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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2A/multiple/lleq_lleq.ma".
16 include "basic_2A/reduction/lpx_lleq.ma".
17 include "basic_2A/computation/cpxs_lreq.ma".
18 include "basic_2A/computation/lpxs_drop.ma".
19 include "basic_2A/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,l. L1 ≡[T, l] L2 →
27 ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ≡[T, l] 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 #l #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_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) →
35 ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, g] L & L1 ≡[T, l] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L0 & L0 ≡[T, l] L2.
36 #h #g #G #L1 #L2 #T #l #H @(lpxs_ind_dx … H) -L1
38 | #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L l) #H
39 [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lleq_trans/ -H12
40 #L0 #L3 #H1 #H2 #H3 #H4 lapply (lleq_nlleq_trans … H … H2) -H2
41 #H2 elim (lleq_lpx_trans … H1 … H) -L
42 #L #H1 #H lapply (nlleq_lleq_div … H … H2) -H2
43 #H2 elim (lleq_lpxs_trans … H3 … H) -L0
44 /3 width=8 by lleq_trans, ex4_2_intro/
45 | -H12 -IH2 /3 width=6 by ex4_2_intro/
50 lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
51 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
52 ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
53 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
54 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
55 #K0 #V0 #H1KL1 #_ #H destruct
56 elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
57 #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
58 /2 width=4 by fqu_lref_O, ex3_intro/
59 | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
60 [ elim (lleq_inv_bind … H)
61 | elim (lleq_inv_flat … H)
62 ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
63 | #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
64 /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
65 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
66 /2 width=4 by fqu_flat_dx, ex3_intro/
67 | #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
68 elim (drop_O1_le (Ⓕ) (m+1) K1)
69 [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
70 #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
71 #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
72 /3 width=4 by fqu_drop, ex3_intro/
73 | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
74 lapply (lleq_fwd_length … H2KL1) //
79 lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
80 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
81 ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
82 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
83 elim (fquq_inv_gen … H) -H
84 [ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
85 /3 width=4 by fqu_fquq, ex3_intro/
86 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
90 lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
91 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
92 ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
93 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
94 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
95 /3 width=4 by fqu_fqup, ex3_intro/
96 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
97 #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
98 /3 width=5 by fqup_strap1, ex3_intro/
102 lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
103 ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
104 ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
105 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
106 elim (fqus_inv_gen … H) -H
107 [ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
108 /3 width=4 by fqup_fqus, ex3_intro/
109 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
113 fact lreq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ →
114 ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
115 ∃∃L. L ⩬[l, m] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
116 (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
117 #h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
118 [ #l #m #_ #L2 #H >(lpxs_inv_atom1 … H) -H
119 /3 width=5 by ex3_intro, conj/
120 | #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
121 | #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
122 elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
123 lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
124 elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
125 @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_pair/
126 #T elim (IH T) #HL0dx #HL0sn
127 @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
128 | #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
129 elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
130 elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
131 @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
132 #T elim (IH T) #HL0dx #HL0sn
133 @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
137 lemma lreq_lpxs_trans_lleq: ∀h,g,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
138 ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
139 ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
140 (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
141 /2 width=1 by lreq_lpxs_trans_lleq_aux/ qed-.