Conservativity of embeddings in the lambda Pi calculus modulo rewriting (long version)

The lambda Pi calculus can be extended with rewrite rules to embed any functional pure type system. In this paper, we show that the embedding is conservative by proving a relative form of normalization, thus justifying the use of the lambda Pi calculus modulo rewriting as a logical framework for logics based on pure type systems. This result was previously only proved under the condition that the target system is normalizing. Our approach does not depend on this condition and therefore also works when the source system is not normalizing.Comment: Long version of TLCA 2015 pape

Type‐Preserving CPS Translation of Σ and Π Types is Not Not Possible

International audienceDependently typed languages like Coq are used to specify and prove functional correctness of source programs,but what we ultimately need are guarantees about correctness of compiled code. By preserving dependenttypes through each compiler pass, we could preserve source-level specifications and correctness proofs intothe generated target-language programs. Unfortunately, type-preserving compilation of dependent types isnontrivial. In 2002, Barthe and Uustalu showed that type-preserving CPS is not possible for languages likeCoq. Specifically, they showed that for strong dependent pairs (Σ types), the standard typed call-by-name CPSis not type preserving. They further proved that for dependent case analysis on sums, a class of typed CPStranslations—including the standard translation—is not possible. In 2016, Morrisett noticed a similar problemwith the standard call-by-value CPS translation for dependent functions (Π types). In essence, the problem isthat the standard typed CPS translation by double-negation, in which computations are assigned types of theform (A → ⊥) → ⊥, disrupts the term/type equivalence that is used during type checking in a dependentlytyped language.In this paper, we prove that type-preserving CPS translation for dependently typed languages is not notpossible. We develop both call-by-name and call-by-value CPS translations from the Calculus of Constructionswith both Π and Σ types (CC) to a dependently typed target language, and prove type preservation andcompiler correctness of each translation. Our target language is CC extended with an additional equivalencerule and an additional typing rule, which we prove consistent by giving a model in the extensional Calculus ofConstructions. Our key observation is that we can use a CPS translation that employs answer-type polymorphism,where CPS-translated computations have type ∀α.(A → α) → α. This type justifies, by a free theorem,the new equality rule in our target language and allows us to recover the term/type equivalences that CPStranslation disrupts. Finally, we conjecture that our translation extends to dependent case analysis on sums,despite the impossibility result, and provide a proof sketch