Definition resource_decay (nextb: block) (phi1 phi2: rmap) :=
(level phi1 >= level phi2)%nat /\
forall l: address,
((fst l >= nextb)%positive -> phi1 @ l = NO Share.bot bot_unreadable) /\
(resource_fmap (approx (level phi2)) (approx (level phi2)) (phi1 @ l) = (phi2 @ l) \/
(exists sh, exists (wsh: writable0_share sh), exists v, exists v',
resource_fmap (approx (level phi2)) (approx (level phi2)) (phi1 @ l) =
YES sh (writable0_readable wsh) (VAL v) NoneP /\
phi2 @ l = YES sh (writable0_readable wsh) (VAL v') NoneP)
\/ ((fst l >= nextb)%positive /\ exists v, phi2 @ l = YES Share.top readable_share_top (VAL v) NoneP)
\/ (exists v, exists pp, phi1 @ l = YES Share.top readable_share_top (VAL v) pp
/\ phi2 @ l = NO Share.bot bot_unreadable)).
Definition resource_nodecay (nextb: block) (phi1 phi2: rmap) :=
(level phi1 >= level phi2)%nat /\
forall l: address,
((fst l >= nextb)%positive -> phi1 @ l = NO Share.bot bot_unreadable) /\
(resource_fmap (approx (level phi2)) (approx (level phi2)) (phi1 @ l) = (phi2 @ l) \/
(exists sh, exists (wsh: writable0_share sh), exists v, exists v',
resource_fmap (approx (level phi2)) (approx (level phi2)) (phi1 @ l) = YES sh (writable0_readable wsh) (VAL v) NoneP
/\ phi2 @ l = YES sh (writable0_readable wsh) (VAL v') NoneP)).

Lemma age1_PURE_2 {phi phi' l k P} :
age1 phi = Some phi' ->
phi' @ l = PURE k P ->
(exists P, phi @ l = PURE k P).