matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_lift.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 "delayed_updating/substitution/fsubst.ma".
  16 include "delayed_updating/substitution/lift_prototerm_eq.ma".
  17 (*
  18 include "delayed_updating/syntax/preterm.ma".
  19 include "delayed_updating/syntax/prototerm_proper.ma".
  20 *)
  21 (* FOCALIZED SUBSTITUTION ***************************************************)
  22
  23 (* Constructions with lift for preterm **************************************)
  24
  25 lemma lift_term_fsubst_sn (f) (t) (u) (p):
  26       (↑[f]t)[⋔(↑[f]p)←↑[↑[p]f]u] ⊆ ↑[f](t[⋔p←u]).
  27 #f #t #u #p #ql * *
  28 [ #rl * #r #Hr #H1 #H2 destruct
  29   >lift_path_append
  30   /4 width=3 by in_comp_lift_path_term, or_introl, ex2_intro/
  31 | * #q #Hq #H1 #H0
  32   @(ex2_intro … H1) @or_intror @conj // -Hq #r #H2 destruct
  33   /2 width=2 by/
  34 ]
  35 qed-.
  36
  37 lemma lift_term_fsubst_dx (f) (t) (u) (p):
  38       ↑[f](t[⋔p←u]) ⊆ (↑[f]t)[⋔(↑[f]p)←↑[↑[p]f]u].
  39 #f #t #u #p #ql * #q * *
  40 [ #r #Hu #H1 #H2 destruct
  41   @or_introl @ex2_intro
  42   [|| <lift_path_append // ]
  43   /2 width=3 by ex2_intro/
  44 | #Hq #H0 #H1 destruct
  45   @or_intror @conj [ /2 width=1 by in_comp_lift_path_term/ ] -Hq #r #Hr
  46   elim (lift_path_inv_append_dx … Hr) -Hr #s1 #s2 #Hs1 #_ #H1 destruct
  47   lapply (lift_path_inj … Hs1) -Hs1 #H1 destruct
  48   /2 width=2 by/
  49 ]
  50 qed-.
  51
  52 lemma lift_term_fsubst (f) (t) (u) (p):
  53       (↑[f]t)[⋔(↑[f]p)←↑[↑[p]f]u] ⇔ ↑[f](t[⋔p←u]).
  54 /3 width=1 by lift_term_fsubst_sn, conj, lift_term_fsubst_dx/ qed.