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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
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15 include "basic_2/relocation/lexs_length.ma".
16 include "basic_2/static/lfxs_drops.ma".
18 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
20 (* Forward lemmas with length for local environments ************************)
22 (* Basic_2A1: uses: llpx_sn_fwd_length *)
23 lemma lfxs_fwd_length (R): ∀L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|.
24 #R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/
27 (* Properties with length for local environments ****************************)
29 (* Basic_2A1: uses: llpx_sn_sort *)
30 lemma lfxs_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤*[R, ⋆s] L2.
32 [ #Y #H #s >(length_inv_zero_sn … H) -H //
33 | #K1 #I1 #IH #Y #H #s
34 elim (length_inv_succ_sn … H) -H #I2 #K2 #HK12 #H destruct
35 /3 width=1 by lfxs_sort/
39 (* Basic_2A1: uses: llpx_sn_gref *)
40 lemma lfxs_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤*[R, §l] L2.
42 [ #Y #H #s >(length_inv_zero_sn … H) -H //
43 | #K1 #I1 #IH #Y #H #s
44 elim (length_inv_succ_sn … H) -H #I2 #K2 #HK12 #H destruct
45 /3 width=1 by lfxs_gref/
49 lemma lfxs_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
50 /3 width=3 by lfxs_unit, lexs_length_isid/ qed.
52 (* Basic_2A1: uses: llpx_sn_lift_le llpx_sn_lift_ge *)
53 lemma lfxs_lifts_bi (R): d_liftable2_sn … lifts R →
54 ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤*[R, T] K2 →
55 ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
56 ∀U. ⬆*[f] T ≘ U → L1 ⪤*[R, U] L2.
57 #R #HR #L1 #L2 #HL12 #K1 #K2 #T * #f1 #Hf1 #HK12 #b #f #HLK1 #HLK2 #U #HTU
58 elim (frees_total L1 U) #f2 #Hf2
59 lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
60 /4 width=12 by lexs_length_cfull, lexs_liftable_co_dedropable_bi, cext2_d_liftable2_sn, cfull_lift_sn, ex2_intro/
63 (* Inversion lemmas with length for local environment ***********************)
65 lemma lfxs_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤*[R, #0] Y2 →
67 | ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
68 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
69 | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
70 #R #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
71 /4 width=9 by lexs_fwd_length, ex4_5_intro, ex3_3_intro, or3_intro2, or3_intro1, or3_intro0, conj/