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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 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
17 include "basic_2/relocation/lifts_tdeq.ma".
18 include "basic_2/s_computation/fqus_fqup.ma".
19 include "basic_2/rt_transition/cpx_drops.ma".
21 (* Properties on supclosure *************************************************)
23 lemma fqu_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
24 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
25 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
26 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
27 /3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
28 [ #I #G #L2 #V2 #X2 #HVX2
29 elim (lifts_total X2 (𝐔❴1❵))
30 /3 width=3 by fqu_drop, cpx_delta, ex2_intro/
31 | #I #G #L2 #V #T2 #X2 #HTX2 #U2 #HTU2
32 elim (cpx_lifts_sn … HTU2 (Ⓣ) … (L2.ⓑ{I}V) … HTX2)
33 /3 width=3 by fqu_drop, drops_refl, drops_drop, ex2_intro/
37 lemma fquq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
38 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
39 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
40 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
41 [ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
42 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
46 lemma fqup_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
47 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
48 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
49 #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
50 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
51 /3 width=3 by fqu_fqup, ex2_intro/
52 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
53 elim (fqu_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
54 elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
58 lemma fqus_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
59 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
60 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
61 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
62 [ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
63 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
67 lemma fqu_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
68 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
69 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
70 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
71 [ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
72 #U2 #HVU2 @(ex3_intro … U2)
73 [1,3: /3 width=7 by cpx_delta, fqu_drop/
74 | #H lapply (tdeq_inv_lref1 … H) -H
75 #H destruct /2 width=5 by lifts_inv_lref2_uni_lt/
77 | #I #G #L #V1 #T #V2 #HV12 #H0 @(ex3_intro … (②{I}V2.T))
78 [1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
79 | #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
81 | #p #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓑ{p,I}V.T2))
82 [1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
83 | #H elim (tdeq_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, cpx_flat/
87 | #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
89 | #I #G #L #V #T1 #U1 #HTU1 #T2 #HT12 #H0
90 elim (cpx_lifts_sn … HT12 (Ⓣ) … (L.ⓑ{I}V) … HTU1) -HT12
91 /4 width=6 by fqu_drop, drops_refl, drops_drop, tdeq_inv_lifts_bi, ex3_intro/
95 lemma fquq_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
96 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
97 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
98 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
99 [ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_ntdeq … H12 … HTU2 H) -T2
100 /3 width=4 by fqu_fquq, ex3_intro/
101 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
105 lemma fqup_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
106 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
107 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
108 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
109 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_ntdeq … H12 … HTU2 H) -T2
110 /3 width=4 by fqu_fqup, ex3_intro/
111 | #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
112 #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_ntdeq … H1 … HTU1 H) -T1
113 /3 width=8 by fqup_strap2, ex3_intro/
117 lemma fqus_cpx_trans_ntdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
118 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≡[h, o] U2 → ⊥) →
119 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
120 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
121 [ #H12 elim (fqup_cpx_trans_ntdeq … H12 … HTU2 H) -T2
122 /3 width=4 by fqup_fqus, ex3_intro/
123 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/