Definition initial_mem (m: mem) lev (IOK: initial_rmap_ok m lev) : juicy_mem.Definition juicy_mem_level (j: juicy_mem) (lev: nat) :=
  level (m_phi j) = lev.

Lemma initial_mem_level: forall lev m j IOK, j = initial_mem m lev IOK -> juicy_mem_level j (level lev).