Codemist Standard Lisp 3.54 for DEC Alpha: May 23 1994
Dump file created: Mon May 23 10:39:11 1994
REDUCE 3.5, 15-Oct-93 ...
Memory allocation: 6023424 bytes
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*** ~ 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));
sghg(1) := ghg(3,2,@(?a,?b,?c),
1 + ?a + ?b
@(?d _= ?d=-------------,?e _= ?e=2*?c),1)->(
2
- ?a - ?b + 2*?c + 1 2*?c + 1
gamma(-----------------------)*gamma(----------)
2 2
?a + ?b + 1 1
*gamma(-------------)*gamma(---))/(
2 2
- ?a + 2*?c + 1 - ?b + 2*?c + 1
gamma(------------------)*gamma(------------------)
2 2
?a + 1 ?b + 1
*gamma(--------)*gamma(--------))
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)->(
gamma(?a - ?b + 1)*gamma(?a - ?c + 1)
?a - 2*?b - 2*?c + 2 ?a + 2
*gamma(----------------------)*gamma(--------))/(
2 2
gamma(?a - ?b - ?c + 1)*gamma(?a + 1)
?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;
(pmrules 2283 0)
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