-lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶* V2 →
- 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e.
- L1 [d, e] ▶* L2 → L2 ⊢ V1 [d, e] ▶* V2 →
- L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #H @(tpss_ind … H) -V2
-[ /2 width=1/
-| #V #V2 #_ #HV2 #IHV @(ltpss_trans_eq … IHV) /2 width=1/
+lemma ltpss_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
+ L1 [d1, e1] ▶* L0 → L0 ⊢ T2 [d2, e2] ▶* U2 →
+ ∃∃T. L1 ⊢ T2 [d2, e2] ▶* T & L0 ⊢ T [d1, e1] ▶* U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
+[ /2 width=3/
+| #U #U2 #_ #HU2 * #T #HT2 #HTU
+ elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
+ elim (ltpss_tps_trans … HTU … HL10) -HTU -HL10 #X #HTX #HXU
+ lapply (tpss_trans_eq … HXU HU2) -U /3 width=3/