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15 include "basic_2/reducibility/tpr_tpss.ma".
17 (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
19 (* Properties concerning parallel unfold on local environments **************)
21 lemma ltpr_ltps_conf: ∀L1,K1,d,e. L1 [d, e] ▶ K1 → ∀L2. L1 ➡ L2 →
22 ∃∃K2. L2 [d, e] ▶* K2 & K1 ➡ K2.
23 #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
26 elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
27 | #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
28 elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
29 elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
30 elim (tpr_tps_ltpr … HV12 … HVW1 … HK12) -V1 /3 width=5/
31 | #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
32 elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
33 elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
34 elim (tpr_tps_ltpr … HV12 … HVW1 … HK12) -V1 /3 width=5/
38 lemma ltpr_ltpss_conf: ∀L1,K1,d,e. L1 [d, e] ▶* K1 → ∀L2. L1 ➡ L2 →
39 ∃∃K2. L2 [d, e] ▶* K2 & K1 ➡ K2.
40 #L1 #K1 #d #e #H @(ltpss_ind … H) -K1 /2 width=3/
41 #K #K1 #_ #HK1 #IHK #L2 #HL12
42 elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
43 elim (ltpr_ltps_conf … HK1 … HK2) -K #K #HK2 #HK1
44 lapply (ltpss_trans_eq … HLK2 HK2) -K2 /2 width=3/