Proof using.
induction tr; intros; simpl in *; repeat break_match; subst; intuition; eauto.
Qed.

(* Specifications: *)
Lemma last_holder'_no_out_extend :
forall tr h c n,
last_holder' h tr = Some n ->
last_holder' h (tr ++ [(c, inr [])]) = Some n.
Lemma last_holder'_no_out_extend :
forall tr h c n,
last_holder' h tr = Some n ->
last_holder' h (tr ++ [(c, inr [])]) = Some n.