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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "ground_2/ynat/ynat_plus.ma".
16 include "basic_2/substitution/drop.ma".
18 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
20 | llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i →
21 ⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
22 llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2
24 (* Basic forward lemmas *****************************************************)
26 lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
27 ∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2.
28 #R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
29 #HL12 lapply (drop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by drop_O1_le/
32 lemma llpx_sn_fwd_drop_dx: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
33 ∀K2,i. ⬇[i] L2 ≡ K2 → ∃K1. ⬇[i] L1 ≡ K1.
34 #R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length … H) -H
35 #HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/
38 lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
39 ∨∨ |L1| ≤ i ∧ |L2| ≤ i
41 | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
43 llpx_sn R (yinj 0) V1 K1 K2 &
44 R K1 V1 V2 & l ≤ yinj i.
45 /2 width=3 by llpx_sn_fwd_lref_aux/ qed-.