A modest proposal: explicit support for foundational pluralism

Introduction. This repo contains a basic Scala 3 implementation of an LCF-style interactive theorem prover for the HOL(C) logic introduced in the paper

A modest proposal: explicit support for foundational pluralism

by Dominic Mulligan and Martin Berger. The paper is available at arxiv.org/abs/2302.10137.

HOL(C) is a natural deduction presentation of HOL, but with a ternary judgement $\Gamma \vdash \phi : l$, adding a notion of taint $l$ to the more common binary judgements $\Gamma \vdash \phi$ between assumptions $\Gamma$ and formulae $\phi$. Taint can be seen as a kind of 'typing system' for proofs that explicitly tracks how classically or constructive a proof is, e.g. is $\phi$ deduced fom $\Gamma$ using Excluded Middle, or Reductio Ad Absurdum, or with Dependent Choice, or not? The purpose of this implementation is to serve as a proof-of-concept for fine-tuning the logic and for reviewing the paper. It is not intended to enable large-scale proof in HOL(C).

Compiling, testing and running the code. All relevant code is in the src directory and can be compiled and executed with the scalac compiler. The code is more easily used with sbt, using the provided build.sbt file, by invoking

sbt compile

for compilation. Invoking

sbt test

executes a few property-based unit tests. Invoking

sbt run

executes integration tests, which are small theorems, including Peirce's law, proved using basic tactics.