matita/matita/contribs/lambdadelta/delayed_updating/substitution/fsubst_lift.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "delayed_updating/substitution/fsubst.ma".
  16 include "delayed_updating/substitution/lift_prototerm_eq.ma".
  17
  18 (* FOCALIZED SUBSTITUTION ***************************************************)
  19
  20 (* Constructions with lift for preterm **************************************)
  21
  22 lemma lift_term_fsubst_sn (f) (t) (u) (p):
  23       (🠡[f]t)[⋔(🠡[f]p)←🠡[🠢[f]p]u] ⊆ 🠡[f](t[⋔p←u]).
  24 #f #t #u #p #ql * *
  25 [ #rl * #r #Hr #H1 #H2 destruct
  26   >lift_path_append
  27   /4 width=3 by in_comp_lift_path_term, or_introl, ex2_intro/
  28 | * #q #Hq #H1 #H0
  29   @(ex2_intro … H1) @or_intror @conj // -Hq #r #H2 destruct
  30   /2 width=2 by/
  31 ]
  32 qed-.
  33
  34 lemma lift_term_fsubst_dx (f) (t) (u) (p):
  35       🠡[f](t[⋔p←u]) ⊆ (🠡[f]t)[⋔(🠡[f]p)←🠡[🠢[f]p]u].
  36 #f #t #u #p #ql * #q * *
  37 [ #r #Hu #H1 #H2 destruct
  38   @or_introl @ex2_intro
  39   [|| <lift_path_append // ]
  40   /2 width=3 by ex2_intro/
  41 | #Hq #H0 #H1 destruct
  42   @or_intror @conj [ /2 width=1 by in_comp_lift_path_term/ ] -Hq #r #Hr
  43   elim (lift_path_inv_append_sn … Hr) -Hr #s1 #s2 #Hs1 #_ #H1 destruct
  44   lapply (lift_path_inj … Hs1) -Hs1 #H1 destruct
  45   /2 width=2 by/
  46 ]
  47 qed-.
  48
  49 lemma lift_term_fsubst (f) (t) (u) (p):
  50       (🠡[f]t)[⋔(🠡[f]p)←🠡[🠢[f]p]u] ⇔ 🠡[f](t[⋔p←u]).
  51 /3 width=1 by lift_term_fsubst_sn, conj, lift_term_fsubst_dx/ qed.