+fact ltpss_tpss_trans_eq_aux: ∀Y1,X2,L1,T2,U2,d,e.
+ L1 ⊢ T2 ▶* [d, e] U2 → ∀L0. L0 ▶* [d, e] L1 →
+ Y1 = L1 → X2 = T2 → L0 ⊢ T2 ▶* [d, e] U2.
+#Y1 #X2 @(cw_wf_ind … Y1 X2) -Y1 -X2 #Y1 #X2 #IH
+#L1 #T2 #U2 #d #e #H @(tpss_ind_alt … H) -L1 -T2 -U2 -d -e
+[ //
+| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #HV12 #HVW2 #_ #L0 #HL01 #H1 #H2 destruct
+ lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
+ elim (ltpss_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
+ elim (ltpss_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
+ lapply (tpss_fwd_tw … HV01) #H2
+ lapply (transitive_le (#[K1] + #[V0]) … H1) -H1 /2 width=1/ -H2 #H
+ lapply (tpss_trans_eq … HV01 HV12) -V1 #HV02
+ lapply (IH … HV02 … HK01 ? ?) -IH -HV02 -HK01
+ [1,3: // |2,4: skip | normalize /2 width=1/ | /2 width=6/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #L0 #HL0 #H1 #H2 destruct
+ lapply (tpss_lsubs_conf … HT12 (L. ⓑ{I} V1) ?) -HT12 /2 width=1/ #HT12
+ lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=2/ ] #HV12
+ lapply (IH … HT12 (L0. ⓑ{I} V1) ? ? ?) -IH -HT12 [1,3,5: /2 width=2/ |2,4: skip | normalize // ] -HL0 #HT12
+ lapply (tpss_lsubs_conf … HT12 (L0. ⓑ{I} V2) ?) -HT12 /2 width=1/
+| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #L0 #HL0 #H1 #H2 destruct
+ lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=3/ ]
+ lapply (IH … HT12 … HL0 ? ?) -IH -HT12 [1,3,5: normalize // |2,4: skip ] -HL0 /2 width=1/
+]
+qed.
+
+lemma ltpss_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
+ ∀L0. L0 ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ltpss_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ▶* [d, e] L1 →
+ L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
+/3 width=3/ qed.
+