matita/matita/contribs/lambdadelta/apps_2/models/deq_cpr.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 "basic_2/rt_transition/cpr.ma".
  16 include "apps_2/models/veq_lifts.ma".
  17 include "apps_2/models/deq.ma".
  18
  19 (* DENOTATIONAL EQUIVALENCE  ************************************************)
  20
  21 (* Forward lemmas with context-sensitive parallel reduction for terms *******)
  22
  23 lemma cpr_fwd_deq (h) (M): is_model M → is_extensional M →
  24                            ∀G,L,T1,T2. ❨G,L❩ ⊢ T1 ➡[h,0] T2 → ❨G,L❩ ⊢ T1 ≗{M} T2.
  25 #h #M #H1M #H2M #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
  26 [ /2 width=2 by deq_refl/
  27 | #G #K #V1 #V2 #W2 #_ #IH #HVW2 #gv #v #H
  28   elim (li_inv_abbr … H) -H // #lv #HK #H
  29   @(mq … H1M) [4,5: @(ti_comp … H) /2 width=2 by veq_refl/ |1,2: skip ] -v
  30   @(mq … H1M)
  31   [4: /3 width=1 by seq_sym, ml/ | skip
  32   |5: /2 width=2 by lifts_SO_fwd_vpush/ | skip ] -W2
  33   >vpush_eq /2 width=1 by/
  34 | #I #G #K #T #U #i #_ #IH #HTU #gv #v #H
  35   elim (li_fwd_bind … H) // -H #lv #d #HK #H
  36   @(mq … H1M) [4,5: @(ti_comp … H) /2 width=2 by veq_refl/ |1,2: skip ] -v
  37   @(mq … H1M)
  38   [4: /3 width=1 by seq_sym, ml/ | skip
  39   |5: /2 width=2 by lifts_SO_fwd_vpush/ | skip ] -U
  40   >vpush_gt /3 width=5 by ml, mq, mr/
  41 | #p * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #gv #lv #Hlv
  42   [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, md/ |1,2: skip ]
  43     @mc [3:|*: /2 width=1 by/ ] -p
  44     @(seq_trans … H1M) [2: @IHT /2 width=1 by li_abbr/ | skip ] -T1
  45     /4 width=1 by ti_comp, vpush_comp, (* 2x *) veq_refl/
  46   | /4 width=1 by li_abst, mx/
  47   ]
  48 | * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #gv #lv #Hlv
  49   [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, ma/ |1,2: skip ]
  50     /3 width=1 by mp/
  51   | @(mq … H1M) [4,5: /3 width=2 by seq_sym, me/ |1,2: skip ]
  52     /2 width=1 by/
  53   ]
  54 | #G #L #V #U1 #T1 #T2 #HTU1 #_ #IH #gv #lv #Hlv
  55   @(mq … H1M)
  56   [4: /3 width=2 by seq_sym, md/ | skip
  57   |3: @(seq_trans … H1M) [2: @mz // | skip ]
  58   ] /3 width=3 by lifts_SO_fwd_vpush, seq_sym/
  59 | #G #L #V #T1 #T2 #_ #IH #gv #lv #Hlv
  60   @(seq_trans … H1M) [2: @(me … H1M) | skip ]
  61   /2 width=1 by/
  62 | #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV #_ #IHT #gv #lv #Hlv
  63   @(mq … H1M) [4,5: /3 width=2 by seq_sym, ma, md/ |1,2: skip ]
  64   @(seq_trans … H1M) [3:|*: /2 width=2 by mb/ ]
  65   @mc // -p [ /4 width=5 by seq_trans, seq_sym, me/ ]
  66   @(seq_trans … H1M) [2: @IHT /2 width=1 by li_abst/ | skip ] -T1
  67   @ti_comp /2 width=1 by veq_refl/
  68   @vpush_comp /2 width=1 by veq_refl/
  69   /4 width=5 by seq_trans, seq_sym, me/
  70 | #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV #IHW #IHT #HV2 #gv #lv #Hlv
  71   @(mq … H1M) [4,5: /3 width=2 by seq_sym, ma, md/ |1,2: skip ]
  72   @(mq … H1M)
  73   [4: /4 width=2 by seq_sym, md, mp/ |1: skip
  74   |5: /4 width=2 by seq_sym, ma, mc/ |2: skip
  75   ]
  76   @(seq_trans … H1M) [2: @mh // | skip ]
  77   @mc [3:|*: /2 width=1 by mr/ ]
  78   @mp [3:|*: /2 width=1 by lifts_SO_fwd_vpush/ ]
  79   @(seq_trans … H1M) [2: @IHT /2 width=1 by li_abbr/ | skip ] -T1
  80   /4 width=1 by ti_comp, vpush_comp, (* 2x *) veq_refl/
  81 ]
  82 qed-.