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14
15 include "basic_2/substitution/lleq_leq.ma".
16 include "basic_2/reduction/llpx_lleq.ma".
17 include "basic_2/computation/cpxs_lleq.ma".
18 include "basic_2/computation/llpxs_cpxs.ma".
19
20 (* LAZY SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS *************)
21
22 (* Properties on lazy equivalence for local environments ********************)
23
24 lemma llpxs_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L2.
25 /3 width=1 by llpx_lrefl, llpx_llpxs/ qed-.
26
27 lemma lleq_llpxs_trans: ∀h,g,G,L2,L,T,d. ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L →
28                         ∀L1. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L.
29 #h #g #G #L2 #L #T #d #H @(llpxs_ind … H) -L
30 /3 width=3 by llpxs_strap1, llpxs_lrefl/
31 qed-.
32
33 lemma lleq_llpxs_conf: ∀h,g,G,L1,L,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g, T, d] L →
34                        ∀L2. L1 ⋕[T, d] L2 → ⦃G, L2⦄ ⊢ ➡*[h, g, T, d] L.
35 /3 width=3 by lleq_llpxs_trans, lleq_sym/ qed-.
36 (*
37 foct leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
38                               ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
39                               ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
40                                    (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
41 #h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
42 [ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
43   /3 width=5 by ex3_intro, conj/
44 | #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
45 | #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
46   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
47   lapply (ysucc_inv_Y_dx … He) -He #He
48   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
49   @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
50   #T elim (IH T) #HL0dx #HL0sn
51   @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
52 | #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
53   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
54   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
55   @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
56   #T elim (IH T) #HL0dx #HL0sn
57   @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
58 ]
59 qed-.
60
61 lamma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
62                            ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
63                            ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
64                                 (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
65 /2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
66 *)