Definition mod_after_alloc (phi: rmap) (lo hi: Z) (b: block) :=
  proj1_sig (make_rmap (mod_after_alloc' phi lo hi b) (ghost_of phi)
    _
    (mod_after_alloc'_ok phi lo hi b) (ghost_of_approx phi)).

Lemma can_age1_juicy_mem: forall j r, age (m_phi j) r -> exists j', age1 j = Some j'.