+
+lemma cnv_lpr_trans (a) (h) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ T ![a,h] → ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T ![a,h].
+/2 width=6 by cnv_cpm_trans_lpr/ qed-.
+
+lemma cnv_lprs_trans (a) (h) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ T ![a,h] → ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T ![a,h].
+#a #h #G #L1 #T #HT #L2 #H
+@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
+qed-.