International audienceA family of syntactic models for the calculus of construction with universes (CC ω) is described, all of them preserving conversion of the calculus definitionally, and thus giving rise directly to a program transformation of CC ω into itself. Those models are based on the remark that negative type constructors (e.g., dependent product, coinductive types or universes) are underspecified in type theory—which leaves some freedom on extra intensional specifications. The model construction can be seen as a compilation phase from a complex type theory into a simpler type theory. Such models can be used to derive (the negative part of) independence results with respect to CC ω , such as functional extensional-ity, propositional extensionality, univalence or the fact that bisimulation on a coinductive type may not coincide with equality. They can also be used to add new principles to the theory, which we illustrate by defining a version of CC ω with ad-hoc polymorphism that shows in particular that para-metricity is not an implicit requirement of type theory. The correctness of some of the models/program transformations have been checked in the COQ proof assistant and have been instrumented as a COQ plugin

This paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper; ii) one call-by-name, whose intuitions already appear in Krivine and Miquel's work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions, which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom.

There is a critical tension between substitution, dependent elimination and effects in type theory. In this paper, we crystallize this tension in the form of a no-go theorem that constitutes the fire triangle of type theory. To release this tension, we propose ∂CBPV, an extension of call-by-push-value (CBPV) Ða general calculus of effectsÐto dependent types. Then, by extending to ∂CBPV the well-known decompositions of call-by-name and call-by-value into CBPV, we show why, in presence of effects, dependent elimination must be restricted in call-by-name, and substitution must be restricted in call-by-value. To justify ∂CBPV and show that it is general enough to interpret many kinds of effects, we define various effectful syntactic translations from ∂CBPV to Martin-Löf type theory: the reader, weaning and forcing translations. CCS Concepts: • Theory of computation → Type theory.

We define a monadic translation of type theory, called weaning translation, that allows for a large range of effects in dependent type theory-such as exceptions, non-termination, non-determinism or writing operation. Through the light of a call-by-push-value decomposition, we explain why the traditional approach fails with type dependency and justify our approach. Crucially, the construction requires that the universe of algebras of the monad forms itself an algebra. The weaning translation applies to a version of the Calculus of Inductive Constructions (CIC) with a restricted version of dependent elimination. Finally, we show how to recover a translation of full CIC by mixing parametricity techniques with the weaning translation. This provides the first effectful version of CIC.Proposition 10. If I is a first-order type, then there is a term eval I :By Proposition 8, full dependent elimination is interpreted, so that by putting all these properties together we recover this interesting extraction result by taking E • := I.Theorem 2. If I is a first-order type and there is a proof of ¬¬I in CIC, then there is a proof of I in CIC. C. Non-determinismWe sketch a model of BTT with non-determinism here.Definition 11. The non-empty list monad is defined as T A := A × list A together with El (X, (nil _)) := X El (X, [X 1 ; . . . ; X n ]) := (X.π 1 × X 1 .π 1 . . . × X n .π 1 , µ)

Traditional approaches to compensate for the lack of exceptions in type theories for proof assistants have severe drawbacks from both a programming and a reasoning perspective. Pédrot and Tabareau recently extended the Calculus of Inductive Constructions (CIC) with exceptions. The new exceptional type theory is interpreted by a translation into CIC, covering full dependent elimination, decidable type-checking and canonicity. However, the exceptional theory is inconsistent as a logical system. To recover consistency, Pédrot and Tabareau propose an additional translation that uses parametricity to enforce that all exceptions are caught locally. While this enforcement brings logical expressivity gains over CIC, it completely prevents reasoning about exceptional programs such as partial functions. This work addresses the dilemma between exceptions and consistency in a more flexible manner, with the Reasonably Exceptional Type Theory (RETT). RETT is structured in three layers: (a) the exceptional layer, in which all terms can raise exceptions; (b) the mediation layer, in which exceptional terms must be provably parametric; (c) the pure layer, in which terms are non-exceptional, but can refer to exceptional terms. We present the general theory of RETT, where each layer is realized by a predicative hierarchy of universes, and develop an instance of RETT in Coq: the impure layer corresponds to the predicative universe hierarchy, the pure layer is realized by the impredicative universe of propositions, and the mediation layer is reified via a parametricity type class. RETT is the first full dependent type theory to support consistent reasoning about exceptional terms, and the CoqRETT plugin readily brings this ability to Coq programmers.