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sage: bool(c^2 - u*v >= 0)

"False" here means "not known to be true" -- it seems that bool just can't make the following observation:

c^2 > c*v > u*v

Although if you test each inequality separately, it will recognize them as True. More powerful symbolic manipulation is available through Maxima; see for example the parts of the reference manual having to do with Symbolic Calculus.

Unfortunately you might have to work harder for this -- I tried test_relation_maxima and got the same unsatisfactory results as with bool. You could try looking through the Maxima reference manual to see if there is something more useful there. You can use Maxima directly through Sage with maxima_console():

sage: maxima_console()
;;; Loading #P"/Applications/sage/local/lib/ecl/sb-bsd-sockets.fas"
;;; Loading #P"/Applications/sage/local/lib/ecl/sockets.fas"
;;; Loading #P"/Applications/sage/local/lib/ecl/defsystem.fas"
;;; Loading #P"/Applications/sage/local/lib/ecl/cmp.fas"
Maxima 5.23.2
using Lisp ECL 11.1.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) factor(10!);
                                   8  4  2
(%o1)                             2  3  5  7