = true) b2 d (LOC: local_of mu b1 = Some (b2, d)), REACH m2 (exportedTgt mu vals2) b2 = true.
Lemma REACH_extern_REACH: forall mu (WD: SM_wd mu) m1 m2 vals1 vals2 (MemInjMu : Mem.inject (as_inj mu) m1 m2) (ValInjMu : Forall2 (val_inject (as_inj mu)) vals1 vals2) b1 (R : REACH m1 (exportedSrc mu vals1) b1 = true) (locBSrc : locBlocksSrc mu b1 = false), exists b2 d, extern_of mu b1 = Some (b2, d) /\ REACH m2 (exportedTgt mu vals2) b2 = true.