Record unchanged_on (m_before m_after: mem) : Prop := mk_unchanged_on {
  unchanged_on_nextblock:
    Ple (nextblock m_before) (nextblock m_after);
  unchanged_on_perm:
    forall b ofs k p,
    P b ofs -> valid_block m_before b ->
    (perm m_before b ofs k p <-> perm m_after b ofs k p);
  unchanged_on_contents:
    forall b ofs,
    P b ofs -> perm m_before b ofs Cur Readable ->
    ZMap.get ofs (PMap.get b m_after.(mem_contents)) =
    ZMap.get ofs (PMap.get b m_before.(mem_contents))
}.Record unchanged_on (m_before m_after: mem) : Prop := mk_unchanged_on {
  unchanged_on_nextblock:
    Ple (nextblock m_before) (nextblock m_after);
  unchanged_on_perm:
    forall b ofs k p,
    P b ofs -> valid_block m_before b ->
    (perm m_before b ofs k p <-> perm m_after b ofs k p);
  unchanged_on_contents:
    forall b ofs,
    P b ofs -> perm m_before b ofs Cur Readable ->
    ZMap.get ofs (PMap.get b m_after.(mem_contents)) =
    ZMap.get ofs (PMap.get b m_before.(mem_contents))
}.Record unchanged_on (m_before m_after: mem) : Prop := mk_unchanged_on {
  unchanged_on_nextblock:
    Ple (nextblock m_before) (nextblock m_after);
  unchanged_on_perm:
    forall b ofs k p,
    P b ofs -> valid_block m_before b ->
    (perm m_before b ofs k p <-> perm m_after b ofs k p);
  unchanged_on_contents:
    forall b ofs,
    P b ofs -> perm m_before b ofs Cur Readable ->
    ZMap.get ofs (PMap.get b m_after.(mem_contents)) =
    ZMap.get ofs (PMap.get b m_before.(mem_contents))
}.Record unchanged_on (m_before m_after: mem) : Prop := mk_unchanged_on {
  unchanged_on_nextblock:
    Ple (nextblock m_before) (nextblock m_after);
  unchanged_on_perm:
    forall b ofs k p,
    P b ofs -> valid_block m_before b ->
    (perm m_before b ofs k p <-> perm m_after b ofs k p);
  unchanged_on_contents:
    forall b ofs,
    P b ofs -> perm m_before b ofs Cur Readable ->
    ZMap.get ofs (PMap.get b m_after.(mem_contents)) =
    ZMap.get ofs (PMap.get b m_before.(mem_contents))
}.Record unchanged_on (m_before m_after: mem) : Prop := mk_unchanged_on {
  unchanged_on_nextblock:
    Ple (nextblock m_before) (nextblock m_after);
  unchanged_on_perm:
    forall b ofs k p,
    P b ofs -> valid_block m_before b ->
    (perm m_before b ofs k p <-> perm m_after b ofs k p);
  unchanged_on_contents:
    forall b ofs,
    P b ofs -> perm m_before b ofs Cur Readable ->
    ZMap.get ofs (PMap.get b m_after.(mem_contents)) =
    ZMap.get ofs (PMap.get b m_before.(mem_contents))
}.

Lemma unchanged_on_trans: forall m1 m2 m3, unchanged_on m1 m2 -> unchanged_on m2 m3 -> unchanged_on m1 m3.