-lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
-#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
-#T #T2 #_ #HT2 #IHT1 #L1 #HL12
-lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
+lemma s_r_conf1_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (LTC … S) R.
+#A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3 by/
+qed-.
+
+lemma s_r_trans_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R →
+ s_r_transitive A B S R → s_rs_transitive A B S R.
+#A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /2 width=3 by/
+#T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR … HT1 … HL12) -H2SR -HT1
+/4 width=5 by s_r_conf1_LTC1, trans_TC/
+qed-.
+
+lemma s_r_trans_LTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (LTC … R).
+#A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /3 width=3 by inj/
+qed-.
+
+lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (LTC … S) R →
+ s_rs_transitive A B S R.
+#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
+elim (TC_idem … (S L1) … T1 T2)
+#_ #H @H @HSR //
+qed-.
+
+lemma s_rs_to_s_r_trans: ∀A,B,S,R. s_rs_transitive A B S R →
+ s_r_transitive A B (LTC … S) R.
+#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
+elim (TC_idem … (S L1) … T1 T2)
+#H #_ @H @HSR //