Richard, to get rid of the error message you should use Indexed
instead of Part
:
postulate = Sum[Abs[Indexed[x, {i}] - Indexed[y, {i}]], {i, 1, p}]/Sum[Max[Indexed[x, {i}], Indexed[y, {i}]], {i, 1, p}] ==
1 - Sum[Min[Indexed[x, {i}], Indexed[y, {i}]], {i, 1, p}]/(Sum[Indexed[x, {i}], {i, 1, p}] +
Sum[Indexed[y, {i}], {i, 1, p}] - Sum[Min[Indexed[x, {i}], Indexed[y, {i}]], {i, 1, p}])
But I do not know how to prove this remarkable formula, unfortunately. Empirical checks always evaluate to True
, e.g.:
dim = 50;
postulate /. {p -> dim, x -> RandomInteger[{-5, 5}, dim], y -> RandomInteger[{-5, 5}, dim]}
... well, not very helpful, I know!