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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 *)
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11 (* v GNU General Public License Version 2 *)
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15 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
17 (* Properties on supclosure *************************************************)
19 lemma fqu_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
20 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
21 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
22 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
23 /3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
24 [ #I #G #L #V2 #U2 #HVU2
25 elim (lift_total U2 0 1)
26 /4 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop, ex2_intro/
27 | #G #L #K #T1 #U1 #k #HLK1 #HTU1 #T2 #HTU2
28 elim (lift_total T2 0 (k+1))
29 /3 width=11 by cpx_lift, fqu_drop, ex2_intro/
33 lemma fquq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
34 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
35 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
36 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
37 [ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
38 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
42 lemma fqup_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
43 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
44 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
45 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
46 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
47 /3 width=3 by fqu_fqup, ex2_intro/
48 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
49 elim (fqu_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
50 elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
54 lemma fqus_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
55 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 →
56 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
57 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
58 [ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
59 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
63 lemma fqu_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
64 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
65 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
66 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
67 [ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
68 #U2 #HVU2 @(ex3_intro … U2)
69 [1,3: /3 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop/
71 lapply (lift_inv_lref2_be … HVU2 ? ?) -HVU2 //
73 | #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
74 [1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
75 | #H0 destruct /2 width=1 by/
77 | #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
78 [1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
79 | #H0 destruct /2 width=1 by/
81 | #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
82 [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
83 | #H0 destruct /2 width=1 by/
85 | #G #L #K #T1 #U1 #k #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (k+1))
86 #U2 #HTU2 @(ex3_intro … U2)
87 [1,3: /2 width=10 by cpx_lift, fqu_drop/
88 | #H0 destruct /3 width=5 by lift_inj/
92 lemma fquq_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
93 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
94 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
95 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
96 [ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
97 /3 width=4 by fqu_fquq, ex3_intro/
98 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
102 lemma fqup_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
103 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
104 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
105 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
106 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
107 /3 width=4 by fqu_fqup, ex3_intro/
108 | #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
109 #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_neq … H1 … HTU1 H) -T1
110 /3 width=8 by fqup_strap2, ex3_intro/
114 lemma fqus_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
115 ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
116 ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
117 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
118 [ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
119 /3 width=4 by fqup_fqus, ex3_intro/
120 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/