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Datei: tzm.xml Sprache: PVS

Original von: PVS^©

%%-------------------** Term Rewriting System (TRS) **------------------------
%%
%% Authors         : Andre Luiz Galdino
%%                   Universidade Federal de Goiás - Brasil
%%
%%                         and
%%
%%                   Mauricio Ayala Rincon
%%                   Universidade de Brasília - Brasil
%%
%% Last Modified On: September 29, 2009
%%
%%----------------------------------------------------------------------------

replace_positions[variable:TYPE+, symbol: TYPE+, arity: [symbol -> nat]]: THEORY

BEGIN

   ASSUMING

     IMPORTING variables_term[variable,symbol, arity],
               sets_aux@countability[term],
               sets_aux@countable_props[term]

       var_countable: ASSUMPTION is_countably_infinite(V)

   ENDASSUMING

     IMPORTING substitution[variable,symbol, arity]

          s, t, t1, t2: VAR term
     sig, delta, alpha: VAR Sub
                     x: VAR (V)
                     f: VAR symbol
                  args: VAR finite_sequence[term]

%%%% Definition: replacing all subterms of s occurring in a sequence of %%%%%%%
%%%% parallel positions by t.                                           %%%%%%%

replace_pos(s, t, (fssp:SPP(s)) ): RECURSIVE term =
    IF length(fssp) = 0
    THEN
     s
    ELSE
     replace_pos(replaceTerm(s, t, fssp(0)), t, rest(fssp))
    ENDIF
  MEASURE length(fssp)

%%%% Properties %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% compatibility with substitution
replace_pos_is_stable: LEMMA
   FORALL t, s, (fss: SPP(s)), sig:
     ext(sig)(replace_pos(s, t, fss)) = replace_pos(ext(sig)(s),ext(sig)(t),fss)

% Preserves first replaced position
preserv_first_position: LEMMA
   FORALL t, s, (fss: SPP(s)), (q: positions?(s)):
       (FORALL (i: below[length(fss)]): parallel(fss(i),q))
           =>
            positionsOF(replace_pos(s, t, add_first(q, fss)))(q)

% Preserves all parallel positions
preserv_all_parallel_positions: LEMMA
   FORALL t, s, (fss: SPP(s)), (q: positions?(s)):
       (FORALL (i: below[length(fss)]): parallel(fss(i),q))
           =>
            positionsOF(replace_pos(s, t, fss))(q)

% Preserves all replaced positions
preserv_all_replaced_positions: LEMMA
  FORALL t, s, (fssp: SPP(s)), (i: below[fssp`length]):
            positionsOF(replace_pos(s, t, fssp))(fssp(i))

% preserves unchanged subterms
preserv_unchanged_subterms: LEMMA
  FORALL t, s, (fss: SPP(s)), (q: positions?(s)):
   (FORALL (i: below[length(fss)]): parallel(fss(i),q))
      =>
       subtermOF(replace_pos(s, t, fss), q) = subtermOF(s, q)

replace_pos_subterm_to_term: LEMMA
    FORALL t, s, (fss: SPP(s)):
      (FORALL (i: below[length(fss)]): subtermOF(s, fss(i)) = t)
          IMPLIES replace_pos(s, t, fss) = s

replace_pos_to_term : LEMMA  replace_pos(s, x, set2seq(Pos_var(s, x))) = s

  CP_lemma_aux2a: LEMMA
  FORALL t, s, (fss: SPP(s)), (q: positions?(s)):
   (FORALL (i: below[length(fss)]): parallel(fss(i),q))
      =>
  replace_pos(s, t, add_last(fss,q)) = replaceTerm(replace_pos(s, t, fss), t, q)

  CP_lemma_aux2a1: LEMMA
  FORALL t, s, (fss: SPP(s)), (q: positions?(s)):
   (FORALL (i: below[length(fss)]): parallel(fss(i),q))
      =>
  replace_pos(s, t, add_first(q, fss)) = replaceTerm(replace_pos(s, t, fss), t, q)

  CP_lemma_aux2a2: LEMMA
  FORALL t, s, (fss: SPP(s)), (i: below[length(fss)]):
     replace_pos(s, t, fss) =
                      replaceTerm(replace_pos(s, t, delete(fss,i)), t, fss(i))

  CP_lemma_aux2a3: LEMMA
  FORALL t, s, (fss1, fss2: SPP(s)):
    equivalent(fss1, fss2)
       =>
     replace_pos(s, t, fss1) = replace_pos(s, t, fss2)

END replace_positions

¤ Dauer der Verarbeitung: 0.3 Sekunden (vorverarbeitet) ¤

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