matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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/computation/csx_aaa.ma".
  16 include "basic_2/computation/cpds_aaa.ma".
  17 include "basic_2/equivalence/cpcs_aaa.ma".
  18 include "basic_2/dynamic/snv.ma".
  19
  20 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
  21
  22 (* Forward lemmas on atomic arity assignment for terms **********************)
  23
  24 lemma snv_fwd_aaa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
  25 #h #g #G #L #T #H elim H -G -L -T
  26 [ /2 width=2/
  27 | #I #G #L #K #V #i #HLK #_ * /3 width=6/
  28 | #a * #G #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2/
  29 | #a #G #L #V #W #W0 #T #U #l #_ #_ #Hl #HVW #HW0 #HTU * #B #HV * #X #HT
  30   lapply (aaa_cpds_conf h g … HV W0 ?) [ -HTU /3 width=4/ ] -W #HW0 (**) (* auto fail without -HTU *)
  31   lapply (aaa_cpds_conf … HT … HTU) -HTU #H
  32   elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct
  33   lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4/
  34 | #G #L #W #T #U #l #_ #_ #_ #HTU #HUW * #B #HW * #A #HT
  35   lapply (aaa_ssta_conf … HT … HTU) -HTU #H
  36   lapply (aaa_cpcs_mono … HUW … H … HW) -HUW -H #H destruct /3 width=3/
  37 ]
  38 qed-.
  39
  40 lemma snv_fwd_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
  41 #h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
  42 qed-.
  43
  44 (* Advanced forward lemmas **************************************************)
  45
  46 lemma snv_fwd_da: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
  47 #h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_fwd_da/
  48 qed-.
  49
  50 lemma snv_fwd_ssta: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃U. ⦃G, L⦄ ⊢ T •[h, g] U.
  51 #h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2 by aaa_fwd_ssta/
  52 qed-.
  53
  54 lemma snv_lsstas_fwd_correct: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ¡[h, g] → ⦃G, L⦄ ⊢ T1 •* [h, g, l] T2 →
  55                               ∃U2. ⦃G, L⦄ ⊢ T2 •[h, g] U2.
  56 #h #g #G #L #T1 #T2 #l #HT1 #HT12
  57 elim (snv_fwd_ssta … HT1) -HT1 /2 width=5 by lsstas_fwd_correct/
  58 qed-.
  59
  60 (* Advanced properties ******************************************************)
  61
  62 lemma snv_scast: ∀h,g,G,L,V,W,l. ⦃G, L⦄ ⊢ V ¡[h, g] → ⦃G, L⦄ ⊢ W ¡[h, g] →
  63                  scast h g l G L V W → ⦃G, L⦄ ⊢V ▪[h, g] l+1 → ⦃G, L⦄ ⊢ ⓝW.V ¡[h, g].
  64 #h #g #G #L #V #W #l #HV #HW #H #Hl
  65 elim (snv_fwd_ssta … HV) /4 width=6 by snv_cast, ssta_lsstas/
  66 qed-.