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] 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_vlift/ | skip ] -W2
  33   >vlift_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_vlift/ | skip ] -U
  40   >vlift_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 ] -p
  43     @(seq_trans … H1M) [2: @IHT /2 width=1 by li_abbr/ | skip ] -T1
  44     /4 width=1 by ti_comp, vlift_comp, (* 2x *) veq_refl/
  45   | /4 width=1 by li_abst, mx/
  46   ]
  47 | * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #gv #lv #Hlv
  48   [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, ma/ |1,2: skip ]
  49     /3 width=1 by mp/
  50   | @(mq … H1M) [4,5: /3 width=2 by seq_sym, me/ |1,2: skip ]
  51     /2 width=1 by/
  52   ]
  53 | #G #L #V #U1 #U2 #T2 #_ #IH #HTU2 #gv #lv #Hlv
  54   @(mq … H1M)
  55   [4: /3 width=2 by seq_sym, md/ | skip
  56   |3: /3 width=1 by li_abbr/ | skip ] -L -U1
  57   /3 width=1 by lifts_SO_fwd_vlift, seq_sym/
  58 | #G #L #V #T1 #T2 #_ #IH #gv #lv #Hlv
  59   @(seq_trans … H1M) [2: @(me … H1M) | skip ]
  60   /2 width=1 by/
  61 | #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV #_ #IHT #gv #lv #Hlv
  62   @(mq … H1M) [4,5: /3 width=2 by seq_sym, ma, md/ |1,2: skip ]
  63   @(mq … H1M)
  64   [4: /3 width=2 by seq_sym, mb/ | skip
  65   |5: @IHT /2 width=1 by li_abst/ | skip ] -T2
  66   @ti_comp /2 width=1 by veq_refl/
  67   @vlift_comp /2 width=1 by veq_refl/
  68   @(mq … H1M) [4,5: /3 width=2 by seq_sym, me/ |1,2: skip ] -L -V1
  69   /2 width=1 by mr/
  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   @(seq_trans … H1M) [3: /3 width=1 by seq_sym, ma/ | skip ]
  73   @mp // [ @(seq_trans … H1M) /3 width=3 by lifts_SO_fwd_vlift/ ] -V1 -V -V2
  74   @(mq … H1M)
  75   [4: /3 width=2 by seq_sym, md/ | skip
  76   |3: @IHT /2 width=1 by li_abbr/ | skip ] -T1
  77   /4 width=1 by ti_comp, vlift_comp, (* 2x *) veq_refl/
  78 ]
  79 qed-.