  We use dual equivalence to give a short, combinatorial proof that Stanley
symmetric functions are Schur positive. We introduce weak dual equivalence, and
use it to give a short, combinatorial proof that Schubert polynomials are key
positive. To demonstrate further the utility of this new tool, we use weak dual
equivalence to prove a nonnegative Littlewood--Richardson rule for the key
expansion of the product of a key polynomial and a Schur polynomial, and to
introduce skew key polynomials that, when skewed by a partition, expand
nonnegatively in the key basis.
