Semantic Criteria of Correct Formalization

This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context

A Case Study on Logical Relations using Contextual Types

Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Constructing categories and setoids of setoids in type theory

In this paper we consider the problem of building rich categories of setoids, in standard intensional Martin-L\"of type theory (MLTT), and in particular how to handle the problem of equality on objects in this context. Any (proof-irrelevant) family F of setoids over a setoid A gives rise to a category C(A, F) of setoids with objects A. We may regard the family F as a setoid of setoids, and a crucial issue in this article is to construct rich or large enough such families. Depending on closure conditions of F, the category C(A, F) has corresponding categorical constructions. We exemplify this with finite limits. A very large family F may be obtained from Aczel's model construction of CZF in type theory. It is proved that the category so obtained is isomorphic to the internal category of sets in this model. Set theory can thus establish (categorical) properties of C(A, F) which may be used in type theory. We also show that Aczel's model construction may be extended to include the elements of any setoid as atoms or urelements. As a byproduct we obtain a natural extension of CZF, adding atoms. This extension, CZFU, is validated by the extended model. The main theorems of the paper have been checked in the proof assistant Coq which is based on MLTT. A possible application of this development is to integrate set-theoretic and type-theoretic reasoning in proof assistants.Comment: 14 page