Record match_genvs (ge1: t A V) (ge2: t B W): Prop := {
  mge_next:
    genv_next ge2 = genv_next ge1;
  mge_symb:
    forall id, PTree.get id (genv_symb ge2) = PTree.get id (genv_symb ge1);
  mge_defs:
    forall b, option_rel R (PTree.get b (genv_defs ge1)) (PTree.get b (genv_defs ge2))
}.Record match_genvs (ge1: t A V) (ge2: t B W): Prop := {
  mge_next:
    genv_next ge2 = genv_next ge1;
  mge_symb:
    forall id, PTree.get id (genv_symb ge2) = PTree.get id (genv_symb ge1);
  mge_defs:
    forall b, option_rel R (PTree.get b (genv_defs ge1)) (PTree.get b (genv_defs ge2))
}.Record match_genvs (ge1: t A V) (ge2: t B W): Prop := {
  mge_next:
    genv_next ge2 = genv_next ge1;
  mge_symb:
    forall id, PTree.get id (genv_symb ge2) = PTree.get id (genv_symb ge1);
  mge_defs:
    forall b, option_rel R (PTree.get b (genv_defs ge1)) (PTree.get b (genv_defs ge2))
}.Record match_genvs (ge1: t A V) (ge2: t B W): Prop := {
  mge_next:
    genv_next ge2 = genv_next ge1;
  mge_symb:
    forall id, PTree.get id (genv_symb ge2) = PTree.get id (genv_symb ge1);
  mge_defs:
    forall b, option_rel R (PTree.get b (genv_defs ge1)) (PTree.get b (genv_defs ge2))
}.

Lemma add_globals_match: forall gl1 gl2, list_forall2 (fun idg1 idg2 => fst idg1 = fst idg2 /\ R (snd idg1) (snd idg2)) gl1 gl2 -> forall ge1 ge2, match_genvs ge1 ge2 -> match_genvs (add_globals ge1 gl1) (add_globals ge2 gl2).