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15 include "basic_2/substitution/lpx_sn_drop.ma".
16 include "basic_2/multiple/llpx_sn.ma".
18 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
20 (* Properties on pointwise extensions ***************************************)
22 lemma lpx_sn_llpx_sn: ∀R. (∀L. reflexive … (R L)) →
23 ∀T,L1,L2,l. lpx_sn R L1 L2 → llpx_sn R l T L1 L2.
24 #R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
26 [ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
27 | -HR #i elim (lt_or_ge i (|L1|))
28 [2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
29 #Hi #Hx #L2 #l elim (ylt_split i l)
30 [ -x /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
31 #Hli #HL12 elim (drop_O1_lt (Ⓕ) L1 i) //
32 #I #K1 #V1 #HLK1 elim (lpx_sn_drop_conf … HL12 … HLK1) -HL12
33 /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
34 | -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
35 | /4 width=1 by llpx_sn_bind, lpx_sn_pair/
36 | -HR /3 width=1 by llpx_sn_flat/