Abstract The $$\lambda \varPi $$ λ Π -calculus modulo theory is an extension of simply typed $$\lambda $$ λ -calculus with dependent types and user-defined rewrite rules. We show that it is possible to replace the rewrite rules of a theory of the $$\lambda \varPi $$ λ Π -calculus modulo theory by equational axioms, when this theory features the notions of proposition and proof, while maintaining the same expressiveness. To do so, we introduce in the target theory a heterogeneous equality, and we build a translation that replaces each use of the conversion rule by the insertion of a transport. At the end, the theory with rewrite rules is a conservative extension of the theory with axioms.

In the context of interactive theorem provers based on a dependent type theory, automation tactics (dedicated decision procedures, call of automated solvers, ...) are often limited to goals which are exactly in some expected logical fragment. This very often prevents users from applying these tactics in other contexts, even similar ones.

We present a generalization of Spector’s bar recursion to the Diller-Nahm variant of Gödel’s Dialectica interpretation. This generalized bar recursion collects witnesses of universal formulas in sets of approximation sequences to provide an interpretation to the double-negation shift principle. The interpretation is presented in a fully computational way, implementing sets via lists. We also present a demand-driven version of this extended bar recursion manipulating partial sequences rather than initial segments. We explain why in a Diller-Nahm context there seems to be several versions of this demand-driven bar recursion, but no canonical one.

Second-order arithmetic has two kinds of computational interpretations: via Spector’s bar recursion of via Girard’s polymorphic lambda-calculus. Bar recursion interprets the negative translation of the axiom of choice which, combined with an interpretation of the negative translation of the excluded middle, gives a computational interpretation of the negative translation of the axiom scheme of comprehension. It is then possible to instantiate universally quantified sets with arbitrary formulas (second-order elimination). On the other hand, polymorphic lambda-calculus interprets directly second-order elimination by means of polymorphic types. The present work aims at bridging the gap between these two interpretations by interpreting directly second-order elimination through update recursion, which is a variant of bar recursion.

We describe a dependent type theory, and a denotational model for it, that incorporates both intensional and extensional semantic universes. In the former, terms and types are interpreted as strategies on certain graph games, which are concrete data structures of a generalized form, and in the latter as stable functions on event domains.

There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from these two interpretations have a fundamentally different computational behavior and their relationship is not well understood. We make a step towards a comparison by defining the first translation of system F into a simply-typed total language with a variant of bar recursion. This translation relies on a realizability interpretation of second-order arithmetic. Due to Gödel's incompleteness theorem there is no proof of termination of system F within second-order arithmetic. However, for each individual term of system F there is a proof in second-order arithmetic that it terminates, with its realizability interpretation providing a bound on the number of reduction steps to reach a normal form. Using this bound, we compute the normal form through primitive recursion. Moreover, since the normalization proof of system F proceeds by induction on typing derivations, the translation is compositional. The flexibility of our method opens the possibility of getting a more direct translation that will provide an alternative approach to the study of polymorphism, namely through bar recursion.

We build a realizability model for Peano arithmetic based on winning conditions for HON games. Our winning conditions are sets of desequentialized interactions which we call positions. We define a notion of winning strategies on arenas equipped with winning conditions. We prove that the interpretation of a classical proof of a formula is a winning strategy on the arena with winning condition corresponding to the formula. Finally we apply this to Peano arithmetic with relativized quantifications and give the example of witness extraction for Π20-formulas.

In intuitionistic realizability like Kleene's or Kreisel's, the axiom of choice is trivially realized. It is even provable in Martin-Löf's intuitionistic type theory. In classical logic, however, even the weaker axiom of countable choice proves the existence of non-computable functions. This logical strength comes at the price of a complicated computational interpretation which involves strong recursion schemes like bar recursion. We take the best from both worlds and define a realizability model for arithmetic and the axiom of choice which encompasses both intuitionistic and classical reasoning. In this model two versions of the axiom of choice can co-exist in a single proof: intuitionistic choice and classical countable choice. We interpret intuitionistic choice efficiently, however its premise cannot come from classical reasoning. Conversely, our version of classical choice is valid in full classical logic, but it is restricted to the countable case and its realizer involves bar recursion. Having both versions allows us to obtain efficient extracted programs while keeping the provability strength of classical logic.