$\mathbb {Q}$ muni de l'arithmétique faible de Penzin est décidable

We prove the decidability of the additive ordered group $\mathbb {Q}$ equipped with a predicate for $2^{\mathbb {Z}}$, the multiplication restricted to $2^{\mathbb {Z}}\times \mathbb {Q}$ and the $2$-adic valuation ranging in $2^{\mathbb {Z}}$.