matita/matita/contribs/lambdadelta/apps_2/models/vpushs_fold.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 "static_2/syntax/fold.ma".
  16 include "apps_2/models/vpushs.ma".
  17
  18 (* MULTIPLE PUSH FOR MODEL EVALUATIONS **************************************)
  19
  20 (* Properties with fold for restricted closures *****************************)
  21
  22 lemma vpushs_fold (M): is_model M → is_extensional M →
  23                        ∀L,T1,T2,gv,lv.
  24                        (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v]) →
  25                        ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv].
  26 #M #H1M #H2M #L elim L -L [| #K * [| * ]]
  27 [ #T1 #T2 #gv #lv #H12
  28   >fold_atom >fold_atom
  29   /4 width=1 by vpushs_atom, veq_refl/
  30 | #I #IH #T1 #T2 #gv #lv #H12
  31   >fold_unit >fold_unit
  32   /5 width=1 by vpushs_unit, mx, mr/
  33 | #V #IH #T1 #T2 #gv #lv #H12
  34   >fold_pair >fold_pair
  35   @IH -IH #v #Hv
  36   @(mq … H1M) [3:|*: /3 width=2 by seq_sym, md/ ]
  37   /4 width=1 by vpushs_abbr, mc, mr/
  38 | #W #IH #T1 #T2 #gv #lv #H12
  39   >fold_pair >fold_pair
  40   /5 width=1 by vpushs_abst, mx, mr/
  41 ]
  42 qed.
  43
  44 (* Inversion lemmas with fold for restricted closures ***********************)
  45
  46 lemma vpushs_inv_fold (M): is_model M → is_injective M →
  47                            ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv] →
  48                            ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v].
  49 #M #H1M #H2M #L elim L -L [| #K * [| * ]]
  50 [ #T1 #T2 #gv #lv
  51   >fold_atom >fold_atom #H12 #v #H
  52   lapply (vpushs_inv_atom … H) -H // #Hv
  53   /4 width=7 by ti_comp, veq_refl, mq/
  54 | #I #IH #T1 #T2 #gv #lv
  55   >fold_unit >fold_unit #H12 #y #H
  56   elim (vpushs_inv_unit … H) -H // #v #d #Hlv #Hv
  57   lapply (IH … H12 … Hlv) -IH -H12 -Hlv #H12
  58   /4 width=7 by ti_comp, ti_fwd_mx_dx, veq_refl, mq/
  59 | #V #IH #T1 #T2 #gv #lv
  60   >fold_pair >fold_pair #H12 #y #H
  61   elim (vpushs_inv_abbr … H) -H // #v #Hlv #Hv
  62   lapply (IH … H12 … Hlv) -IH -H12 -Hlv #H12
  63   @(mq … H1M) [4,5: @(ti_comp … Hv) /3 width=2 by veq_refl/ |1,2: skip ]
  64   /4 width=7 by ti_comp, ti_fwd_abbr_dx, veq_refl, mq/
  65 | #W #IH #T1 #T2 #gv #lv
  66   >fold_pair >fold_pair #H12 #y #H
  67   elim (vpushs_inv_abst … H) -H // #v #d #Hlv #Hv
  68   lapply (IH … H12 … Hlv) -IH -H12 -Hlv #H12
  69   /4 width=7 by ti_comp, ti_fwd_mx_dx, veq_refl, mq/
  70 ]
  71 qed-.