Module Renamings.
Definition memren := block -> option block.
Definition ren_incr f1 f2 :=
forall (b b' : block),
f1 b = Some b' -> f2 b = Some b'.
Definition ren_separated (f f' : memren) m1 m2 :=
forall (b1 b2 : block),
f b1 = None ->
f' b1 = Some b2 ->
~ Mem.valid_block m1 b1 /\ ~ Mem.valid_block m2 b2.
Definition ren_domain_incr (f1 f2: memren) :=
forall b,
f1 b -> f2 b.
(** Defining the domain of a renaming with respect to a memory*)
Definition domain_memren (f: memren) m :=
forall b, Mem.valid_block m b <-> isSome (f b).
(** Results about id injections*)
Definition id_ren m :=
fun b => if is_left (valid_block_dec m b) then Some b else None.
Hint Unfold id_ren.

Lemma incr_domain_id:
forall m f f'
(Hincr: ren_incr f f')
(Hf_id: forall b b', f b = Some b' -> b = b')
(Hdomain_f: domain_memren f' m),
ren_incr f (id_ren m).