+
+lemma cnv_cpm_trans (h) (a) (G) (L):
+ ∀T1. ❪G,L❫ ⊢ T1 ![h,a] →
+ ∀n,T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → ❪G,L❫ ⊢ T2 ![h,a].
+/2 width=6 by cnv_cpm_trans_lpr/ qed-.
+
+(* Note: this is the preservation property *)
+lemma cnv_cpms_trans (h) (a) (G) (L):
+ ∀T1. ❪G,L❫ ⊢ T1 ![h,a] →
+ ∀n,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 → ❪G,L❫ ⊢ T2 ![h,a].
+/2 width=6 by cnv_cpms_trans_lpr/ qed-.
+
+lemma cnv_lpr_trans (h) (a) (G):
+ ∀L1,T. ❪G,L1❫ ⊢ T ![h,a] → ∀L2. ❪G,L1❫ ⊢ ➡[h] L2 → ❪G,L2❫ ⊢ T ![h,a].
+/2 width=6 by cnv_cpm_trans_lpr/ qed-.
+
+lemma cnv_lprs_trans (h) (a) (G):
+ ∀L1,T. ❪G,L1❫ ⊢ T ![h,a] → ∀L2. ❪G,L1❫ ⊢ ➡*[h] L2 → ❪G,L2❫ ⊢ T ![h,a].
+#h #a #G #L1 #T #HT #L2 #H
+@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
+qed-.