matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbg_fpbs.ma

   1 (**************************************************************************)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "static_2/static/feqg_fqup.ma".
  16 include "static_2/static/feqg_feqg.ma".
  17 include "basic_2/rt_transition/fpbq_fpb.ma".
  18 include "basic_2/rt_computation/fpbs_fqup.ma".
  19 include "basic_2/rt_computation/fpbg.ma".
  20
  21 (* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
  22
  23 (* Advanced forward lemmas **************************************************)
  24
  25 lemma fpbg_fwd_fpbs:
  26       ∀G1,G2,L1,L2,T1,T2.
  27       ❪G1,L1,T1❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
  28 #G1 #G2 #L1 #L2 #T1 #T2 *
  29 /3 width=5 by fpbs_strap2, fpb_fpbq/
  30 qed-.
  31
  32 (* Advanced properties with sort-irrelevant equivalence on closures *********)
  33
  34 (* Basic_2A1: uses: fleq_fpbg_trans *)
  35 lemma feqx_fpbg_trans:
  36       ∀G,G2,L,L2,T,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ →
  37       ∀G1,L1,T1. ❪G1,L1,T1❫ ≅ ❪G,L,T❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
  38 #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
  39 elim (feqg_fpb_trans …  H1 … H0) -G -L -T
  40 /4 width=9 by fpbs_strap2, fpbq_feqx, ex2_3_intro/
  41 qed-.
  42
  43 (* Properties with parallel proper rst-reduction on closures ****************)
  44
  45 lemma fpb_fpbg_trans:
  46       ∀G1,G,G2,L1,L,L2,T1,T,T2.
  47       ❪G1,L1,T1❫ ≻ ❪G,L,T❫ → ❪G,L,T❫ > ❪G2,L2,T2❫ →
  48       ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
  49 /3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
  50
  51 (* Properties with parallel rst-reduction on closures ***********************)
  52
  53 lemma fpbq_fpbg_trans:
  54       ∀G1,G,G2,L1,L,L2,T1,T,T2.
  55       ❪G1,L1,T1❫ ≽ ❪G,L,T❫ → ❪G,L,T❫ > ❪G2,L2,T2❫ →
  56       ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
  57 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
  58 elim (fpbq_inv_fpb … H1) -H1
  59 /2 width=5 by feqx_fpbg_trans, fpb_fpbg_trans/
  60 qed-.
  61
  62 (* Properties with parallel rst-compuutation on closures ********************)
  63
  64 lemma fpbs_fpbg_trans:
  65       ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ →
  66       ∀G2,L2,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
  67 #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
  68 qed-.
  69
  70 (* Advanced properties with plus-iterated structural successor for closures *)
  71
  72 lemma fqup_fpbg_trans:
  73       ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ⬂+ ❪G,L,T❫ →
  74       ∀G2,L2,T2. ❪G,L,T❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
  75 /3 width=5 by fpbs_fpbg_trans, fqup_fpbs/ qed-.
  76
  77 (* Advanced inversion lemmas of parallel rst-computation on closures ********)
  78
  79 (* Basic_2A1: was: fpbs_fpbg *)
  80 lemma fpbs_inv_fpbg:
  81       ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ →
  82       ∨∨ ❪G1,L1,T1❫ ≅ ❪G2,L2,T2❫
  83        | ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
  84 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
  85 [ /3 width=1 by feqg_refl, or_introl/
  86 | #G #G2 #L #L2 #T #T2 #_ #H2 * #H1
  87   elim (fpbq_inv_fpb … H2) -H2 #H2
  88   [ /3 width=5 by feqg_trans, or_introl/
  89   | elim (feqg_fpb_trans … H1 … H2) -G -L -T
  90     /4 width=5 by ex2_3_intro, or_intror, feqx_fpbs/
  91   | /3 width=5 by fpbg_feqx_trans, or_intror/
  92   | /4 width=5 by fpbg_fpbq_trans, fpb_fpbq, or_intror/
  93   ]
  94 ]
  95 qed-.
  96
  97 (* Advanced properties of parallel rst-computation on closures **************)
  98
  99 lemma fpbs_fpb_trans:
 100       ∀F1,F2,K1,K2,T1,T2. ❪F1,K1,T1❫ ≥ ❪F2,K2,T2❫ →
 101       ∀G2,L2,U2. ❪F2,K2,T2❫ ≻ ❪G2,L2,U2❫ →
 102       ∃∃G1,L1,U1. ❪F1,K1,T1❫ ≻ ❪G1,L1,U1❫ & ❪G1,L1,U1❫ ≥ ❪G2,L2,U2❫.
 103 #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
 104 [ #H12 #G2 #L2 #U2 #H2 elim (feqg_fpb_trans … H12 … H2) -F2 -K2 -T2
 105   /3 width=5 by feqx_fpbs, ex2_3_intro/
 106 | * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
 107   @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
 108 ]
 109 qed-.