(** * Invariance properties between two memory states *)
Section UNCHANGED_ON.
Variable P: block -> Z -> Prop.
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))
}.

(* Specifications: *)
Lemma unchanged_on_refl:
forall m, unchanged_on m m.
Lemma valid_block_unchanged_on:
forall m m' b,
unchanged_on m m' -> valid_block m b -> valid_block m' b.
Lemma perm_unchanged_on:
forall m m' b ofs k p,
unchanged_on m m' -> P b ofs ->
perm m b ofs k p -> perm m' b ofs k p.
Lemma perm_unchanged_on_2:
forall m m' b ofs k p,
unchanged_on m m' -> P b ofs -> valid_block m b ->
perm m' b ofs k p -> perm m b ofs k p.
Lemma unchanged_on_trans:
forall m1 m2 m3, unchanged_on m1 m2 -> unchanged_on m2 m3 -> unchanged_on m1 m3.
Lemma loadbytes_unchanged_on_1:
forall m m' b ofs n,
unchanged_on m m' ->
valid_block m b ->
(forall i, ofs <= i < ofs + n -> P b i) ->
loadbytes m' b ofs n = loadbytes m b ofs n.
Lemma loadbytes_unchanged_on:
forall m m' b ofs n bytes,
unchanged_on m m' ->
(forall i, ofs <= i < ofs + n -> P b i) ->
loadbytes m b ofs n = Some bytes ->
loadbytes m' b ofs n = Some bytes.
Lemma load_unchanged_on_1:
forall m m' chunk b ofs,
unchanged_on m m' ->
valid_block m b ->
(forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
load chunk m' b ofs = load chunk m b ofs.
Lemma load_unchanged_on:
forall m m' chunk b ofs v,
unchanged_on m m' ->
(forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
load chunk m b ofs = Some v ->
load chunk m' b ofs = Some v.
Lemma store_unchanged_on:
forall chunk m b ofs v m',
store chunk m b ofs v = Some m' ->
(forall i, ofs <= i < ofs + size_chunk chunk -> ~ P b i) ->
unchanged_on m m'.
Lemma storebytes_unchanged_on:
forall m b ofs bytes m',
storebytes m b ofs bytes = Some m' ->
(forall i, ofs <= i < ofs + Z.of_nat (length bytes) -> ~ P b i) ->
unchanged_on m m'.
Lemma alloc_unchanged_on:
forall m lo hi m' b,
alloc m lo hi = (m', b) ->
unchanged_on m m'.
Lemma free_unchanged_on:
forall m b lo hi m',
free m b lo hi = Some m' ->
(forall i, lo <= i < hi -> ~ P b i) ->
unchanged_on m m'.
Lemma drop_perm_unchanged_on:
forall m b lo hi p m',
drop_perm m b lo hi p = Some m' ->
(forall i, lo <= i < hi -> ~ P b i) ->
unchanged_on m m'.
Lemma unchanged_on_refl:
forall m, unchanged_on m m.
Lemma valid_block_unchanged_on:
forall m m' b,
unchanged_on m m' -> valid_block m b -> valid_block m' b.
Lemma perm_unchanged_on:
forall m m' b ofs k p,
unchanged_on m m' -> P b ofs ->
perm m b ofs k p -> perm m' b ofs k p.
Lemma perm_unchanged_on_2:
forall m m' b ofs k p,
unchanged_on m m' -> P b ofs -> valid_block m b ->
perm m' b ofs k p -> perm m b ofs k p.
Lemma unchanged_on_trans:
forall m1 m2 m3, unchanged_on m1 m2 -> unchanged_on m2 m3 -> unchanged_on m1 m3.
Lemma loadbytes_unchanged_on_1:
forall m m' b ofs n,
unchanged_on m m' ->
valid_block m b ->
(forall i, ofs <= i < ofs + n -> P b i) ->
loadbytes m' b ofs n = loadbytes m b ofs n.
Lemma loadbytes_unchanged_on:
forall m m' b ofs n bytes,
unchanged_on m m' ->
(forall i, ofs <= i < ofs + n -> P b i) ->
loadbytes m b ofs n = Some bytes ->
loadbytes m' b ofs n = Some bytes.
Lemma load_unchanged_on_1:
forall m m' chunk b ofs,
unchanged_on m m' ->
valid_block m b ->
(forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
load chunk m' b ofs = load chunk m b ofs.
Lemma load_unchanged_on:
forall m m' chunk b ofs v,
unchanged_on m m' ->
(forall i, ofs <= i < ofs + size_chunk chunk -> P b i) ->
load chunk m b ofs = Some v ->
load chunk m' b ofs = Some v.
Lemma store_unchanged_on:
forall chunk m b ofs v m',
store chunk m b ofs v = Some m' ->
(forall i, ofs <= i < ofs + size_chunk chunk -> ~ P b i) ->
unchanged_on m m'.
Lemma storebytes_unchanged_on:
forall m b ofs bytes m',
storebytes m b ofs bytes = Some m' ->
(forall i, ofs <= i < ofs + Z.of_nat (length bytes) -> ~ P b i) ->
unchanged_on m m'.
Lemma alloc_unchanged_on:
forall m lo hi m' b,
alloc m lo hi = (m', b) ->
unchanged_on m m'.
Lemma free_unchanged_on:
forall m b lo hi m',
free m b lo hi = Some m' ->
(forall i, lo <= i < hi -> ~ P b i) ->
unchanged_on m m'.
Lemma drop_perm_unchanged_on:
forall m b lo hi p m',
drop_perm m b lo hi p = Some m' ->
(forall i, lo <= i < hi -> ~ P b i) ->
unchanged_on m m'.