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Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To separate static and dynamic code, several static analyses and type systems have been put forward. We consider Pfenning's type theory with irrelevant quantification which is compatible with a type-based notion of equality that respects η-laws. We extend Pfenning's theory to universes and large eliminations and develop its meta-theory. Subject reduction, normalization and consistency are obtained by a Kripke model over the typed equality judgement. Finally, a type-directed equality algorithm is described whose completeness is proven by a second Kripke model.

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Type theory should be able to handle its own meta-theory, both to justify its foundational claims and to obtain a verified implementation. At the core of a type checker for intensional type theory lies an algorithm to check equality of types, or in other words, to check whether two types are convertible. We have formalized in Agda a practical conversion checking algorithm for a dependent type theory with one universe à la Russell, natural numbers, and η-equality for Π types. We prove the algorithm correct via a Kripke logical relation parameterized by a suitable notion of equivalence of terms. We then instantiate the parameterized fundamental lemma twice: once to obtain canonicity and injectivity of type formers, and once again to prove the completeness of the algorithm. Our proof relies on inductive-recursive definitions, but not on the uniqueness of identity proofs. Thus, it is valid in variants of intensional Martin-Löf Type Theory as long as they support induction-recursion, for instance, Extensional, Observational, or Homotopy Type Theory. algorithm correctly implements equality of types is based on a Kripke logical relation and follows Abel and Scherer [2012] and Abel et al. [2016].Our novel contribution is a full formalization of the algorithm and its proof of soundness, completeness, and termination, in Martin-Löf Type Theory [Martin-Löf 1975] with intensional equality, plus some well-understood extensions such as induction-recursion [Dybjer 2000]. As a mechanization language, we use Agda itself in its latest version, using the language variant without Streicher's Axiom K [1993]. 1 Consequently, our proof is directly transferable to related foundations such as Homotopy Type Theory extended by induction-recursion.The Agda formalization is quite sizeable: around 10.000 lines of code (500.000 characters). Especially the proof of the fundamental theorem of logical relations is substantial (5.000 lines). The congruence rule for the recursor for natural numbers alone requires a lemma that stretches over more than 500 lines. It is not that the proof is mathematically deep, once the right definition of the logical relation and the right formulation of the fundamental theorem are in placeÐit is just that a formalization requires us to get all the technicalities right. In research articles with pen and paper proofs only, the proof of the fundamental theorem is often skipped or reduced to the single sentence łproof by induction on the typing and equality derivationsž. Yet checking that each case of the induction goes through would require a reviewer many hours of disciplined technical reasoning. Written out, the proof would stretch over many pages.In previous works [Abel et al. 2007;Abel and Scherer 2012;Harper and Pfenning 2005], two logical relations and two fundamental theorems are needed for the meta theory: one that entails soundness of the algorithmic equality, and one for completeness. While in pen-and-paper proofs we could get away with remarks like łproof analogousž, a formalization would require us to do ...