Naive cubical type theory

Abstract This article proposes a way of doing type theory informally, assuming cubical style reasoning. It can thus be viewed as first step toward alternative to the program informalization carried out in homotopy book for dependent augmented with axioms univalence and higher inductive types. We adopt cartesian proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, Licata implicit foundation, confining our presentation elementary results such function extensionality, derivation weak connections path induction, groupoid structure types, Eckmman–Hilton duality.