Shallow Embedding of Type Theory is Morally Correct

TL;DR

This paper demonstrates that shallow embedding of type theory in proof assistants is morally correct by ensuring injectivity and soundness, enabling formal reasoning about syntax with minimal technical overhead.

Contribution

It introduces a method to shallowly embed type theory that preserves syntactic distinctions and avoids illegal equalities, applicable in all major dependent type theory proof assistants.

Findings

Shallow embedding is injective up to definitional equality.

Implementation hiding prevents illegal propositional equalities.

Technique is demonstrated with formalizations of canonicity and parametricity.

Abstract

There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-L\"of type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.