The definition of simple rational terms (Definition 2.5) is meaningful in the compositional syntax of Section 3: the simple rational terms with outputs in $B^*$ are the rational terms built from the empty list $[\,] : 1 \to B^*$ using co-pairing $\langle f_1, f_2 \rangle$, concatenation $f_1 \odot f_2$, the homomorphism combinator $\mathrm{hom}\, f$, and prepending and appending a value, $w \cdot f$ and $f \cdot w$, where $w : 1 \to B^*$.

For two rational terms $f, g : A \to B^*$ and a value $w : 1 \to B^*$, Section 6 considers three conjugacy relations, in increasing order of strength:

1. $f$ and $g$ are denotationally $w$-conjugate if $[\![ f \cdot w ]\!] = [\![ w \cdot g ]\!]$;
2. $f$ and $g$ are provably $w$-conjugate if $f \cdot w \equiv w \cdot g$;
3. $f$ and $g$ are syntactically $w$-conjugate if $f = w \cdot h$ and $g = h \cdot w$ for some term $h$.

The length of a list value $w$ is the length of the list it produces.