Definition fixup_trace (trace: AV.address -> option (pshare * AV.kind))
                                    (f: AV.address -> resource) : AV.address -> resource :=
   fun x => match trace x, f x with
                   | None, PURE k pp => PURE k pp
                   | Some(sh,k), PURE _ pp => YES sh k pp
                   | Some (sh,k), YES _ _ pp => YES sh k pp
                   | Some (sh, k), NO => YES sh k NoneP
                   | None, _ => NO
                   end.

Lemma fixup_trace_rmap: forall (tr: sig AV.valid) (f: rmap), {phi: rmap | level phi = level f /\ resource_at phi = fixup_trace (proj1_sig tr) (resource_at f)}.