matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqup.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/s_computation/fqup.ma".
  16 include "basic_2/s_computation/fqus.ma".
  17
  18 (* STAR-ITERATED SUPCLOSURE *************************************************)
  19
  20 (* Alternative definition with plus-iterated supclosure *********************)
  21
  22 lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
  23 #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: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
  29                      ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
  30 #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 * /2 width=1 by or_introl/
  35   * /2 width=1 by or_intror/
  36 ]
  37 qed-.
  38
  39 (* Advanced properties ******************************************************)
  40
  41 lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
  42                        ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
  43 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
  44 [ /2 width=5 by fqup_strap1/
  45 | * /2 width=1 by fqu_fqup/
  46 ]
  47 qed-.
  48
  49 lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
  50                        ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
  51 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
  52 [ /2 width=5 by fqup_strap2/
  53 | * /2 width=1 by fqu_fqup/
  54 ]
  55 qed-.
  56
  57 lemma fqus_fqup_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ →
  58                        ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
  59 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
  60 /2 width=5 by fqus_strap1_fqu, fqup_strap1/
  61 qed-.
  62
  63 lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ →
  64                        ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
  65 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
  66 /3 width=5 by fqus_strap2_fqu, fqup_strap2/
  67 qed-.
  68
  69 (* Advanced inversion lemmas for plus-iterated supclosure *******************)
  70
  71 lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
  72                         ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
  73 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
  74 #G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
  75 qed-.