Definition meminj_no_overlap (f: meminj) (m: mem) : Prop :=
forall b1 b1' delta1 b2 b2' delta2 ofs1 ofs2,
b1 <> b2 ->
f b1 = Some (b1', delta1) ->
f b2 = Some (b2', delta2) ->
perm m b1 ofs1 Max Nonempty ->
perm m b2 ofs2 Max Nonempty ->
b1' <> b2' \/ ofs1 + delta1 <> ofs2 + delta2.

(* Specifications: *)
Lemma store_mapped_inj:
forall f chunk m1 b1 ofs v1 n1 m2 b2 delta v2,
mem_inj f m1 m2 ->
store chunk m1 b1 ofs v1 = Some n1 ->
meminj_no_overlap f m1 ->
f b1 = Some (b2, delta) ->
Val.inject f v1 v2 ->
exists n2,
store chunk m2 b2 (ofs + delta) v2 = Some n2
/\ mem_inj f n1 n2.
Lemma store_unmapped_inj:
forall f chunk m1 b1 ofs v1 n1 m2,
mem_inj f m1 m2 ->
store chunk m1 b1 ofs v1 = Some n1 ->
f b1 = None ->
mem_inj f n1 m2.
Lemma store_outside_inj:
forall f m1 m2 chunk b ofs v m2',
mem_inj f m1 m2 ->
(forall b' delta ofs',
f b' = Some(b, delta) ->
perm m1 b' ofs' Cur Readable ->
ofs <= ofs' + delta < ofs + size_chunk chunk -> False) ->
store chunk m2 b ofs v = Some m2' ->
mem_inj f m1 m2'.
Lemma storebytes_mapped_inj:
forall f m1 b1 ofs bytes1 n1 m2 b2 delta bytes2,
mem_inj f m1 m2 ->
storebytes m1 b1 ofs bytes1 = Some n1 ->
meminj_no_overlap f m1 ->
f b1 = Some (b2, delta) ->
list_forall2 (memval_inject f) bytes1 bytes2 ->
exists n2,
storebytes m2 b2 (ofs + delta) bytes2 = Some n2
/\ mem_inj f n1 n2.
Lemma storebytes_unmapped_inj:
forall f m1 b1 ofs bytes1 n1 m2,
mem_inj f m1 m2 ->
storebytes m1 b1 ofs bytes1 = Some n1 ->
f b1 = None ->
mem_inj f n1 m2.
Lemma storebytes_outside_inj:
forall f m1 m2 b ofs bytes2 m2',
mem_inj f m1 m2 ->
(forall b' delta ofs',
f b' = Some(b, delta) ->
perm m1 b' ofs' Cur Readable ->
ofs <= ofs' + delta < ofs + Z.of_nat (length bytes2) -> False) ->
storebytes m2 b ofs bytes2 = Some m2' ->
mem_inj f m1 m2'.
Lemma storebytes_empty_inj:
forall f m1 b1 ofs1 m1' m2 b2 ofs2 m2',
mem_inj f m1 m2 ->
storebytes m1 b1 ofs1 nil = Some m1' ->
storebytes m2 b2 ofs2 nil = Some m2' ->
mem_inj f m1' m2'.
Lemma store_mapped_inj:
forall f chunk m1 b1 ofs v1 n1 m2 b2 delta v2,
mem_inj f m1 m2 ->
store chunk m1 b1 ofs v1 = Some n1 ->
meminj_no_overlap f m1 ->
f b1 = Some (b2, delta) ->
Val.inject f v1 v2 ->
exists n2,
store chunk m2 b2 (ofs + delta) v2 = Some n2
/\ mem_inj f n1 n2.
Lemma store_unmapped_inj:
forall f chunk m1 b1 ofs v1 n1 m2,
mem_inj f m1 m2 ->
store chunk m1 b1 ofs v1 = Some n1 ->
f b1 = None ->
mem_inj f n1 m2.
Lemma store_outside_inj:
forall f m1 m2 chunk b ofs v m2',
mem_inj f m1 m2 ->
(forall b' delta ofs',
f b' = Some(b, delta) ->
perm m1 b' ofs' Cur Readable ->
ofs <= ofs' + delta < ofs + size_chunk chunk -> False) ->
store chunk m2 b ofs v = Some m2' ->
mem_inj f m1 m2'.
Lemma storebytes_mapped_inj:
forall f m1 b1 ofs bytes1 n1 m2 b2 delta bytes2,
mem_inj f m1 m2 ->
storebytes m1 b1 ofs bytes1 = Some n1 ->
meminj_no_overlap f m1 ->
f b1 = Some (b2, delta) ->
list_forall2 (memval_inject f) bytes1 bytes2 ->
exists n2,
storebytes m2 b2 (ofs + delta) bytes2 = Some n2
/\ mem_inj f n1 n2.
Lemma storebytes_unmapped_inj:
forall f m1 b1 ofs bytes1 n1 m2,
mem_inj f m1 m2 ->
storebytes m1 b1 ofs bytes1 = Some n1 ->
f b1 = None ->
mem_inj f n1 m2.
Lemma storebytes_outside_inj:
forall f m1 m2 b ofs bytes2 m2',
mem_inj f m1 m2 ->
(forall b' delta ofs',
f b' = Some(b, delta) ->
perm m1 b' ofs' Cur Readable ->
ofs <= ofs' + delta < ofs + Z.of_nat (length bytes2) -> False) ->
storebytes m2 b ofs bytes2 = Some m2' ->
mem_inj f m1 m2'.
Lemma storebytes_empty_inj:
forall f m1 b1 ofs1 m1' m2 b2 ofs2 m2',
mem_inj f m1 m2 ->
storebytes m1 b1 ofs1 nil = Some m1' ->
storebytes m2 b2 ofs2 nil = Some m2' ->
mem_inj f m1' m2'.