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Datei: Sprache: PVS

Original von: PVS^©

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

unification[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]

                 Vs: VAR set[(V)]
             V1, V2: VAR finite_set[(V)]
                 V3: VAR finite_set[term]
            x, y, z: VAR (V)
    tau, sig, sigma,
  delta, rho, theta: VAR Sub
            st, stp: VAR finseq[term]
    r, s, t, t1, t2: VAR term
                  n: VAR nat
       p, q, p1, p2: VAR position
                  R: VAR pred[[term, term]]


%%%% Defining an instance of a term %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

instance(t, s): bool = EXISTS sigma: ext(sigma)(s) = t;

%%%% Defining substitution more general "<=" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

<=(theta, sigma): bool = EXISTS tau: sigma = comp(tau, theta)

mg_po: LEMMA preorder?(<=)

%%%% Defining unification between two terms %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

unifier(sigma)(s,t): bool = ext(sigma)(s) = ext(sigma)(t)

unifiable(s,t): bool = EXISTS sigma: unifier(sigma)(s,t)

U(s,t): set[Sub] = {sigma: Sub | unifier(sigma)(s,t)}

%%%% Defining a most general unifier "mgu" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

mgu(theta)(s,t): bool = member(theta, U(s,t)) &
                    FORALL sigma: member(sigma, U(s,t)) IMPLIES theta <= sigma

%%%% Initial auxiliary lemma %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

uni_diff_equal_length_arg : LEMMA
      FORALL (s: term, (t: term | unifiable(s, t) & s /= t), f: symbol,
             st: {args: finite_sequence[term] | args`length = arity(f)}):
       NOT st`length = 0 AND s = app(f, st) IMPLIES
       (FORALL (fp: symbol, stp: {args: finite_sequence[term] |
               args`length = arity(fp)}): t = app(fp, stp) IMPLIES
        (f = fp & st`length = stp`length))

%%%% Position of the first difference between %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% two unifiable and different terms %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

resolving_diff(s : term, (t : term | unifiable(s,t) & s /= t ) ):
   RECURSIVE position =
    (CASES s OF
       vars(s) : empty_seq,
       app(f, st) :
       IF length(st) = 0 THEN empty_seq
       ELSE
        (CASES t OF
           vars(t) : empty_seq,
           app(fp, stp) :
           LET k : below[length(stp)] =
             min({kk : below[length(stp)] |
                         subtermOF(s,#(kk+1)) /= subtermOF(t,#(kk+1))}) IN
             add_first(k+1,
                      resolving_diff(subtermOF(s,#(k+1)),subtermOF(t,#(k+1))))
         ENDCASES)
       ENDIF
     ENDCASES)
MEASURE s BY <<

%%%% Lemmas about resolving_diff %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

resol_diff_nonempty_implies_funct_terms : LEMMA
    FORALL (s: term, (t: term | unifiable(s, t) & s /= t)):
           resolving_diff(s,t) /= empty_seq IMPLIES
           (app?(s) AND app?(t))

resol_diff_to_rest_resol_diff : LEMMA
    FORALL (s: term, (t: term | unifiable(s, t) & s /= t)):
        LET rd =  resolving_diff(s,t) IN
            rd /= empty_seq IMPLIES
            resolving_diff(subtermOF(s,#(first(rd))),
                           subtermOF(t,#(first(rd)))) = rest(rd)

position_s_resolving_diff : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t, p : position):
        p = resolving_diff(s, t) IMPLIES positionsOF(s)(p);

position_t_resolving_diff : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t, p : position):
        p = resolving_diff(s, t) IMPLIES positionsOF(t)(p);

resolving_diff_has_diff_argument : LEMMA
    FORALL (s : term, t : term | unifiable(s,t) & s /= t,
            p : position | positionsOF(s)(p) & positionsOF(t)(p)):
        p = resolving_diff(s, t) IMPLIES
          subtermOF(s, p) /= subtermOF(t, p)

resolving_diff_has_unifiable_argument : LEMMA
    FORALL (s : term, t : term | unifiable(s,t) & s /= t,
            p : position | positionsOF(s)(p) & positionsOF(t)(p)):
        p = resolving_diff(s, t) IMPLIES
          unifiable(subtermOF(s, p), subtermOF(t, p))

resolving_diff_vars : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t,
     p : position | positionsOF(s)(p) & positionsOF(t)(p)):
        p = resolving_diff(s, t) IMPLIES
    vars?(subtermOF(s, p)) OR vars?(subtermOF(t, p))

%%%% Auxiliary lemmas about substitutions and unifiers %%%%%%%%%%%%%%%%%%%%%%%%

unifier_o : LEMMA
        member(sig, U(ext(theta)(s), ext(theta)(t))) IMPLIES
        member(comp(sig, theta), U(s, t))

mgu_o : LEMMA
        sig <= rho IMPLIES comp(sig, theta) <= comp(rho, theta)

unifier_and_subs : LEMMA
        member(theta, U(s, t)) IMPLIES
        (FORALL (sig: Sub): member(comp(sig, theta), U(s, t)))

idemp_mgu_iff_all_unifier : LEMMA
    FORALL (theta: Sub | member(theta, U(s, t))):
        mgu(theta)(s, t) & idempotent_sub?(theta) IFF
        (FORALL (sig: Sub | member(sig, U(s, t))): sig = comp(sig, theta))

unifiable_terms_unifiable_args : LEMMA
    FORALL (s : term, t : term,
            p : position | positionsOF(s)(p) & positionsOF(t)(p)):
        member(sig, U(s, t)) IMPLIES
           member(sig, U(subtermOF(s, p), subtermOF(t, p)))

var_term_unifiable_not_var_in_term : LEMMA
    FORALL (s : term, t : term ):
        vars?(s) & unifiable(s, t) & s /= t IMPLIES
           NOT member(s, Vars(t))

%%%% Substitution to fix the %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% first difference %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

sub_of_frst_diff(s : term , (t : term | unifiable(s,t) & s /= t )) : Sub =
  LET k : position = resolving_diff(s,t) IN
     LET sp = subtermOF(s,k) , tp = subtermOF(t,k) IN
        IF vars?(sp)
        THEN (LAMBDA (x : (V)) : IF x = sp THEN tp ELSE x ENDIF)
        ELSE (LAMBDA (x : (V)) : IF x = tp THEN sp ELSE x ENDIF)
        ENDIF

%%%% Lemmas about "sub_of_frst_diff" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

dom_sub_of_frst_diff_is : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t, sig : Sub):
        sig = sub_of_frst_diff(s, t) AND p = resolving_diff(s, t)
        IMPLIES
           IF vars?(subtermOF(s, p))
             THEN Dom(sig) = singleton(subtermOF(s, p))
             ELSE Dom(sig) = singleton(subtermOF(t, p))
           ENDIF

var_sub_1stdiff_not_member_term : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        LET sig = sub_of_frst_diff(s ,t) IN
           FORALL ( x | member(x,Dom(sig)), r | member(r,Ran(sig) )) :
               NOT member(x, Vars(r))

sub_of_frst_diff_unifier_o : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        member(rho, U(s, t)) IMPLIES
           LET sig = sub_of_frst_diff(s, t) IN
              EXISTS theta : rho = comp(theta, sig)

ext_sub_of_frst_diff_unifiable : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        LET sig = sub_of_frst_diff(s, t) IN
           unifiable(ext(sig)(s), (ext(sig)(t)))

sub_of_frst_diff_remove_x : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        LET sig = sub_of_frst_diff(s, t) IN
           Dom(sig)(x) IMPLIES
               (NOT member(x, Vars(ext(sig)(s)))) AND
               (NOT member(x, Vars(ext(sig)(t))))

vars_sub_of_frst_diff_s_is_subset_union : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        LET sig = sub_of_frst_diff(s, t) IN
           subset?(Vars(ext(sig)(s)), union( Vars(s), Vars(t)))

vars_sub_of_frst_diff_t_is_subset_union : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        LET sig = sub_of_frst_diff(s, t) IN
           subset?(Vars(ext(sig)(t)), union( Vars(s), Vars(t)))

union_vars_ext_sub_of_frst_diff : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t) :
        LET sig = sub_of_frst_diff(s, t) IN
           union(Vars(ext(sig)(s)), Vars(ext(sig)(t)))
           = difference(union( Vars(s), Vars(t)), Dom(sig))

vars_ext_sub_of_frst_diff_decrease : LEMMA
    FORALL (s : term, t : term | unifiable(s, t) & s /= t):
        LET sig = sub_of_frst_diff(s, t) IN
           Card(union( Vars(ext(sig)(s)), Vars(ext(sig)(t))))
           < Card(union( Vars(s), Vars(t)))

%%%% Function to compute a unifier of %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% two unifiable terms %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

unification_algorithm(s : term, (t : term | unifiable(s,t))) : RECURSIVE Sub =
    IF s = t THEN identity
    ELSE LET sig = sub_of_frst_diff(s, t) IN
            comp( unification_algorithm(ext(sig)(s) , ext(sig)(t)) , sig)
    ENDIF
MEASURE Card(union(Vars(s), Vars(t)))

%%%% Lemmas about "unification_algorithm" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

unification_algorithm_gives_unifier : LEMMA
    unifiable(s,t) IMPLIES member(unification_algorithm(s, t), U(s, t))

unification_algorithm_gives_mg_subs : LEMMA
    member(rho, U(s, t)) IMPLIES unification_algorithm(s, t) <= rho

%%%% Existence of a most general unifier %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

unification : LEMMA
    unifiable(s,t) => EXISTS theta : mgu(theta)(s,t)

END unification

¤ Dauer der Verarbeitung: 0.19 Sekunden (vorverarbeitet) ¤

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