+(* Note: this does not hold since L = Y.β§, U = #0, f = β«―g requires T = #(-1) *)
+lemma frees_inv_drops: βf2,L,U. L β’ π
*β¦Uβ¦ β‘ f2 β
+ βf,K. β¬*[β, f] L β‘ K β βf1. f ~β f1 β‘ f2 β
+ ββT. K β’ π
*β¦Tβ¦ β‘ f1 & β¬*[f] T β‘ U.
+#f2 #L #U #H lapply (frees_fwd_isfin β¦ H) elim H -f2 -L -U
+[ #f2 #I #Hf2 #_ #f #K #H1 #f1 #H2
+ lapply (coafter_fwd_isid2 β¦ H2 ??) -H2 // -Hf2 #Hf1
+ elim (drops_inv_atom1 β¦ H1) -H1 #H #Hf destruct
+ /4 width=3 by frees_atom, lifts_refl, ex2_intro/
+| #f2 #I #L #s #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_fwd_push β¦ Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxp β¦ H2) -H2 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 β¦ H1) -H1 #Z #K #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 β¦ H1) -H1 #HLK
+ ]
+ elim (IH β¦ HLK β¦ Hf2) -L // -f2 #X #Hg1 #HX
+ lapply (lifts_inv_sort2 β¦ HX) -HX #H destruct
+ /3 width=3 by frees_sort, lifts_sort, ex2_intro/
+| #f2 #I #L #W #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_inv_next β¦ Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxn β¦ H2) -H2 [ |*: // ] #g #g1 #Hf2 #H0 #H destruct
+ elim (drops_inv_skip1 β¦ H1) -H1 #J #K #HLK #HJI #H destruct
+ elim (liftsb_inv_pair_dx β¦ HJI) -HJI #V #HVW #H destruct
+ elim (IH β¦ HLK β¦ Hf2) -L // -f2 #X #Hg1 #HX
+ lapply (lifts_inj β¦ HX β¦ HVW) -W #H destruct
+ /3 width=3 by frees_zero, lifts_lref, ex2_intro/
+| #f2 #L #Hf2 #_ #f #Y #H1 #f1 #H2
+ lapply (coafter_fwd_isid2 β¦ H2 ??) -H2 // -Hf2 #Hf1
+ elim (pn_split f) * #g #H destruct
+ [ elim (drops_inv_skip1 β¦ H1) -H1 #J #K #HLK #HJI #H destruct
+ lapply (liftsb_inv_unit_dx β¦ HJI) -HJI #H destruct
+ /3 width=3 by frees_void, lifts_lref, ex2_intro/
+ | lapply (drops_inv_drop1 β¦ H1) -H1 #H1
+| #f2 #I #L #j #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_fwd_push β¦ Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxp β¦ H2) -H2 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 β¦ H1) -H1 #J #K #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 β¦ H1) -H1 #HLK (* cannot continue *)
+ ]
+ elim (IH β¦ HLK β¦ Hf2) -L // -f2 #X #Hg1 #HX
+ elim (lifts_inv_lref2 β¦ HX) -HX #i #Hij #H destruct
+ /4 width=7 by frees_lref, lifts_lref, at_S1, at_next, ex2_intro/
+| #f2 #I #L #l #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+ lapply (isfin_fwd_push β¦ Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+ elim (coafter_inv_xxp β¦ H2) -H2 [1,3: * |*: // ]
+ [ #g #g1 #Hf2 #H #H0 destruct
+ elim (drops_inv_skip1 β¦ H1) -H1 #J #K #HLK #_ #H destruct
+ | #g #Hf2 #H destruct
+ lapply (drops_inv_drop1 β¦ H1) -H1 #HLK
+ ]
+ elim (IH β¦ HLK β¦ Hf2) -L // -f2 #X #Hg1 #HX
+ lapply (lifts_inv_gref2 β¦ HX) -HX #H destruct
+ /3 width=3 by frees_gref, lifts_gref, ex2_intro/
+| #f2W #f2U #f2 #p #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
+ elim (sor_inv_isfin3 β¦ H1f2) // #H1f2W #H
+ lapply (isfin_inv_tl β¦ H) -H #H1f2U
+ elim (coafter_inv_sor β¦ H2 β¦ H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H #Hf1
+ elim (coafter_inv_tl0 β¦ H) -H #g1 #H2f2U #H destruct
+ elim (IHW β¦ H1 β¦ H2f2W) -IHW -H2f2W // -H1f2W #V #Hf1W #HVW
+ elim (IHU β¦ H2f2U) -IHU -H2f2U
+ /3 width=5 by frees_bind, drops_skip, lifts_bind, ext2_pair, ex2_intro/
+| #f2W #f2U #f2 #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
+ elim (sor_inv_isfin3 β¦ H1f2) // #H1f2W #H1f2U
+ elim (coafter_inv_sor β¦ H2 β¦ H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H2f2U #Hf1
+ elim (IHW β¦ H1 β¦ H2f2W) -IHW -H2f2W // -H1f2W
+ elim (IHU β¦ H1 β¦ H2f2U) -L -H2f2U
+ /3 width=5 by frees_flat, lifts_flat, ex2_intro/
+]
+qed-.