matita/matita/contribs/lambdadelta/basic_2/static/lsubf_frees.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/static/lsubf.ma".
  16
  17 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
  18
  19 (* Properties with context-sensitive free variables *************************)
  20
  21 lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 → ∀f,L1. ⦃L1, f⦄ ⫃𝐅* ⦃L2, f2⦄ →
  22                          ∃∃f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 & f1 ⊆ f.
  23 #f2 #L2 #T #H elim H -f2 -L2 -T
  24 [ #f2 #I #Hf2 #f #L1 #H elim (lsubf_inv_atom2 … H) -H
  25   #H #_ destruct /3 width=3 by frees_atom, sle_isid_sn, ex2_intro/
  26 | #f2 #I #K2 #W #s #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  27   [ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
  28   destruct elim (IH … H12) -K2
  29   /3 width=3 by frees_sort, sle_inv_tl_dx, ex2_intro/
  30 | #f2 #I #K2 #W #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  31   [ #K1 #H elim (sle_inv_nx … H ??) -H [ <tl_next_rew |*: // ]
  32     #g2 #_ #H1 #H12 #H2 destruct elim (IH … H12) -K2
  33     /3 width=7 by frees_zero, sle_next, ex2_intro/
  34   | #g #K1 #V #Hg #Hf #H elim (sle_inv_nx … H ??) -H [ <tl_next_rew |*: // ]
  35     #g2 #_ #H1 #H12 #H2 #H3 destruct elim (IH … H12) -K2
  36     #f1 #Hf1 elim (sor_isfin_ex … f1 g ??)
  37     /4 width=7 by frees_fwd_isfin, frees_flat, frees_zero, sor_inv_sle, sle_next, ex2_intro/
  38   ]
  39 | #f2 #I #K2 #W #i #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  40   [ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
  41   destruct elim (IH … H12) -K2
  42   /3 width=3 by frees_lref, sle_inv_tl_dx, ex2_intro/
  43 | #f2 #I #K2 #W #l #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  44   [ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
  45   destruct elim (IH … H12) -K2
  46   /3 width=3 by frees_gref, sle_inv_tl_dx, ex2_intro/
  47 | #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f #L1 #H12
  48   elim (IHV f L1) -IHV [2: /3 width=4 by lsubf_sle_div, sor_inv_sle_sn/ ]
  49   elim (IHT (⫯f) (L1.ⓑ{I}V)) -IHT [2: /4 width=4 by lsubf_sle_div, lsubf_pair_nn, sor_inv_sle_dx, sor_inv_tl_dx/ ]
  50   -f2V -f2T -f2 -L2 #f1T #HT #Hf1T #f1V #HV #Hf1V elim (sor_isfin_ex … f1V (⫱f1T) ??)
  51   /4 width=9 by frees_fwd_isfin, frees_bind, sor_inv_sle, sle_xn_tl, isfin_tl, ex2_intro/
  52 | #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f #L1 #H12
  53   elim (IHV f L1) -IHV [2: /3 width=4 by lsubf_sle_div, sor_inv_sle_sn/ ]
  54   elim (IHT f L1) -IHT [2: /3 width=4 by lsubf_sle_div, sor_inv_sle_dx/ ]
  55   -f2V -f2T -f2 -L2 #f1T #HT #Hf1T #f1V #HV #Hf1V elim (sor_isfin_ex … f1V f1T ??)
  56   /3 width=7 by frees_fwd_isfin, frees_flat, sor_inv_sle, ex2_intro/
  57 ]
  58 qed-.