Ltac assoc_destruct :=
    match goal with
    | [ |- context [assoc (assoc_set _ ?k0' _) ?k0 ] ] =>
      destruct (K_eq_dec k0 k0'); [subst k0'; rewrite get_set_same with (k := k0)|
                                   rewrite get_set_diff with (k' := k0) by auto]
    end.Ltac assoc_rewrite :=
    match goal with
    | [  |- context [assoc (assoc_set _ ?k0' _) ?k0 ] ] =>
      first [rewrite get_set_same with (k := k0) by auto |
             rewrite get_set_diff with (k' := k0) by auto ]
    end.

Lemma assoc_default_a_equiv : forall l l' (k : K) (v : V), a_equiv l l' -> assoc_default l k v = assoc_default l' k v.