Problem
Research questions and friction points this paper is trying to address.
proof theory
dependent type theory
interactive theorem provers
sequent calculus
binding
Proof Theory and Dependent Type Theory: Distinct Foundations for Designing Proof Assistants
🤖 AI Summary
Current interactive theorem provers rely heavily on dependent type theory, often overlooking the potential of structural proof theory. This work systematically compares sequent calculus and dependent type theory in the context of proof assistant design, arguing that proof-theoretic frameworks offer distinct advantages in separating logic from proof representation, supporting nondeterministic proof strategies, and simplifying type mechanisms. The paper innovatively integrates lambda-tree syntax with the nabla quantifier to intrinsically handle binding structures and implements this approach within the Abella system for validation. Empirical results demonstrate that the proposed method significantly enhances conciseness and expressiveness in formalizing metatheory and reasoning about complex binding constructs.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
sequent calculus
dependent type theory
proof reconstruction
lambda-tree syntax
nabla-quantifier
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