Decidability of conversion for type theory in type theory

Abstract

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 ...

Canonicity for Cubical Type Theory

“…Appendix A] using the IH. This proves (1) in this case; for (2) one uses that all notions for giving a 1 and t 1 above preserve equality, and thus I u 1…”

“…Thus we can apply the Expansion Lemma and obtain I u 1 : A↓(i1) and I u 1 = u 1 ↓ : A↓(i1), and hence also I u 1 : A(i1) and I u 1 = u 1 ↓ : A(i1). By IH, we also have I, ϕ u 1 = u 1 ↓ = u(i1) : A↓(i1) = A(i1).…”

“…Here pred is the usual predecessor function defined using natrec. 1 1 This trick allows us that we never have to reduce in the system of a composition when defining composition for natural numbers, which also gives that reduction over Γ never refers to reduction in a restricted context Γ, ϕ (given that Γ is not restricted). If we would instead directly require u above to be of the form S u , we would have to explain reductions for systems like [(i = 0) (S t), (i = 1) (S t )] and more generally how reduction and systems interact.…”

“…In a way, our technique can be considered as a presheaf extension of the computability argument given in [2,1]; the latter being an adaption of the former using a typed reduction relation instead. A similar extension of this technique has been used to show the independence of Markov's principle in type theory [8].…”

Towards Certified Meta-Programming with Typed Template-Coq

“…Most solutions are based on extensions of Martin-Löf Intensional Type Theory with inductive-recursive or quotient inductive-inductive types [2,9], therefore extending the meta-theory. Recent work on verifying soundness and completeness of the conversion algorithm of a dependent type theory (with natural numbers, dependent products and a universe) in a type theory with IR types [1] gives us hope that this path can nonetheless be taken to provide the strongest guarantees on our conversion algorithm. The intrinsically-typed syntax used there is quite close to our typing derivations.…”