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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS *************)
17 include "basic_2/rt_transition/cpx_fqus.ma".
18 include "basic_2/rt_computation/cpxs_drops.ma".
19 include "basic_2/rt_computation/cpxs_lsubr.ma".
20 include "basic_2/rt_computation/cpxs_cpxs.ma".
22 (* Properties on supclosure *************************************************)
24 lemma fqu_cpxs_trans (b):
25 ∀G1,G2,L1,L2,T2,U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 →
26 ∀T1. ❨G1,L1,T1❩ ⬂[b] ❨G2,L2,T2❩ →
27 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & ❨G1,L1,U1❩ ⬂[b] ❨G2,L2,U2❩.
28 #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
29 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
30 #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
33 lemma fquq_cpxs_trans (b):
34 ∀G1,G2,L1,L2,T2,U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 →
35 ∀T1. ❨G1,L1,T1❩ ⬂⸮[b] ❨G2,L2,T2❩ →
36 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & ❨G1,L1,U1❩ ⬂⸮[b] ❨G2,L2,U2❩.
37 #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
38 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fquq_cpx_trans … HT1 … HT2) -T
39 #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
42 lemma fqup_cpxs_trans (b):
43 ∀G1,G2,L1,L2,T2,U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 →
44 ∀T1. ❨G1,L1,T1❩ ⬂+[b] ❨G2,L2,T2❩ →
45 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & ❨G1,L1,U1❩ ⬂+[b] ❨G2,L2,U2❩.
46 #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
47 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
48 #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
51 lemma fqus_cpxs_trans (b):
52 ∀G1,G2,L1,L2,T2,U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 →
53 ∀T1. ❨G1,L1,T1❩ ⬂*[b] ❨G2,L2,T2❩ →
54 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & ❨G1,L1,U1❩ ⬂*[b] ❨G2,L2,U2❩.
55 #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
56 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqus_cpx_trans … HT1 … HT2) -T
57 #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
60 (* Note: a proof based on fqu_cpx_trans_tneqx might exist *)
61 (* Basic_2A1: uses: fqu_cpxs_trans_neq *)
62 lemma fqu_cpxs_trans_tneqg (S) (b):
63 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂[b] ❨G2,L2,T2❩ →
64 ∀U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
65 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & T1 ≛[S] U1 → ⊥ & ❨G1,L1,U1❩ ⬂[b] ❨G2,L2,U2❩.
66 #S #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
67 [ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❨1❩)
68 #U2 #HVU2 @(ex3_intro … U2)
69 [1,3: /3 width=7 by cpxs_delta, fqu_drop/
70 | #H lapply (teqg_inv_lref1 … H) -H
71 #H destruct /2 width=5 by lifts_inv_lref2_uni_lt/
73 | #I #G #L #V1 #T #V2 #HV12 #H0 @(ex3_intro … (②[I]V2.T))
74 [1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
75 | #H elim (teqg_inv_pair … H) -H /2 width=1 by/
77 | #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
78 [1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
79 | #H elim (teqg_inv_pair … H) -H /2 width=1 by/
81 | #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
82 [1,3: /4 width=4 by lsubr_cpxs_trans, cpxs_bind, lsubr_unit, fqu_clear/
83 | #H elim (teqg_inv_pair … H) -H /2 width=1 by/
85 | #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓕ[I]V.T2))
86 [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
87 | #H elim (teqg_inv_pair … H) -H /2 width=1 by/
89 | #I #G #L #T1 #U1 #HTU1 #T2 #HT12 #H0
90 elim (cpxs_lifts_sn … HT12 (Ⓣ) … (L.ⓘ[I]) … HTU1) -HT12
91 /4 width=6 by fqu_drop, drops_refl, drops_drop, teqg_inv_lifts_bi, ex3_intro/
95 (* Basic_2A1: uses: fquq_cpxs_trans_neq *)
96 lemma fquq_cpxs_trans_tneqg (S) (b):
97 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂⸮[b] ❨G2,L2,T2❩ →
98 ∀U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
99 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & T1 ≛[S] U1 → ⊥ & ❨G1,L1,U1❩ ⬂⸮[b] ❨G2,L2,U2❩.
100 #S #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
101 [ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tneqg … H12 … HTU2 H) -T2
102 /3 width=4 by fqu_fquq, ex3_intro/
103 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
107 (* Basic_2A1: uses: fqup_cpxs_trans_neq *)
108 lemma fqup_cpxs_trans_tneqg (S) (b):
109 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂+[b] ❨G2,L2,T2❩ →
110 ∀U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
111 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & T1 ≛[S] U1 → ⊥ & ❨G1,L1,U1❩ ⬂+[b] ❨G2,L2,U2❩.
112 #S #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
113 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tneqg … H12 … HTU2 H) -T2
114 /3 width=4 by fqu_fqup, ex3_intro/
115 | #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
116 #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_tneqg … H1 … HTU1 H) -T1
117 /3 width=8 by fqup_strap2, ex3_intro/
121 (* Basic_2A1: uses: fqus_cpxs_trans_neq *)
122 lemma fqus_cpxs_trans_tneqg (S) (b):
123 ∀G1,G2,L1,L2,T1,T2. ❨G1,L1,T1❩ ⬂*[b] ❨G2,L2,T2❩ →
124 ∀U2. ❨G2,L2❩ ⊢ T2 ⬈* U2 → (T2 ≛[S] U2 → ⊥) →
125 ∃∃U1. ❨G1,L1❩ ⊢ T1 ⬈* U1 & T1 ≛[S] U1 → ⊥ & ❨G1,L1,U1❩ ⬂*[b] ❨G2,L2,U2❩.
126 #S #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
127 [ #H12 elim (fqup_cpxs_trans_tneqg … H12 … HTU2 H) -T2
128 /3 width=4 by fqup_fqus, ex3_intro/
129 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/