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14
15 include "Basic-2/reduction/tpr_lift.ma".
16 include "Basic-2/reduction/ltpr.ma".
17
18 (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
19
20 (* Basic-1: was: wcpr0_drop *)
21 lemma ltpr_drop_conf: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀L2. L1 ⇒ L2 →
22                       ∃∃K2. ↓[d, e] L2 ≡ K2 & K1 ⇒ K2.
23 #L1 #K1 #d #e #H elim H -H L1 K1 d e
24 [ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/
25 | #K1 #I #V1 #X #H
26   elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
27 | #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
28   elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X;
29   elim (IHLK1 … HL12) -IHLK1 HL12 /3/
30 | #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
31   elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X;
32   elim (tpr_inv_lift … HV12 … HWV1) -HV12 HWV1;
33   elim (IHLK1 … HL12) -IHLK1 HL12 /3 width=5/
34 ]
35 qed.
36
37 (* Basic-1: was: wcpr0_drop_back *)
38 lemma ltpr_drop_trans: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀K2. K1 ⇒ K2 →
39                        ∃∃L2. ↓[d, e] L2 ≡ K2 & L1 ⇒ L2.
40 #L1 #K1 #d #e #H elim H -H L1 K1 d e
41 [ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/
42 | #K1 #I #V1 #X #H
43   elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
44 | #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
45   elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/
46 | #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
47   elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct -X;
48   elim (lift_total W2 d e) #V2 #HWV2
49   lapply (tpr_lift … HW12 … HWV1 … HWV2) -HW12 HWV1;
50   elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/
51 ]
52 qed.