Hint Extern 1 (lc_co (g_AbsCong _ _ _)) =>
  let x1 := fresh in
  pick_fresh x1;
  apply (lc_g_AbsCong_exists x1).Hint Extern 1 (lc_co (g_CAbsCong _ _ _)) =>
  let c1 := fresh in
  pick_fresh c1;
  apply (lc_g_CAbsCong_exists c1).

Lemma tm_subst_tm_tm_open_tm_wrt_tm_rec_tm_subst_tm_brs_open_brs_wrt_tm_rec_tm_subst_tm_co_open_co_wrt_tm_rec_tm_subst_tm_constraint_open_constraint_wrt_tm_rec_mutual : (forall a3 a1 a2 x1 n1, lc_tm a1 -> tm_subst_tm_tm a1 x1 (open_tm_wrt_tm_rec n1 a2 a3) = open_tm_wrt_tm_rec n1 (tm_subst_tm_tm a1 x1 a2) (tm_subst_tm_tm a1 x1 a3)) /\ (forall brs1 a1 a2 x1 n1, lc_tm a1 -> tm_subst_tm_brs a1 x1 (open_brs_wrt_tm_rec n1 a2 brs1) = open_brs_wrt_tm_rec n1 (tm_subst_tm_tm a1 x1 a2) (tm_subst_tm_brs a1 x1 brs1)) /\ (forall g1 a1 a2 x1 n1, lc_tm a1 -> tm_subst_tm_co a1 x1 (open_co_wrt_tm_rec n1 a2 g1) = open_co_wrt_tm_rec n1 (tm_subst_tm_tm a1 x1 a2) (tm_subst_tm_co a1 x1 g1)) /\ (forall phi1 a1 a2 x1 n1, lc_tm a1 -> tm_subst_tm_constraint a1 x1 (open_constraint_wrt_tm_rec n1 a2 phi1) = open_constraint_wrt_tm_rec n1 (tm_subst_tm_tm a1 x1 a2) (tm_subst_tm_constraint a1 x1 phi1)).