User blog:C7X/Question About Normal Forms

Is it a good idea to create a notation where all expressions in the set of terms are already in normal form?

Example: CNF
Define a set of terms \(T\): \(0\in T \\ \forall m\in\mathbb{N}(\forall (\alpha_0,\alpha_1,\cdots,\alpha_m)\in \mathbb{N}^m(\omega^{\alpha_0}+\omega^{\alpha_1}+\cdots+\omega^{\alpha_m}\in T\iff \alpha_0\ge\alpha_1\ge\cdots\ge\alpha_m))\)

And then there would be a comparison algorithm for \(a<b\) for terms \(a\) and \(b\) here

There are no normal forms that are specially defined, instead when \(T\) is constructed, all non-normal cases (e.g. \(\omega^2+\omega^3\)) are ignored and therefore not in \(T\).