matita/matita/contribs/lambdadelta/static_2/s_computation/fqus_fqup.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "static_2/s_computation/fqup.ma".
  16 include "static_2/s_computation/fqus.ma".
  17
  18 (* STAR-ITERATED SUPCLOSURE *************************************************)
  19
  20 (* Alternative definition with plus-iterated supclosure *********************)
  21
  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/
  25 qed.
  26
  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 //
  35 ]
  36 qed-.
  37
  38 (* Advanced properties ******************************************************)
  39
  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/
  45 ]
  46 qed-.
  47
  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/
  53 ]
  54 qed-.
  55
  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/
  60 qed-.
  61
  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/
  66 qed-.
  67
  68 (* Advanced inversion lemmas for plus-iterated supclosure *******************)
  69
  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/
  74 qed-.