#C25.txt: April 23, 2020 Conway's  Surreal numbers

Help:=proc(): print(` GE(x,y),LE(x,y), EQ(x,y)`): 
print(` GT(x,y) ,  LT(x,y), IsNumber(x) ,ADD(x,y) `):
print(` SN(N), Minus(x) , MUL(X,Y) `):
end:

#GE(x,y): inputs two (genuine) (surreal) numbers
#in John Horton Conway's sense and outputs true
#iff x>=y
GE:=proc(x,y) local xR, yL:
#xR is a typical member of x[2] and yL is a typical
#member of y[1]

evalb(
{true,seq( evalb( not GE(y,xR)), xR in x[2])}={true}
and 
{true,seq( evalb( not GE(yL,x)), yL in y[1])}={true} ):
    
end:

#LE(x,y) is x<=y?
LE:=proc(x,y):GE(y,x): end:

#EQ(x,y): is x=y in Conway's sense
EQ:=proc(x,y): GE(x,y) and LE(x,y):end:


#GT(x,y): x>y ?
GT:=proc(x,y): GE(x,y) and not GE(y,x) : end:


#LT(x,y): x<y ?
LT:=proc(x,y): GT(y,x) : end:

#IsNumber(x): Is x a legal number?
IsNumber:=proc(x) local xL,xR:

if not (type(x,list) and nops(x)=2 and type(x[1],set) and
       type(x[2],set) ) then
 RETURN(false):
fi:

if member(false, {seq(IsNumber(xL), xL in x[1])}) then
 RETURN(false):
fi:

if member(false, {seq(IsNumber(xR), xR in x[2])}) then
 RETURN(false):
fi:



#condition is: no member of L (alias x[1]) is 
#>= member of R (x[2])

if member(true,{seq(seq(GE(xL,xR), xL in x[1]), xR in x[2])})
 then
 RETURN(false):
fi:

true:

end:

#ADD(x,y): x+y=[{xL+y,x+yL}, {xR+y,x+yR}]
ADD:=proc(x,y) local xL, xR, yL, yR:
option remember:

[ { seq( ADD(xL,y), xL in x[1]), 
    seq( ADD(x,yL), yL in y[1])},
{ seq( ADD(xR,y), xR in x[2]), 
    seq( ADD(x,yR), yR in y[2])}
]:
end:

#SN(N): the surreal rendition of the pos. integer N
SN:=proc(N) option remember:
if N=0 then
 [{},{}]:
else
 [ {SN(N-1)}, {}]:
fi:
end:
 
Minus:=proc(x) local xR, xL:
option remember:
[ {seq(Minus(xR), xR in x[2])},
  {seq(Minus(xL), xL in x[1])}
]:

end:
 













#CODE BY CHARLES KENNEY (Added after class)
#MUL(X,Y) multiplies two surreal numbers X and Y
MUL:=proc(X,Y) local i,x,y:
[{seq(seq(ADD(ADD(MUL(x,Y),MUL(X,y)),Minus(MUL(x,y))),x in X[1]),y in Y[1]), seq(seq(ADD(ADD(MUL(x,Y),MUL(X,y)),Minus(MUL(x,y))),x in X[2]),y in Y[2])},
{seq(seq(ADD(ADD(MUL(x,Y),MUL(X,y)),Minus(MUL(x,y))),y in Y[2]),x in X[1]) , seq(seq(ADD(ADD(MUL(x,Y),MUL(X,y)),Minus(MUL(x,y))),y in Y[1]),x in X[2])}]:
end: