Proof Theory and Dependent Type Theory: Distinct Foundations for Designing Proof Assistants

Abstract

This paper examines the foundational distinctions between proof theory and dependent type theory (DTT) in the design of interactive theorem provers. While several implemented systems are designed using the dependently typed λ-calculus to represent proofs, no major proof assistant is designed using modern structural proof theory, even though, as I will argue here, the sequent calculus offers a compelling alternative framework. Six specific topics are proposed where the proof-theoretic perspective is arguably superior to the DTT perspective. These topics include the separation of logic from proof structure, the strategic use of non-determinism in proof reconstruction, and the avoidance of complex typing-discipline issues such as universe levels and proof irrelevance. The final topic -- the treatment of bindings -- is further developed to demonstrate how a natural, intensional approach is achieved through the mobility of binders. This methodology is illustrated via the Abella theorem prover, which leverages lambda-tree syntax and the nabla-quantifier to provide an elegant environment for reasoning about the meta-theory of languages and logics involving complex binding.