reduce-historical/r36/xlog/PMRULES.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...



*** ~ already defined as operator 
% Tests of PM.

% TESTS OF BASIC CONSTRUCTS.

operator f, h$



% A "literal" template.

m(f(a),f(a));


t


% Not literally equal.

m(f(a),f(b));



%Nested operators.

m(f(a,h(b)),f(a,h(b)));


t


% A "generic" template.

m(f(a,b),f(a,?a));


{?a->b}

m(f(a,b),f(?a,?b));


{?a->a,?b->b}


% ??a takes "rest" of arguments.

m(f(a,b),f(??a));


{??a->[a,b]}


% But ?a does not.

m(f(a,b),f(?a));



% Conditional matches.

m(f(a,b),f(?a,?b _=(?a=?b)));



m(f(a,a),f(?a,?b _=(?a=?b)));


{?a->a,?b->a}


% "plus" is symmetric.

m(a+b+c,c+?a+?b);


{?a->a,?b->b}


%It is also associative.

m(a+b+c,c+?a);


{?a->a + b}


% Note the effect of using multi-generic symbol is different.

m(a+b+c,c+??c);


{??c->[a,b]}


%Flag h as associative.

flag('(h),'assoc);



m(h(a,b,d,e),h(?a,d,?b));


{?a->h(a,b),?b->e}


% Substitution tests.

s(f(a,b),f(a,?b)->?b^2);


 2
b


s(a+b,a+b->a*b);


a*b


% "associativity" is used to group a+b+c in to (a+b) + c.

s(a+b+c,a+b->a*b);


a*b + c


% Only substitute top at top level.

s(a+b+f(a+b),a+b->a*b,inf,0);


f(a + b) + a*b



% SIMPLE OPERATOR DEFINITIONS.

% Numerical factorial.

operator nfac$



s(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)},1);


3*nfac(2)


s(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)},2);


6*nfac(1)


si(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)});


6


% General factorial.

operator gamma,fac;



fac(?x _=Natp(?x)) ::- ?x*fac(?x-1);


hold(?x*fac(?x - 1))


fac(0)  :- 1;


1


fac(?x) :- Gamma(?x+1);


gamma(?x + 1)


fac(3);


6


fac(3/2);


       5
gamma(---)
       2


% Legendre polynomials in ?x of order ?n, ?n a natural number.

operator legp;



legp(?x,0) :- 1;


1


legp(?x,1) :- ?x;


?x


legp(?x,?n _=natp(?n))
   ::- ((2*?n-1)*?x*legp(?x,?n-1)-(?n-1)*legp(?x,?n-2))/?n;


      (2*?n - 1)*?x*legp(?x,?n - 1) - (?n - 1)*legp(?x,?n - 2)
hold(----------------------------------------------------------)
                                 ?n


legp(z,5);


        4       2
 z*(63*z  - 70*z  + 15)
------------------------
           8


legp(a+b,3);


    3       2           2            3
 5*a  + 15*a *b + 15*a*b  - 3*a + 5*b  - 3*b
---------------------------------------------
                      2


legp(x,y);


legp(x,y)



% TESTS OF EXTENSIONS TO BASIC PATTERN MATCHER.

comment *: MSet[?exprn,?val] or ?exprn ::: ?val
        assigns the value ?val to the projection ?exprn in such a way
        as to store explicitly each form of ?exprn requested. *;

 
Nosimp('mset,(t t));



Newtok '((!: !: !: !-) Mset);



infix :::-;



precedence Mset,RSetd;



?exprn :::- ?val ::- (?exprn ::- (?exprn :- ?val ));


hold(?exprn::-(?exprn:-?val))


scs := sin(?x)^2 + Cos(?x)^2 -> 1;


              2          2
scs := cos(?x)  + sin(?x) ->1


% The following pattern substitutes the rule sin^2 + cos^2 into a sum of
% such terms.  For 2n terms (ie n sin and n cos) the pattern has a worst
% case complexity of O(n^3).

operator trig,u;



trig(?i) :::- Ap(+, Ar(?i,sin(u(?1))^2+Cos(u(?1))^2));


                                       2             2
hold(trig(?i):-ap(plus,ar(?i,sin(u(?1))  + cos(u(?1)) )))


if si(trig 1,scs) = 1 then write("Pm ok") else Write("PM failed");


Pm ok


if si(trig 10,scs) = 10 then write("Pm ok") else Write("PM failed");


Pm ok


% The next one takes about 70 seconds on an HP 9000/350, calling UNIFY
% 1927 times.

% if si(trig 50,scs) = 50 then write("Pm ok") else Write("PM failed");

% Hypergeometric Function simplification.

newtok '((!#) !#);


*** # redefined 


flag('(#), 'symmetric);



operator #,@,ghg;



xx := ghg(4,3,@(a,b,c,d),@(d,1+a-b,1+a-c),1);


xx := ghg(4,3,@(a,b,c,d),@(d,a - b + 1,a - c + 1),1)


S(xx,sghg(3));


*** sghg declared operator 

ghg(4,3,@(a,b,c,d),@(d,a - b + 1,a - c + 1),1)


s(ws,sghg(2));


ghg(4,3,@(a,b,c,d),@(d,a - b + 1,a - c + 1),1)


yy := ghg(3,2,@(a-1,b,c/2),@((a+b)/2,c),1);


                         c      a + b
yy := ghg(3,2,@(a - 1,b,---),@(-------,c),1)
                         2        2


S(yy,sghg(1));


                   c      a + b
ghg(3,2,@(a - 1,b,---),@(-------,c),1)
                   2        2


yy := ghg(3,2,@(a-1,b,c/2),@(a/2+b/2,c),1);


                         c      a + b
yy := ghg(3,2,@(a - 1,b,---),@(-------,c),1)
                         2        2


S(yy,sghg(1));


                   c      a + b
ghg(3,2,@(a - 1,b,---),@(-------,c),1)
                   2        2


% Some Ghg theorems.

flag('(@), 'symmetric);



% Watson's Theorem.

SGhg(1) := Ghg(3,2,@(?a,?b,?c),@(?d _=?d=(1+?a+?b)/2,?e _=?e=2*?c),1) -> 
     Gamma(1/2)*Gamma(?c+1/2)*Gamma((1+?a+?b)/2)*Gamma((1-?a-?b)/2+?c)/
     (Gamma((1+?a)/2)*Gamma((1+?b)/2)*Gamma((1-?a)/2+?c)
        *Gamma((1-?b)/2+?c));


                                           1 + ?a + ?b
sghg(1) := ghg(3,2,@(?a,?b,?c),@(?d _= ?d=-------------,?e _= ?e=2*?c),1)->(
                                                2

                      - ?a - ?b + 2*?c + 1          2*?c + 1
              gamma(-----------------------)*gamma(----------)
                               2                       2

                      ?a + ?b + 1          1             - ?a + 2*?c + 1
              *gamma(-------------)*gamma(---))/(gamma(------------------)
                           2               2                   2

                       - ?b + 2*?c + 1          ?a + 1          ?b + 1
              *gamma(------------------)*gamma(--------)*gamma(--------))
                             2                    2               2


% Dixon's theorem.

SGhg(2) := Ghg(3,2,@(?a,?b,?c),@(?d _=?d=1+?a-?b,?e _=?e=1+?a-?c),1) -> 
     Gamma(1+?a/2)*Gamma(1+?a-?b)*Gamma(1+?a-?c)*Gamma(1+?a/2-?b-?c)/
     (Gamma(1+?a)*Gamma(1+?a/2-?b)*Gamma(1+?a/2-?c)*Gamma(1+?a-?b-?c));


sghg(2) := ghg(3,2,@(?a,?b,?c),@(?d _= ?d=1 + ?a - ?b,?e _= ?e=1 + ?a - ?c),1)->

                                                         ?a - 2*?b - 2*?c + 2
           (gamma(?a - ?b + 1)*gamma(?a - ?c + 1)*gamma(----------------------)
                                                                  2

                    ?a + 2
            *gamma(--------))/(gamma(?a - ?b - ?c + 1)*gamma(?a + 1)
                      2

                      ?a - 2*?b + 2          ?a - 2*?c + 2
              *gamma(---------------)*gamma(---------------))
                            2                      2


SGhg(3) := Ghg(?p,?q,@(?a,??b),@(?a,??c),?z)
                   -> Ghg(?p-1,?q-1,@(??b),@(??c),?z);


sghg(3) := 

ghg(?p,?q,@(??b,?a),@(??c,?a),?z)->ghg(?p - 1,?q - 1,@(??b),@(??c),?z)


SGhg(9) := Ghg(1,0,@(?a),?b,?z )       ->  (1-?z)^(-?a);


                                        1
sghg(9) := ghg(1,0,@(?a),?b,?z)->---------------
                                             ?a
                                  ( - ?z + 1)

SGhg(10) := Ghg(0,0,?a,?b,?z)          ->  E^?z;


                                ?z
sghg(10) := ghg(0,0,?a,?b,?z)->e

SGhg(11) := Ghg(?p,?q,@(??t),@(??b),0) ->  1;


sghg(11) := ghg(?p,?q,@(??t),@(??b),0)->1


% If one of the bottom parameters is zero or a negative integer the
% hypergeometric functions may be singular, so the presence of a
% functions of this type causes a warning message to be printed.

% Note it seems to have an off by one level spec., so this may need
% changing in future.
%
% Reference: AS 15.1; Slater, Generalized Hypergeometric Functions,
%     Cambridge University Press,1966.

s(Ghg(3,2,@(a,b,c),@(b,c),z),SGhg(3));


ghg(2,1,@(a,b),@(b),z)


si(Ghg(3,2,@(a,b,c),@(b,c),z),{SGhg(3),Sghg(9)});


      1
-------------
           a
 ( - z + 1)


S(Ghg(3,2,@(a-1,b,c),@(a-b,a-c),1),sghg 2);


                                  a - 2*b - 2*c + 1          a + 1
 gamma(a - b)*gamma(a - c)*gamma(-------------------)*gamma(-------)
                                          2                    2
---------------------------------------------------------------------
                         a - 2*b + 1          a - 2*c + 1
 gamma(a - b - c)*gamma(-------------)*gamma(-------------)*gamma(a)
                              2                    2


end;
(TIME:  pmrules 650 650)