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15 include "basic_2/substitution/lleq_leq.ma".
16 include "basic_2/reduction/lpx_lleq.ma".
17 include "basic_2/computation/cpxs_leq.ma".
18 include "basic_2/computation/lpxs_ldrop.ma".
19 include "basic_2/computation/lpxs_cpxs.ma".
21 (* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
23 (* Properties on lazy equivalence for local environments ********************)
25 lemma lleq_lpxs_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
26 ∀L1,T,d. L1 ⋕[T, d] L2 →
27 ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ⋕[T, d] K2.
28 #h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
29 #K #K2 #_ #HK2 #IH #L1 #T #d #HT elim (IH … HT) -L2
30 #L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K
31 /3 width=3 by lpxs_strap1, ex2_intro/
34 lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
35 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
36 ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
37 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
38 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
39 #K0 #V0 #H1KL1 #_ #H destruct
40 elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
41 #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
42 /2 width=4 by fqu_lref_O, ex3_intro/
43 | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
44 [ elim (lleq_inv_bind … H)
45 | elim (lleq_inv_flat … H)
46 ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
47 | #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
48 /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
49 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
50 /2 width=4 by fqu_flat_dx, ex3_intro/
51 | #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
52 elim (ldrop_O1_le (e+1) K1)
53 [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
54 #H2KL elim (lpxs_ldrop_trans_O1 … H1KL1 … HL1) -L1
55 #K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct
56 /3 width=4 by fqu_drop, ex3_intro/
57 | lapply (ldrop_fwd_length_le2 … HL1) -L -T1 -g
58 lapply (lleq_fwd_length … H2KL1) //
63 lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
64 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
65 ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
66 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
67 elim (fquq_inv_gen … H) -H
68 [ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
69 /3 width=4 by fqu_fquq, ex3_intro/
70 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
74 lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
75 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
76 ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
77 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
78 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
79 /3 width=4 by fqu_fqup, ex3_intro/
80 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
81 #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
82 /3 width=5 by fqup_strap1, ex3_intro/
86 lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
87 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
88 ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
89 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
90 elim (fqus_inv_gen … H) -H
91 [ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
92 /3 width=4 by fqup_fqus, ex3_intro/
93 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
97 fact leq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
98 ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
99 ∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
100 (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
101 #h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
102 [ #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
103 /3 width=5 by ex3_intro, conj/
104 | #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
105 | #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
106 elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
107 lapply (ysucc_inv_Y_dx … He) -He #He
108 elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
109 @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
110 #T elim (IH T) #HL0dx #HL0sn
111 @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
112 | #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
113 elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
114 elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
115 @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
116 #T elim (IH T) #HL0dx #HL0sn
117 @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
121 lemma leq_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
122 ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
123 ∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
124 (∀T. L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
125 /2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.