@antoinechambertloir Yes, all the standard examples of monomial orders are *weight orders* for some concrete rational weight vectors. For these it is easy to see that they are well-founded. (I'm not 100% sure everything works fine with non-rational weights because of the constructive difficulties with real numbers.)

Appearantly https://link.springer.com/chapter/10.1007/3-540-15984-3_321 contains a proof that every ordering is a weight ordering, but unfortunately I can't access the paper. I doubt the proof will be constructive.