The next 700 syntactical models of type theory

Abstract

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

The next 700 relational program logics

“…In order to recover rules dealing with such a context, we apply the same recipe internally to a relational dependent type theory as described by Tonelli [2013]. In practice, this type theory is described as a syntactic model in the sense of Boulier et al [2017], that is a translation from a source type theory to a target type theory that we take to be our ambient type theory, where a type in the source theory is translated to a pair of types and a relation between them. We call the resulting source type theory RDTT and describe part of its construction in Figure 3.…”

“…Relating monadic expressions is natural and very wide-spread in proof assistants like Coq, Isabelle [Lochbihler 2018], or F ⋆ [Grimm et al 2018], with various degrees of automation. Boulier et al [2017]; Casinghino et al [2014]; Pédrot and Tabareau [2018] extend dependent type theory with a few selected primitive effects: partiality, exceptions, reader. The resulting theory allows to some extent to reason directly on pairs of effectful programs, without resorting to a monadic encoding.…”

Towards Certified Meta-Programming with Typed Template-Coq

“…In this section, we describe three examples of such plugins: (i) a plugin that adds a constructor to an inductive type, (ii) a re-implementation of Lasson's parametricity plugin 3 , and (iii) an implementation of a plugin that provides an extension of CIC-using a syntactic translation-in which it is possible to prove the negation of functional extensionality [8].…”

“…This is a simple example of syntactical translation which enriches the logical power of Coq, in the sense that new theorems can be proven (as opposed to the parametricity translation which is conservative over CIC). See [8] for an introduction to syntactical translations and a complete description of the intensional function translation. Even if the translation is very simple as it just adds a boolean to every function (Fig.…”

“…By correctness we mean computational soundness and typing soundness (see [8]), and both can be stated in Template-Coq. Anand has made substantial attempts in this direction to prove the computational soundness, in Template-Coq, of a variant of parametricity providing stronger theorems for free on propositions [3].…”