+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R).
+#R #HR #L elim L -L // /2 width=1/
+qed.
+
+lemma lpx_trans: ∀R. Transitive ? R → Transitive … (lpx R).
+#R #HR #L1 #L #H elim H -L //
+#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
+elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
+qed.
+
+lemma lpx_conf: ∀R. Confluent ? R → Confluent … (lpx R).
+#R #HR #L0 #L1 #H elim H -L1
+[ #X #H >(lpx_inv_atom1 … H) -X /2 width=3/
+| #I #K0 #K1 #V0 #V1 #_ #HV01 #IHK01 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK02 #HV02 #H destruct
+ elim (IHK01 … HK02) -K0 #K #HK1 #HK2
+ elim (HR … HV01 … HV02) -HR -V0 /3 width=5/
+]
+qed.
+
+lemma lpx_TC_inj: ∀R,L1,L2. lpx R L1 L2 → lpx (TC … R) L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
+qed.
+
+lemma lpx_TC_step: ∀R,L1,L. lpx (TC … R) L1 L →
+ ∀L2. lpx R L L2 → lpx (TC … R) L1 L2.
+#R #L1 #L #H elim H -L /2 width=1/
+#I #K1 #K #V1 #V #_ #HV1 #IHK1 #X #H
+elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK2 #HV2 #H destruct /3 width=3/
+qed.
+
+lemma TC_lpx_pair_dx: ∀R. reflexive ? R →
+ ∀I,K,V1,V2. TC … R V1 V2 →
+ TC … (lpx R) (K.ⓑ{I}V1) (K.ⓑ{I}V2).
+#R #HR #I #K #V1 #V2 #H elim H -V2
+/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
+qed.
+
+lemma TC_lpx_pair_sn: ∀R. reflexive ? R →
+ ∀I,V,K1,K2. TC … (lpx R) K1 K2 →
+ TC … (lpx R) (K1.ⓑ{I}V) (K2.ⓑ{I}V).
+#R #HR #I #V #K1 #K2 #H elim H -K2
+/4 width=5 by lpx_refl, lpx_pair, inj, step/ (**) (* too slow without trace *)
+qed.
+
+lemma lpx_TC: ∀R,L1,L2. TC … (lpx R) L1 L2 → lpx (TC … R) L1 L2.
+#R #L1 #L2 #H elim H -L2 /2 width=1/ /2 width=3/
+qed.
+
+lemma lpx_inv_TC: ∀R. reflexive ? R →
+ ∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2 /2 width=1/ /3 width=3/
+qed.