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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-TRANSITION FOR TERMS **************)
17 include "static_2/relocation/lifts_teqx.ma".
18 include "static_2/s_computation/fqus_fqup.ma".
19 include "basic_2/rt_transition/cpx_drops.ma".
20 include "basic_2/rt_transition/cpx_lsubr.ma".
22 (* Properties on supclosure *************************************************)
24 lemma fqu_cpx_trans (b):
25 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
26 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
27 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂[b] ❪G2,L2,U2❫.
28 #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
29 /3 width=3 by cpx_pair_sn, cpx_bind, cpx_flat, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex2_intro/
30 [ #I #G #L2 #V2 #X2 #HVX2
31 elim (lifts_total X2 (𝐔❨1❩))
32 /3 width=3 by fqu_drop, cpx_delta, ex2_intro/
33 | /5 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit, fqu_clear, ex2_intro/
34 | #I #G #L2 #T2 #X2 #HTX2 #U2 #HTU2
35 elim (cpx_lifts_sn … HTU2 (Ⓣ) … (L2.ⓘ[I]) … HTX2)
36 /3 width=3 by fqu_drop, drops_refl, drops_drop, ex2_intro/
40 lemma fquq_cpx_trans (b):
41 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂⸮[b] ❪G2,L2,T2❫ →
42 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
43 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂⸮[b] ❪G2,L2,U2❫.
44 #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
45 [ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
46 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
50 lemma fqup_cpx_trans (b):
51 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ →
52 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
53 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂+[b] ❪G2,L2,U2❫.
54 #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
55 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
56 /3 width=3 by fqu_fqup, ex2_intro/
57 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
58 elim (fqu_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
59 elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
63 lemma fqus_cpx_trans (b):
64 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫ →
65 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 →
66 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & ❪G1,L1,U1❫ ⬂*[b] ❪G2,L2,U2❫.
67 #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
68 [ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
69 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
73 lemma fqu_cpx_trans_tneqx (b):
74 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂[b] ❪G2,L2,T2❫ →
75 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
76 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂[b] ❪G2,L2,U2❫.
77 #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
78 [ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❨1❩)
79 #U2 #HVU2 @(ex3_intro … U2)
80 [1,3: /3 width=7 by cpx_delta, fqu_drop/
81 | #H lapply (teqx_inv_lref1 … H) -H
82 #H destruct /2 width=5 by lifts_inv_lref2_uni_lt/
84 | #I #G #L #V1 #T #V2 #HV12 #H0 @(ex3_intro … (②[I]V2.T))
85 [1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
86 | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
88 | #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
89 [1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
90 | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
92 | #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ[p,I]V.T2))
93 [1,3: /4 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit, fqu_clear/
94 | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
96 | #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓕ[I]V.T2))
97 [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
98 | #H elim (teqx_inv_pair … H) -H /2 width=1 by/
100 | #I #G #L #T1 #U1 #HTU1 #T2 #HT12 #H0
101 elim (cpx_lifts_sn … HT12 (Ⓣ) … (L.ⓘ[I]) … HTU1) -HT12
102 /4 width=6 by fqu_drop, drops_refl, drops_drop, teqx_inv_lifts_bi, ex3_intro/
106 lemma fquq_cpx_trans_tneqx (b):
107 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂⸮[b] ❪G2,L2,T2❫ →
108 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
109 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂⸮[b] ❪G2,L2,U2❫.
110 #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
111 [ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tneqx … H12 … HTU2 H) -T2
112 /3 width=4 by fqu_fquq, ex3_intro/
113 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
117 lemma fqup_cpx_trans_tneqx (b):
118 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ →
119 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
120 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂+[b] ❪G2,L2,U2❫.
121 #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
122 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tneqx … H12 … HTU2 H) -T2
123 /3 width=4 by fqu_fqup, ex3_intro/
124 | #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
125 #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_tneqx … H1 … HTU1 H) -T1
126 /3 width=8 by fqup_strap2, ex3_intro/
130 lemma fqus_cpx_trans_tneqx (b):
131 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫ →
132 ∀U2. ❪G2,L2❫ ⊢ T2 ⬈ U2 → (T2 ≛ U2 → ⊥) →
133 ∃∃U1. ❪G1,L1❫ ⊢ T1 ⬈ U1 & T1 ≛ U1 → ⊥ & ❪G1,L1,U1❫ ⬂*[b] ❪G2,L2,U2❫.
134 #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
135 [ #H12 elim (fqup_cpx_trans_tneqx … H12 … HTU2 H) -T2
136 /3 width=4 by fqup_fqus, ex3_intro/
137 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/