(** * Memory extensions *)

(**  A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
  by increasing the sizes of the memory blocks of [m1] (decreasing
  the low bounds, increasing the high bounds), and replacing some of
  the [Vundef] values stored in [m1] by more defined values stored
  in [m2] at the same locations. *)

Record extends' (m1 m2: mem) : Prop :=
  mk_extends {
    mext_next: nextblock m1 = nextblock m2;
    mext_inj:  mem_inj inject_id m1 m2;
    mext_perm_inv: forall b ofs k p,
      perm m2 b ofs k p ->
      perm m1 b ofs k p \/ ~perm m1 b ofs Max Nonempty
  }.(** * Memory extensions *)

(**  A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
  by increasing the sizes of the memory blocks of [m1] (decreasing
  the low bounds, increasing the high bounds), and replacing some of
  the [Vundef] values stored in [m1] by more defined values stored
  in [m2] at the same locations. *)

Record extends' (m1 m2: mem) : Prop :=
  mk_extends {
    mext_next: nextblock m1 = nextblock m2;
    mext_inj:  mem_inj inject_id m1 m2;
    mext_perm_inv: forall b ofs k p,
      perm m2 b ofs k p ->
      perm m1 b ofs k p \/ ~perm m1 b ofs Max Nonempty
  }.(** * Memory extensions *)

(**  A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
  by increasing the sizes of the memory blocks of [m1] (decreasing
  the low bounds, increasing the high bounds), and replacing some of
  the [Vundef] values stored in [m1] by more defined values stored
  in [m2] at the same locations. *)

Record extends' (m1 m2: mem) : Prop :=
  mk_extends {
    mext_next: nextblock m1 = nextblock m2;
    mext_inj:  mem_inj inject_id m1 m2;
    mext_perm_inv: forall b ofs k p,
      perm m2 b ofs k p ->
      perm m1 b ofs k p \/ ~perm m1 b ofs Max Nonempty
  }.(** * Memory extensions *)

(**  A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
  by increasing the sizes of the memory blocks of [m1] (decreasing
  the low bounds, increasing the high bounds), and replacing some of
  the [Vundef] values stored in [m1] by more defined values stored
  in [m2] at the same locations. *)

Record extends' (m1 m2: mem) : Prop :=
  mk_extends {
    mext_next: nextblock m1 = nextblock m2;
    mext_inj:  mem_inj inject_id m1 m2;
    mext_perm_inv: forall b ofs k p,
      perm m2 b ofs k p ->
      perm m1 b ofs k p \/ ~perm m1 b ofs Max Nonempty
  }.(** * Memory extensions *)

(**  A store [m2] extends a store [m1] if [m2] can be obtained from [m1]
  by increasing the sizes of the memory blocks of [m1] (decreasing
  the low bounds, increasing the high bounds), and replacing some of
  the [Vundef] values stored in [m1] by more defined values stored
  in [m2] at the same locations. *)

Record extends' (m1 m2: mem) : Prop :=
  mk_extends {
    mext_next: nextblock m1 = nextblock m2;
    mext_inj:  mem_inj inject_id m1 m2;
    mext_perm_inv: forall b ofs k p,
      perm m2 b ofs k p ->
      perm m1 b ofs k p \/ ~perm m1 b ofs Max Nonempty
  }.Definition extends := extends'.

Theorem alloc_extends: forall m1 m2 lo1 hi1 b m1' lo2 hi2, extends m1 m2 -> alloc m1 lo1 hi1 = (m1', b) -> lo2 <= lo1 -> hi1 <= hi2 -> exists m2', alloc m2 lo2 hi2 = (m2', b) /\ extends m1' m2'.