matita/matita/contribs/lambdadelta/basic_2/etc_2A1/nta/nta_aaa.etc

   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/computation/csn_aaa.ma".
  16 include "basic_2/equivalence/lcpcs_aaa.ma".
  17 include "basic_2/dynamic/nta.ma".
  18
  19 (* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************)
  20
  21 (* Forward lemmas on atomic arity assignment for terms **********************)
  22
  23 lemma nta_fwd_aaa: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃∃A. L ⊢ T ⁝ A & L ⊢ U ⁝ A.
  24 #h #L #T #U #H elim H -L -T -U
  25 [ /2 width=3/
  26 | #L #K #V #W #U #i #HLK #_ #HWU * #B #HV #HW
  27   lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/
  28 | #L #K #W #V #U #i #HLK #_ #HWU * #B #HW #_ -V
  29   lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/
  30 | * #L #V #W #T #U #_ #_ * #B #HV #HW * #A #HT #HU
  31   [ /3 width=3/ | /3 width=5/ ]
  32 | #L #V #W #T #U #_ #_ * #B #HV #HW * #X #H1 #H2
  33   elim (aaa_inv_abst … H1) -H1 #B1 #A1 #HW1 #HT #H destruct
  34   elim (aaa_inv_abst … H2) -H2 #B2 #A #_ #HU #H destruct
  35   lapply (aaa_mono … HW1 … HW) -HW1 #H destruct /4 width=5/
  36 | #L #V #W #T #U #_ #_ * #X #HT #HUX * #A #H #_ -W
  37   elim (aaa_inv_appl … H) -H #B #HV #HUA
  38   lapply (aaa_mono … HUA … HUX) -HUX #H destruct /3 width=5/
  39 | #L #T #U #_ * #A #HT #HU /3 width=3/
  40 | #L #T #U1 #U2 #V2 #_ #HU12 #_ * #X #HT #HU1 * #A #HU2 #_
  41   lapply (aaa_cpcs_mono … HU12 … HU1 … HU2) -U1 #H destruct /2 width=3/
  42 ]
  43 qed-.
  44
  45 (* Note: this is the stong normalization property *)
  46 (* Basic_1: was only: ty3_sn3 *)
  47 theorem nta_fwd_csn: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → L ⊢ ⬇* T ∧ L ⊢ ⬇* U.
  48 #h #L #T #U #H elim (nta_fwd_aaa … H) -H /3 width=2/
  49 qed-.