Bar induction is compatible with constructive type theory

Powerful yet effective induction principles play an important role in computing, being a paramount component of programming languages, automated reasoning, and program verification systems. The Bar Induction (BI) principle is a fundamental concept of intuitionism, which is equivalent to the standard principle of transfinite induction. In this work, we investigate the compatibility of several variants of BI with Constructive Type Theory (CTT), a dependent type theory in the spirit of Martin-Löf's extensional theory. We first show that CTT is compatible with a BI principle for sequences of numbers. Then, we establish the compatibility of CTT with a more general BI principle for sequences of name-free closed terms. The formalization of the latter principle within the theory involved enriching CTT's term syntax with a limit constructor and showing that consistency is preserved. Furthermore, we provide novel insights regarding BI, such as the non-truncated version of BI on monotone bars being intuitionistically false. These enhancements are carried out formally using the Nuprl proof assistant that implements CTT and the formalization of CTT within the Coq proof assistant presented in previous works.