計算機は囁く

Mathematica は当然

In[5]:= Reduce[Exists[x, 0 <= x <= 1 && y == x^2 - 2 a x], Reals]

Out[5]= (a <= 0 && 0 <= y <= 1 - 2 a) || (0 < a <= 1/2 && -a^2 <= y <= 1 - 2 a) ||
    (1/2 < a <= 1 && -a^2 <= y <= 0) || (a > 1 && 1 - 2 a <= y <= 0)

と答えます．

一方，qepmax+nnsolvexx は

(%i1) qe([[E,x]],"%and"(0<=x,x<=1,y=x^2-2*a*x));

(%o1) (((a > 0) %and (y+1 > 0) %and (y+2*a-1 <= 0))
  %or ((y >= 0) %and (y+2*a-1 <= 0))
  %or ((y <= 0) %and (y+2*a-1 >= 0))) %and (y+a^2 >= 0)

(%i2) nnsolvexx(%);

(%o2) ((a-1 < 0) %and (a > 0) %and (y+2*a-1 <= 0) %and (y+a^2 >= 0))
  %or ((y >= 0) %and (y+2*a-1 <= 0))
  %or ((y <= 0) %and (y+2*a-1 >= 0))

と囁きます．