Hi!
I'm playing around with Sage for manipulating some inequalities of
distances. For example, I may have some points in an unknown metric
space, so I know the triangular inequality holds, but little else. In
some cases, I may also have additional inequalities (such as one
distance being smaller than another).
What I'm wondering is what's possible to solve, here. For example, I'm
interested in finding lower bounds on distances. A trivial example is
the inverted triangular inequality, xz >= xy - yz, which can be
derived from the triangular inequality in a single step. Slightly less
obvious is the hyperplane inequality, for four points x, y, u and v,
where y is assumed to be closer to v and we want to lower-bound xy:
sage: var('uv ux uy vx vy xy')
(uv, ux, uy, vx, vy, xy)
sage: eq1 = vx <= vy + xy # Triangularity
sage: eq2 = uy <= ux + xy # Triangularity
sage: eq3 = vy <= uy # Assumption
sage: eq1 - xy
vx - xy <= vy
sage: (eq1 - xy).substitute(vy==eq2.rhs()) # By eq3...
vx - xy <= ux + xy
sage: _ + xy - ux
-ux + vx <= 2*xy
sage: _ / 2
-1/2*ux + 1/2*vx <= xy
This is, of course, pretty manual. (Sage doesn't even have to know
about eq3, as I just use it myself, when forcibly inserting
eq2.rhs()... :)
My question is, how much of this sort of thing can I automate (i.e.,
to help me discover new inequalities, for example)? And which tools
(such as adding assumptions, using solve() or .simplify()...) should I
use? Is it at all possible?-)
- M
--
Magnus Lie Hetland
http://hetland.org
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