+lemma lexs_sym: ∀RN,RP.
+ (∀L1,L2,T1,T2. RN L1 T1 T2 → RN L2 T2 T1) →
+ (∀L1,L2,T1,T2. RP L1 T1 T2 → RP L2 T2 T1) →
+ ∀f. symmetric … (lexs RN RP f).
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f
+/3 width=2 by lexs_next, lexs_push/
+qed-.
+
+lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2.
+ L1.ⓑ{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 →
+ ∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 →
+ L1.ⓑ{I}W1 ⦻*[RN, RP, f] L2.ⓑ{I}W2.
+#RN #RP #f #I #L1 #L2 #V1 #V2 #HL12 #W1 #W2 #HN #HP
+elim (lexs_fwd_pair … HL12) -HL12 /2 width=1 by lexs_inv_tl/
+qed-.
+
+lemma lexs_co: ∀RN1,RP1,RN2,RP2.
+ (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) →
+ (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
+ ∀f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by lexs_atom, lexs_next, lexs_push/
+qed-.
+
+lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2.
+ (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
+ ∀f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
+ L1 ⦻*[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
+#f #I #K1 #K2 #V1 #V2 #_ #HV12 #IH #H
+[ elim (isid_inv_next … H) -H //
+| /4 width=3 by lexs_push, isid_inv_push/
+]
+qed-.
+
+lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
+ ∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ⦻*[RN, RP, f1] L2.
+#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
+#f2 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH