Extending type theory with syntactic models

Abstract : This thesis is about the metatheory of intuitionnistic type theory. The considered systems are variants of Martin-Löf type theory of Calculus of Constructions, and we are interested in the coherence of those systems and in the independence of axioms with respect to those systems. The common theme of this thesis is the construction of syntactic models, which are models reusing type theory to interpret type theory. In a first part, we introduce type theory by a minimal system and several possible extensions. In a second part, we introduce the syntactic models given by program translation and give several examples. In a third part, we present Template-Coq, a plugin for metaprogramming in Coq. We demonstrate how to use it to implement directly some syntactic models. Last, we consider type theories with two equalities: one strict and one univalent. We propose a re-reading of works of Coquand et.al. and of Orton and Pitts on the cubical model by introducing degenerate fibrancy.
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Simon Pierre Boulier. Extending type theory with syntactic models. Logic in Computer Science [cs.LO]. Ecole nationale supérieure Mines-Télécom Atlantique, 2018. English. ⟨NNT : 2018IMTA0110⟩. ⟨tel-02007839⟩

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