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15 include "basic_2/substitution/llneq_alt.ma".
16 include "basic_2/computation/lpxs_ldrop.ma".
17 include "basic_2/computation/lpxs_cpxs.ma".
19 (* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
21 (* Forward lemmas on negated lazy equivalence for local environments ********)
23 lemma lpxs_llneq_fwd_cpxs: ∀h,g,G,L1,L2,T,d. L1 ⧣⧣[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
24 ∃∃T1,T2. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] T1 & ⦃G, L2⦄ ⊢ T ▶*[d, ∞] T2 &
25 ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & (T1 = T2 → ⊥).
26 #h #g #G #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
27 [ #I1 #I2 #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #_ #HnV12 #HL12
28 elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HY
29 elim (lpxs_inv_pair1 … H) -H #Z #X #_ #HV12 #H destruct
30 lapply (ldrop_mono … HY … HLK2) -HY #H destruct
31 lapply (ldrop_fwd_drop2 … HLK1) #H2LK1
32 elim (lift_total V1 0 (i+1)) #T1 #HVT1
33 elim (lift_total V2 0 (i+1)) #T2 #HVT2
34 @(ex4_2_intro … T1 T2) /3 width=10 by cpxs_lift, cpys_subst, lift_inj/ (**) (* explicit constructor *)
35 | #I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #_ #IHK12 #HL12
36 elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #Y #H #HY
37 elim (lpxs_inv_pair1 … H) -H #Z #X #HK12 #_ #H destruct
38 lapply (ldrop_mono … HY … HLK2) -HY #H destruct
39 elim (IHK12 HK12) -IHK12 -HK12 #V1 #V2 #HV1 #HV2 #HV12 #HnV12
40 lapply (ldrop_fwd_drop2 … HLK1) #H2LK1
41 elim (lift_total V1 0 (i+1)) #T1 #HVT1
42 elim (lift_total V2 0 (i+1)) #T2 #HVT2
43 @(ex4_2_intro … T1 T2) /3 width=10 by cpxs_lift, cpys_subst_Y2, lift_inj/ (**) (* explicit constructor *)
44 | #a #I #L1 #L2 #V #T #d #_ #IHV #HL12 elim (IHV HL12) -IHV -HL12
45 #V1 #V2 #HV1 #HV2 #HV12 #HnV12
46 @(ex4_2_intro … (ⓑ{a,I}V1.T) (ⓑ{a,I}V2.T)) /2 width=1 by cpys_bind, cpxs_pair_sn/
47 #H destruct /2 width=1 by/
48 | #a #I #L1 #L2 #V #T #d #_ #IHT #HL12 elim (IHT ?) /2 width=1 by lpxs_pair_refl/ -IHT -HL12
49 #T1 #T2 #HT1 #HT2 #HT12 #HnT12
50 @(ex4_2_intro … (ⓑ{a,I}V.T1) (ⓑ{a,I}V.T2)) /2 width=1 by cpys_bind, cpxs_bind_dx/
51 #H destruct /2 width=1 by/
52 | #I #L1 #L2 #V #T #d #_ #IHV #HL12 elim (IHV HL12) -IHV -HL12
53 #V1 #V2 #HV1 #HV2 #HV12 #HnV12
54 @(ex4_2_intro … (ⓕ{I}V1.T) (ⓕ{I}V2.T)) /2 width=1 by cpys_flat, cpxs_pair_sn/
55 #H destruct /2 width=1 by/
56 | #I #L1 #L2 #V #T #d #_ #IHT #HL12 elim (IHT HL12) -IHT -HL12
57 #T1 #T2 #HT1 #HT2 #HT12 #HnT12
58 @(ex4_2_intro … (ⓕ{I}V.T1) (ⓕ{I}V.T2)) /2 width=1 by cpys_flat, cpxs_flat_dx/
59 #H destruct /2 width=1 by/
63 lemma lpxs_nlleq_fwd_cpxs: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
65 ∃∃T1,T2. ⦃G, L1⦄ ⊢ T ▶*[d, ∞] T1 & ⦃G, L2⦄ ⊢ T ▶*[d, ∞] T2 &
66 ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & (T1 = T2 → ⊥).
67 /5 width=4 by lpxs_llneq_fwd_cpxs, lpxs_fwd_length, llneq_llneqa, conj/ qed-.