(** * Relating the memory states *)

(** The [magree] predicate is a variant of [Mem.extends] where we
  allow the contents of the two memory states to differ arbitrarily
  on some locations.  The predicate [P] is true on the locations whose
  contents must be in the [lessdef] relation. *)

Definition locset := block -> Z -> Prop.Record magree (m1 m2: mem) (P: locset) : Prop := mk_magree {
  ma_perm:
    forall b ofs k p,
    Mem.perm m1 b ofs k p -> Mem.perm m2 b ofs k p;
  ma_perm_inv:
    forall b ofs k p,
    Mem.perm m2 b ofs k p -> Mem.perm m1 b ofs k p \/ ~Mem.perm m1 b ofs Max Nonempty;
  ma_memval:
    forall b ofs,
    Mem.perm m1 b ofs Cur Readable ->
    P b ofs ->
    memval_lessdef (ZMap.get ofs (PMap.get b (Mem.mem_contents m1)))
                   (ZMap.get ofs (PMap.get b (Mem.mem_contents m2)));
  ma_nextblock:
    Mem.nextblock m2 = Mem.nextblock m1
}.Record magree (m1 m2: mem) (P: locset) : Prop := mk_magree {
  ma_perm:
    forall b ofs k p,
    Mem.perm m1 b ofs k p -> Mem.perm m2 b ofs k p;
  ma_perm_inv:
    forall b ofs k p,
    Mem.perm m2 b ofs k p -> Mem.perm m1 b ofs k p \/ ~Mem.perm m1 b ofs Max Nonempty;
  ma_memval:
    forall b ofs,
    Mem.perm m1 b ofs Cur Readable ->
    P b ofs ->
    memval_lessdef (ZMap.get ofs (PMap.get b (Mem.mem_contents m1)))
                   (ZMap.get ofs (PMap.get b (Mem.mem_contents m2)));
  ma_nextblock:
    Mem.nextblock m2 = Mem.nextblock m1
}.Record magree (m1 m2: mem) (P: locset) : Prop := mk_magree {
  ma_perm:
    forall b ofs k p,
    Mem.perm m1 b ofs k p -> Mem.perm m2 b ofs k p;
  ma_perm_inv:
    forall b ofs k p,
    Mem.perm m2 b ofs k p -> Mem.perm m1 b ofs k p \/ ~Mem.perm m1 b ofs Max Nonempty;
  ma_memval:
    forall b ofs,
    Mem.perm m1 b ofs Cur Readable ->
    P b ofs ->
    memval_lessdef (ZMap.get ofs (PMap.get b (Mem.mem_contents m1)))
                   (ZMap.get ofs (PMap.get b (Mem.mem_contents m2)));
  ma_nextblock:
    Mem.nextblock m2 = Mem.nextblock m1
}.Record magree (m1 m2: mem) (P: locset) : Prop := mk_magree {
  ma_perm:
    forall b ofs k p,
    Mem.perm m1 b ofs k p -> Mem.perm m2 b ofs k p;
  ma_perm_inv:
    forall b ofs k p,
    Mem.perm m2 b ofs k p -> Mem.perm m1 b ofs k p \/ ~Mem.perm m1 b ofs Max Nonempty;
  ma_memval:
    forall b ofs,
    Mem.perm m1 b ofs Cur Readable ->
    P b ofs ->
    memval_lessdef (ZMap.get ofs (PMap.get b (Mem.mem_contents m1)))
                   (ZMap.get ofs (PMap.get b (Mem.mem_contents m2)));
  ma_nextblock:
    Mem.nextblock m2 = Mem.nextblock m1
}.Record magree (m1 m2: mem) (P: locset) : Prop := mk_magree {
  ma_perm:
    forall b ofs k p,
    Mem.perm m1 b ofs k p -> Mem.perm m2 b ofs k p;
  ma_perm_inv:
    forall b ofs k p,
    Mem.perm m2 b ofs k p -> Mem.perm m1 b ofs k p \/ ~Mem.perm m1 b ofs Max Nonempty;
  ma_memval:
    forall b ofs,
    Mem.perm m1 b ofs Cur Readable ->
    P b ofs ->
    memval_lessdef (ZMap.get ofs (PMap.get b (Mem.mem_contents m1)))
                   (ZMap.get ofs (PMap.get b (Mem.mem_contents m2)));
  ma_nextblock:
    Mem.nextblock m2 = Mem.nextblock m1
}.Record magree (m1 m2: mem) (P: locset) : Prop := mk_magree {
  ma_perm:
    forall b ofs k p,
    Mem.perm m1 b ofs k p -> Mem.perm m2 b ofs k p;
  ma_perm_inv:
    forall b ofs k p,
    Mem.perm m2 b ofs k p -> Mem.perm m1 b ofs k p \/ ~Mem.perm m1 b ofs Max Nonempty;
  ma_memval:
    forall b ofs,
    Mem.perm m1 b ofs Cur Readable ->
    P b ofs ->
    memval_lessdef (ZMap.get ofs (PMap.get b (Mem.mem_contents m1)))
                   (ZMap.get ofs (PMap.get b (Mem.mem_contents m2)));
  ma_nextblock:
    Mem.nextblock m2 = Mem.nextblock m1
}.

Lemma magree_loadbytes: forall m1 m2 P b ofs n bytes, magree m1 m2 P -> Mem.loadbytes m1 b ofs n = Some bytes -> (forall i, ofs <= i < ofs + n -> P b i) -> exists bytes', Mem.loadbytes m2 b ofs n = Some bytes' /\ list_forall2 memval_lessdef bytes bytes'.