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In Maxima (which handles our assumptions):

(%i1) assume(b>0);
(%o1)                               [b > 0]
(%i2) assume(b<1);
(%o2)                               [b < 1]
(%i3) assume(a>0);
(%o3)                               [a > 0]
(%i4) assume(a<1);
(%o4)                               [a < 1]
(%i5) is (a*b > 0);
(%o5)                                true
(%i6) is (a*b < 1);
(%o6)                               unknown

And Sage returns false for booleans if it doesn't know for sure that it's true.

I'm not sure whether they would consider this a bug; their assumptions framework is self-described as "weak", though of course it's better than having no assumptions at all!