∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 →
∀L2. L0 ⦻*[RN2, RP2, f] L2 →
L1 ⦻*[RN, RP, f] L2.
-#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -L1 -L0 -f
+#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
-| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
+| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
#K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
-| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
+| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_push1 … H) -H
#K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_push/
]
qed-.
∀f1. L1 ⦻*[RN, RP, f1] L2 →
∀f2. L1 ⦻*[RN, RP, f2] L2 →
∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
-#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH #f2 #H #f #Hf
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
∀f1. L1 ⦻*[RN, RP, f1] L2 →
∀f2. L1 ⦻*[RN, RP, f2] L2 →
∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
-#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH #f2 #H #f #Hf
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)