Reasoning about modular datatypes with Mendler induction

TL;DR

This paper introduces a Mendler-style induction technique for reasoning about impredicative encodings of modular datatypes in proof assistants, simplifying proofs and enabling modular reasoning similar to traditional methods.

Contribution

It presents a novel Mendler-style induction approach that simplifies reasoning about impredicative datatypes in type-theoretic proof assistants, overcoming limitations of existing methods.

Findings

Enables modular proofs for impredicative datatypes

Simplifies inductive reasoning without dependent induction

Demonstrates effectiveness through subject reduction case study

Abstract

In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof assistants that are based on type theory, like Coq. The reason is that it involves type definitions, such as those of type-level fixpoint operators, that are not strictly positive. The known work-around of impredicative encodings is problematic, insofar as it impedes conventional inductive reasoning. Weak induction principles can be used instead, but they considerably complicate proofs. This paper proposes a novel and simpler technique to reason inductively about impredicative encodings, based on Mendler-style induction. This technique involves dispensing with dependent induction, ensuring that datatypes can be lifted to predicates and relying on relational formulations. A case study on proving subject reduction for structural operational semantics illustrates that the approach enables modular proofs, and that these proofs are essentially similar to conventional ones.