matita/matita/contribs/lambdadelta/apps_2/models/veq_lifts.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 "ground_2/relocation/rtmap_basic.ma".
  16 include "static_2/relocation/lifts.ma".
  17 include "apps_2/models/veq.ma".
  18
  19 (* EVALUATION EQUIVALENCE  **************************************************)
  20
  21 (* Forward lemmas with generic relocation ***********************************)
  22
  23 fact lifts_fwd_vlift_aux (M): is_model M → is_extensional M →
  24                               ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀m. 𝐁❴m,1❵ = f →
  25                               ∀gv,lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[m←d]lv].
  26 #M #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
  27 [ #f #s #m #Hf #gv #lv #d
  28   @(mq … H1M) [4,5: /3 width=2 by seq_sym, ms/ |1,2: skip ]
  29   /2 width=1 by mr/
  30 | #f #i1 #i2 #Hi12 #m #Hm #gv #lv #d destruct
  31   @(mq … H1M) [4,5: /3 width=2 by seq_sym, ml/ |1,2: skip ]
  32   elim (lt_or_ge i1 m) #Hi1
  33   [ lapply (at_basic_inv_lt … Hi12) -Hi12 // #H destruct
  34     >vlift_lt /2 width=1 by mr/
  35   | lapply (at_basic_inv_ge … Hi12) -Hi12 // #H destruct
  36     >vlift_gt /2 width=1 by mr, le_S_S/
  37   ]
  38 | #f #l #m #Hf #gv #lv #d
  39   @(mq … H1M) [4,5: /3 width=2 by seq_sym, mg/ |1,2: skip ]
  40   /2 width=1 by mr/
  41 | #f #p * #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #m #Hm #gv #lv #d destruct
  42   [ @(mq … H1M) [4,5: /3 width=2 by seq_sym, md/ |1,2: skip ]
  43     @(seq_trans … H1M)
  44     [3: @ti_comp // | skip ]
  45     [1,2: /2 width=2 by veq_refl/ ]
  46     [2: @(vlift_comp … H1M) | skip ]
  47     [1,2: /2 width=2 by/ |3,4: /2 width=2 by veq_refl/ ] -IHV
  48     @(seq_trans … H1M)
  49     [3: @ti_comp // | skip ]
  50     [1,2: /2 width=2 by veq_refl/ ]
  51     [2: @veq_sym // @vlift_swap // | skip ]
  52     /2 width=1 by/
  53   | @mx // [ /2 width=1 by/ ] -IHV #d0
  54     @(seq_trans … H1M)
  55     [3: @ti_comp // | skip ]
  56     [1,2: /2 width=2 by veq_refl/ ]
  57     [2: @veq_sym // @vlift_swap // | skip ]
  58     /2 width=1 by/
  59   ]
  60 | #f * #V1 #v2 #T1 #T2 #_ #_ #IHV #IHT #m #Hm #gv #lv #d
  61   [ /4 width=5 by seq_sym, ma, mp, mq/
  62   | /4 width=5 by seq_sym, me, mq/
  63   ]
  64 ]
  65 qed-.
  66
  67 lemma lifts_SO_fwd_vlift (M) (gv): is_model M → is_extensional M →
  68                                    ∀T1,T2. ⬆*[1] T1 ≘ T2 →
  69                                    ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[0←d]lv].
  70 /2 width=3 by lifts_fwd_vlift_aux/ qed-.