Solution and de Rham functors for D-cap-modules

We lay the groundwork for a Riemann-Hilbert correspondence for Ardakov-Wadsley's D-cap-modules by introducing corresponding solution and de Rham functors. Our constructions rely on Scholze's $p$-adic Hodge theory for rigid-analytic varieties, but we work over a decompletion of $B_{dR}^{+}$ which we call the positive overconvergent de Rham period ring. The main result of this article is the compatibility of our de Rham functor with Scholze's horizontal sections functor. This may be regarded as a generalisation of the classical non-Archimedean Cauchy Theorem, which roughly states that $p$-adic differential equations on unit discs have nonzero radius of convergence.