Abstract machines for functional languages rely on the notion of environment, a data structure storing the previously encountered and delayed beta-redexes. This paper provides a close analysis of the different approaches to define and implement environments. There are two main styles. The most common one is to have many local environments, one for every piece of code in the data structures of the machine. A minority of works instead uses a single global environment. Up to now, the two approaches have been considered equivalent, in particular at the level of the complexity of the overhead: they have both been used to obtain bilinear bounds, that is, linear in the number of beta steps and in the size of the initial term.

We present an interpretation of a version of dependent type theory where a type is interpreted by a Kan semisimplicial set. This interprets only a weak notion of conversion similar to the one used in the first published version of Martin-Löf type theory. Each truncated version of this model can be carried out internally in dependent type theory, and we have formalized the first truncated level, which is enough to represent isomorphisms of algebraic structure as equality.

Incorporating extensional equality into a dependent intensional type system such as the Calculus of Constructions (CC) provides with stronger type-checking capabilities and makes the proof development closer to intuition. Since strong forms of extensionality generally leads to undecidable type-checking, it seems a reasonable trade-off to extend intensional equality with a decidable first-order theory, as experimented in earlier work on CoqMTU and its implementation CoqMT. In this work, CoqMTU is extended with strong eliminations. The meta-theoretical study, particularly the part relying on semantic arguments, is more complex. A set-theoretical model of the equational theory is the key ingredient to derive the logical consistency of the formalism. Strong normalization, the main lemma from which type-decidability follows, is proved by attaching realizability information to the values of the model. The approach we have followed is to first consider an abstract notion of first-order equational theory, and then instantiate it with a particular instance, Presburger Arithmetic. These results have been formalized using Coq.

We study a complex type theory, a Calculus of Inductive Constructions with a predicative hierarchy of universes and a first-order theory T built in its conversion relation. The theory T is specified abstractly, by a set of constructors, a set of defined symbols, axioms expressing that constructors are free and defined symbols completely defined, and a generic elimination principle relying on crucial properties of first-order structures satisfying the axioms. We first show that CoqMTU enjoys all basic meta-theoretical properties of such calculi, confluence, subject reduction and strong normalization when restricted to weak-elimination, implying the decidability of type-checking in this case as well as consistency. The case of strong elimination is left open.