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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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15 include "static_2/s_computation/fqup.ma".
16 include "static_2/s_computation/fqus.ma".
18 (* STAR-ITERATED SUPCLOSURE *************************************************)
20 (* Alternative definition with plus-iterated supclosure *********************)
22 lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫.
23 #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
24 /3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
27 (* Basic_2A1: was: fqus_inv_gen *)
28 lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G2,L2,T2❫ →
29 ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
30 #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
31 #G #G2 #L #L2 #T #T2 #_ *
32 [ #H2 * /3 width=5 by fqup_strap1, or_introl/
33 * /3 width=1 by fqu_fqup, or_introl/
34 | * #HG #HL #HT destruct //
38 (* Advanced properties ******************************************************)
40 lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G,L,T❫ → ❪G,L,T❫ ⬂[b] ❪G2,L2,T2❫ →
41 ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫.
42 #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
43 [ /2 width=5 by fqup_strap1/
44 | * /2 width=1 by fqu_fqup/
48 lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ⬂[b] ❪G,L,T❫ → ❪G,L,T❫ ⬂*[b] ❪G2,L2,T2❫ →
49 ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫.
50 #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
51 [ /2 width=5 by fqup_strap2/
52 | * /2 width=1 by fqu_fqup/
56 lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ⬂*[b] ❪G,L,T❫ → ❪G,L,T❫ ⬂+[b] ❪G2,L2,T2❫ →
57 ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫.
58 #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
59 /2 width=5 by fqus_strap1_fqu, fqup_strap1/
62 lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G,L,T❫ →
63 ❪G,L,T❫ ⬂*[b] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫.
64 #b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
65 /3 width=5 by fqus_strap2_fqu, fqup_strap2/
68 (* Advanced inversion lemmas for plus-iterated supclosure *******************)
70 lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+[b] ❪G2,L2,T2❫ →
71 ∃∃G,L,T. ❪G1,L1,T1❫ ⬂[b] ❪G,L,T❫ & ❪G,L,T❫ ⬂*[b] ❪G2,L2,T2❫.
72 #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
73 #G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/