I am currently trying to read Colmez' "Série principale unitaire pour $Gl_2(\mathbb{Q}_p)$ et représentations triangulines de dimension 2", that you can find here https://webusers.imj-prg.fr/~pierre.colmez/triangulines . In the proof of Lemma 4.1. at the very end I can not follow anymore.

The statement is the following: For $b\in \mathcal{E}^{\dagger}$ (i.e. an element of the Robba ring that is bounded at 0) apparently can find an element $c$ of $\mathcal{B}^\dagger$ with $\varphi(c)=bc$, where $\varphi$ is the operator with $T\mapsto (1+T)^p-1$. Furthermore there is stated that $\mathcal{B}^\dagger$ is absolutely non-ramified over $\mathcal{E}^\dagger$. In the same paper I can not find a definition of $\mathcal{B}^\dagger$, but in another paper of Colmez, "Représentations cristallines et représentation de hauteur finie", that you can find here https://webusers.imj-prg.fr/~pierre.colmez/hauteurfinie.pdf it is defined. I am still quite new to the theory of $(\varphi,\Gamma)$-modules and did not work with Witt-vectors yet. So I have truble understanding what exactly $\mathcal{B}^\dagger$ is and how those statements would follow.

I am looking for a reference where these two statements are proven. It feels like they follow fast from the definition, so I would also be happy about some reference where $\mathcal{B}^\dagger$ is regarded in greater detail.