+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [dt, et - e] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ -Hdtd -Hidet
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+ | -Hdti -Hdedet
+ lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+ elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_1_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+ ]
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/