Definition Complete (cenv: composite_env) (env: PTree.t A): Prop :=
  forall i,
    (exists co, PTree.get i cenv = Some co) <->
    (exists a, PTree.get i env = Some a).Definition Consistent (cenv: composite_env) (env: PTree.t A): Prop :=
  forall i co a,
    PTree.get i cenv = Some co ->
    PTree.get i env = Some a ->
    a = f_member (co_su co) (map
                              (fun it0: positive * type =>
                                 let (i0, t0) := it0 in
                                 (i0, t0, F env t0))
                              (co_members co)).Definition env_rec (i: positive) (co: composite) (env: PTree.t A): PTree.t A :=
  PTree.set i
    (f_member (co_su co) (map
                              (fun it0: positive * type =>
                                 let (i0, t0) := it0 in (i0, t0, F env t0))
                              (co_members co)))
    env.Definition Env (l: list (positive * composite)): PTree.t A :=
  fold_right
    (fun (ic: positive * composite) =>
       let (i, co) := ic in env_rec i co)
    (PTree.empty A)
    l.

Lemma relative_defined_type_PTree_set: forall t (env: PTree.t A) i a, relative_defined_type (PTree.elements env) t -> relative_defined_type (PTree.elements (PTree.set i a env)) t.