#### pmrules.tst4.0 KB Permalink History Raw

 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213 ``````% Tests of PM. % TESTS OF BASIC CONSTRUCTS. operator f, h\$ % A "literal" template. m(f(a),f(a)); % Not literally equal. m(f(a),f(b)); %Nested operators. m(f(a,h(b)),f(a,h(b))); % A "generic" template. m(f(a,b),f(a,?a)); m(f(a,b),f(?a,?b)); % ??a takes "rest" of arguments. m(f(a,b),f(??a)); % 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))); % "plus" is symmetric. m(a+b+c,c+?a+?b); %It is also associative. m(a+b+c,c+?a); % Note the effect of using multi-generic symbol is different. m(a+b+c,c+??c); %Flag h as associative. flag('(h),'assoc); m(h(a,b,d,e),h(?a,d,?b)); % Substitution tests. s(f(a,b),f(a,?b)->?b^2); s(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); % Only substitute top at top level. s(a+b+f(a+b),a+b->a*b,inf,0); % SIMPLE OPERATOR DEFINITIONS. % Numerical factorial. operator nfac\$ s(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)},1); s(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)},2); si(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)}); % General factorial. operator gamma,fac; fac(?x _=Natp(?x)) ::- ?x*fac(?x-1); fac(0) :- 1; fac(?x) :- Gamma(?x+1); fac(3); fac(3/2); % Legendre polynomials in ?x of order ?n, ?n a natural number. operator legp; legp(?x,0) :- 1; legp(?x,1) :- ?x; legp(?x,?n _=natp(?n)) ::- ((2*?n-1)*?x*legp(?x,?n-1)-(?n-1)*legp(?x,?n-2))/?n; legp(z,5); legp(a+b,3); 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 )); scs := sin(?x)^2 + Cos(?x)^2 -> 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)); if si(trig 1,scs) = 1 then write("Pm ok") else Write("PM failed"); if si(trig 10,scs) = 10 then write("Pm ok") else Write("PM failed"); % 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 '((!#) !#); flag('(#), 'symmetric); operator #,@,ghg; xx := ghg(4,3,@(a,b,c,d),@(d,1+a-b,1+a-c),1); S(xx,sghg(3)); s(ws,sghg(2)); yy := ghg(3,2,@(a-1,b,c/2),@((a+b)/2,c),1); S(yy,sghg(1)); yy := ghg(3,2,@(a-1,b,c/2),@(a/2+b/2,c),1); S(yy,sghg(1)); % 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)); % 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(3) := Ghg(?p,?q,@(?a,??b),@(?a,??c),?z) -> Ghg(?p-1,?q-1,@(??b),@(??c),?z); SGhg(9) := Ghg(1,0,@(?a),?b,?z ) -> (1-?z)^(-?a); SGhg(10) := Ghg(0,0,?a,?b,?z) -> E^?z; 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)); si(Ghg(3,2,@(a,b,c),@(b,c),z),{SGhg(3),Sghg(9)}); S(Ghg(3,2,@(a-1,b,c),@(a-b,a-c),1),sghg 2); end; ``````