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In this paper, we make a substantial step towards an encoding of Cubical TypeTheory (CTT) in the Dedukti logical framework. Type-checking CTT expressionsfeatures a decision procedure in a de Morgan algebra that so far could not beexpressed by the rewrite rules of Dedukti. As an alternative, 2 Layer TypeTheories are variants of Martin-L\"of Type Theory where all or part of thedefinitional equality can be represented in terms of a so-called externalequality. We propose to split the encoding by giving an encoding of 2 LayerType Theories (2LTT) in Dedukti, and a partial encoding of CTT in 2LTT.

Implementation of Two Layers Type Theory in Dedukti and Application to Cubical Type Theory