Proof using.
induction l; intuition.
- subst. right. exists nil. simpl. eauto.
- break_exists. subst. right. eexists. eexists.
rewrite app_comm_cons. eauto.
Qed.

Lemma inputs_1_app :
forall tr1 tr2,
inputs_1 (tr1 ++ tr2) = inputs_1 tr1 ++ inputs_1 tr2.