1 (* Note: this does not hold since L = Y.ⓧ, U = #0, f = ⫯g requires T = #(-1) *)
2 lemma frees_inv_drops: ∀f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
3 ∀f,K. ⬇*[Ⓣ, f] L ≡ K → ∀f1. f ~⊚ f1 ≡ f2 →
4 ∃∃T. K ⊢ 𝐅*⦃T⦄ ≡ f1 & ⬆*[f] T ≡ U.
5 #f2 #L #U #H lapply (frees_fwd_isfin … H) elim H -f2 -L -U
6 [ #f2 #I #Hf2 #_ #f #K #H1 #f1 #H2
7 lapply (coafter_fwd_isid2 … H2 ??) -H2 // -Hf2 #Hf1
8 elim (drops_inv_atom1 … H1) -H1 #H #Hf destruct
9 /4 width=3 by frees_atom, lifts_refl, ex2_intro/
10 | #f2 #I #L #s #_ #IH #Hf2 #f #Y #H1 #f1 #H2
11 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
12 elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
13 [ #g #g1 #Hf2 #H #H0 destruct
14 elim (drops_inv_skip1 … H1) -H1 #Z #K #HLK #_ #H destruct
16 lapply (drops_inv_drop1 … H1) -H1 #HLK
18 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
19 lapply (lifts_inv_sort2 … HX) -HX #H destruct
20 /3 width=3 by frees_sort, lifts_sort, ex2_intro/
21 | #f2 #I #L #W #_ #IH #Hf2 #f #Y #H1 #f1 #H2
22 lapply (isfin_inv_next … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
23 elim (coafter_inv_xxn … H2) -H2 [ |*: // ] #g #g1 #Hf2 #H0 #H destruct
24 elim (drops_inv_skip1 … H1) -H1 #J #K #HLK #HJI #H destruct
25 elim (liftsb_inv_pair_dx … HJI) -HJI #V #HVW #H destruct
26 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
27 lapply (lifts_inj … HX … HVW) -W #H destruct
28 /3 width=3 by frees_zero, lifts_lref, ex2_intro/
29 | #f2 #L #Hf2 #_ #f #Y #H1 #f1 #H2
30 lapply (coafter_fwd_isid2 … H2 ??) -H2 // -Hf2 #Hf1
31 elim (pn_split f) * #g #H destruct
32 [ elim (drops_inv_skip1 … H1) -H1 #J #K #HLK #HJI #H destruct
33 lapply (liftsb_inv_unit_dx … HJI) -HJI #H destruct
34 /3 width=3 by frees_void, lifts_lref, ex2_intro/
35 | lapply (drops_inv_drop1 … H1) -H1 #H1
36 | #f2 #I #L #j #_ #IH #Hf2 #f #Y #H1 #f1 #H2
37 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
38 elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
39 [ #g #g1 #Hf2 #H #H0 destruct
40 elim (drops_inv_skip1 … H1) -H1 #J #K #HLK #_ #H destruct
42 lapply (drops_inv_drop1 … H1) -H1 #HLK (* cannot continue *)
44 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
45 elim (lifts_inv_lref2 … HX) -HX #i #Hij #H destruct
46 /4 width=7 by frees_lref, lifts_lref, at_S1, at_next, ex2_intro/
47 | #f2 #I #L #l #_ #IH #Hf2 #f #Y #H1 #f1 #H2
48 lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
49 elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
50 [ #g #g1 #Hf2 #H #H0 destruct
51 elim (drops_inv_skip1 … H1) -H1 #J #K #HLK #_ #H destruct
53 lapply (drops_inv_drop1 … H1) -H1 #HLK
55 elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
56 lapply (lifts_inv_gref2 … HX) -HX #H destruct
57 /3 width=3 by frees_gref, lifts_gref, ex2_intro/
58 | #f2W #f2U #f2 #p #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
59 elim (sor_inv_isfin3 … H1f2) // #H1f2W #H
60 lapply (isfin_inv_tl … H) -H #H1f2U
61 elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H #Hf1
62 elim (coafter_inv_tl0 … H) -H #g1 #H2f2U #H destruct
63 elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W #V #Hf1W #HVW
64 elim (IHU … H2f2U) -IHU -H2f2U
65 /3 width=5 by frees_bind, drops_skip, lifts_bind, ext2_pair, ex2_intro/
66 | #f2W #f2U #f2 #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
67 elim (sor_inv_isfin3 … H1f2) // #H1f2W #H1f2U
68 elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H2f2U #Hf1
69 elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W
70 elim (IHU … H1 … H2f2U) -L -H2f2U
71 /3 width=5 by frees_flat, lifts_flat, ex2_intro/