We present a set-theoretic, proof-irrelevant model for Calculus of Constructions (CC) with predicative induction and judgmental equality in Zermelo-Fraenkel set theory with an axiom for countably many inaccessible cardinals. We use Aczel's trace encoding which is universally defined for any function type, regardless of being impredicative. Direct and concrete interpretations of simultaneous induction and mutually recursive functions are also provided by extending Dybjer's interpretations on the basis of Aczel's rule sets. Our model can be regarded as a higher-order generalization of the truth-table methods. We provide a relatively simple consistency proof of type theory, which can be used as the basis for a theorem prover.
Informal motivation. The types-as-sets interpretation of type theory in a sufficiently strong classical axiomatic set theory, such as the Zermelo-Fraenkel (ZF) set theory, has been regarded as the most straightforward approach to demonstrating the consistency of type theory (cf. [Aczel(1998)] and [Coquand(1990)]). It can be construed as a higher-order generalization of the truth-table methods. Such a model captures the intuitive meaning of the constructs: the product, λ-abstraction, and application correspond to the ordinary set-theoretic product, function, and application, respectively.
A straightforward model of type theory is very useful for establishing the consistency of type theory, and it can be used to determine the proof-theoretic strength of type theory (cf. [Aczel(1998), Dybjer(1991), Dybjer(2000), Werner(1997)]). However, a higher-order generalization of the trivial Boolean model is not so simple (cf. [Miquel and Werner(2003)]). The main cause of this problem, as identified by [Reynolds(1984)], is the fact that type systems
In particular, Barendregt’s PTS-style β-conversion side condition turns into an explicit judgment. Two objects are not just equal; they are equal with respect to a type (cf. [Nordström et al.(1990) Nordström, Petersson, andSmith, Goguen(1994), Aczel(1998)]).
The type system considered in our study is CC with predicative induction and judgmental equality. It is a type system with the following features: dependent types, impredicative type (Prop) of propositions, a cumulative hierarchy of predicative universes (Type i ), predicative inductions, and judgmental equality.
The main difficulty in the construction of a set-theoretic model of our system stems from the impredicativity of Prop and the subtyping property Prop ≺ Type 0 . Without subtyping, one could use the solution provided by [Miquel and Werner(2003)] and [Werner(2008)], whereby proof-terms are syntactically distinguished from other function terms. Thus, the problem lies in the case distinction between the impredicative type Prop and the predicative types Type i , whereas the subsumption eliminates the difference. An interpretation function f : {0, 1} → V is required, where V is a set universe, that is different from the identity function. See Section 3 for further details.
For a set-theoretic interpretation of the cumulative type universes and predicative inductions, it is sufficient to assume countably many (strongly) inaccessible cardinals. [Werner(1997)] showed that ZF with an axiom guaranteeing the existence of infinitely many inaccessible cardinals is a good candidate. However, it is not clear whether the inaccessible cardinal axiom is necessary for our construction. The required feature of an inaccessible cardinal κ is the closure property of the universe V κ under the powerset operation. This is a necessary condition for the interpretation of inductive types. Following [Dybjer(1991)], we use Aczel’s rule sets to obtain a direct and concrete interpretation of induction and recursion rules.
The remainder of this paper is organized as follows. In Section 2, we provide a formal presentation of CC with predicative induction and judgmental equality. Examples are presented to enable the reader to understand the syntax and typing rules. This section can be regarded as an introduction to the base theory of the proof assistant Coq. Indeed, the syntax we have provided is as close to Coq syntax as that used in practice, except for the judgmental equality and the restriction on predicative inductions. 1 The difficulties in providing set-theoretic interpretations of impredicative or polymorphic types, subtypes, etc., are discussed in Section 3. We use the computational information about the domains saved in the interpretation of a : A to avoid these difficulties. This means that for the construction of set-theoretic models, type systems with judgmental equality are more explicit than systems without it. Using some typical examples, we explain the construction of a set-theoretic interpretation of inductive types and recursive functions.
Finally, in Section 4, we prove the soundness of our interpretation. The proof itself is relatively simple, and it can also be used to verify the consistency of our system. This is because some types such as Π(α : Prop).α will be interpreted as the empty set; hence, they cannot be inhabited in the type system.
In Section 5, we summarize the main results, and we discuss related work for future investigation.
First, we provide the full presentation of the system, i.e., Coquand’s CC with judgmental equality and predicative induction over infinitely many cumulative universes.
2.1. Syntax. We assume an infinite set of countably many variables, and we let x, x i , X, X i , … vary over the variables. We also use special constants Prop and Type i , i ∈ N. They are called sorts. Sorts are usually denoted by s, s i , etc. 2 1 We remark that many impredicative inductive types can be coded by
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