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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "static_2/static/reqx_req.ma".
16 include "basic_2/rt_transition/rpx_reqx.ma".
17 include "basic_2/rt_transition/rpx_lpx.ma".
19 (* EXTENDED PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS **************)
21 (* Properties with sort-irrelevant equivalence for local environments *******)
23 lemma reqx_lpx_trans_rpx (G) (L) (T:term):
24 ∀L1. L1 ≛[T] L → ∀L2. ❪G,L❫ ⊢ ⬈ L2 → ❪G,L❫ ⊢ ⬈[T] L2.
25 /3 width=1 by lpx_rpx, reqx_rpx_trans/ qed.
27 (* Basic_2A1: uses: lleq_lpx_trans *)
28 lemma reqx_lpx_trans (G):
29 ∀L2,K2. ❪G,L2❫ ⊢ ⬈ K2 → ∀L1. ∀T:term. L1 ≛[T] L2 →
30 ∃∃K1. ❪G,L1❫ ⊢ ⬈ K1 & K1 ≛[T] K2.
31 #G #L2 #K2 #HLK2 #L1 #T #HL12
32 lapply (lpx_rpx … T … HLK2) -HLK2 #HLK2
33 lapply (reqx_rpx_trans … HL12 … HLK2) -L2 #H
34 elim (rpx_fwd_lpx_req … H) -H #K1 #HLK1 #HK12
35 /3 width=3 by req_reqx, ex2_intro/
38 (* Inversion lemmas with sort-irrelevant equivalence for local environments *)
40 lemma rpx_inv_reqx_lpx (G) (T):
41 ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T] L2 →
42 ∃∃L. L1 ≛[T] L & ❪G,L❫ ⊢ ⬈ L2.
44 elim (rpx_inv_req_lpx … H) -H #L #HL1 #HL2
45 /3 width=3 by req_reqx, ex2_intro/
48 (* Forward lemmas with sort-irrelevant equivalence for local environments ***)
50 lemma rpx_fwd_lpx_reqx (G) (T):
51 ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T] L2 →
52 ∃∃L. ❪G,L1❫ ⊢ ⬈ L & L ≛[T] L2.
54 elim (rpx_fwd_lpx_req … H) -H #L #HL1 #HL2
55 /3 width=3 by req_reqx, ex2_intro/