Shallow Embedding of Type Theory is Morally Correct

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-Löf type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.

Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

“…In the introduction we have seen Fig. 1, due to Kaposi and others [1], [2], which presents the components of type theory as a generalised algebraic theory. In detail, a model of type theory according to Fig.…”

“…1 consists of four types and type families (Con, Ty, Sub, Tm), together with ten terms that inhabit the four type families in various ways, and together with twelve equations. The original work by Altenkirch and Kaposi [3] (as well as [1], [2]) contains additional components for various type formers. These and the twenty-six components of Fig.…”

“…Fig. 1: The formulation of a category with families (CwF) as a generalised algebraic theory (GAT), as given by Kaposi and others [1], [2]. Presentation-wise, it slightly differs from (but is equivalent to) the similar suggestion by Altenkirch and Kaposi [3].…”

“…the very first sentence by Abel et al [9]), which refers to formalising the (intended) initial model of type theory. 1 Altenkirch and Kaposi [3] call this simply type theory in type theory. To avoid confusion, we refer to the type theory in which these models are implemented as the host theory, and the structure that get implemented as the object theory or simply the model.…”

“…A GAT consists of sorts, operations, and equations. In the case of 1 It is often seen as type-theoretic folklore that the initial model of a type theory coincides with the syntax, i.e. that the syntactic model is initial.…”