Sound and Complete Bidirectional Typechecking for Higher-Rank Polymorphism with Existentials and Indexed Types

TL;DR

This paper presents a sound and complete bidirectional typechecking system for a complex type language including higher-rank polymorphism, existentials, and indexed types, with formal proofs of correctness.

Contribution

It introduces a novel declarative and algorithmic type system for advanced types, extending bidirectional typing principles to indexed types with proof-theoretic foundations.

Findings

Provides a declarative specification based on focalization

Develops an algorithmic typing system that is sound and complete

Supports first-class existential types and pattern matching coverage

Abstract

Bidirectional typechecking, in which terms either synthesize a type or are checked against a known type, has become popular for its applicability to a variety of type systems, its error reporting, and its ease of implementation. Following principles from proof theory, bidirectional typing can be applied to many type constructs. The principles underlying a bidirectional approach to indexed types (generalized algebraic datatypes) are less clear. Building on proof-theoretic treatments of equality, we give a declarative specification of typing based on focalization. This approach permits declarative rules for coverage of pattern matching, as well as support for first-class existential types using a focalized subtyping judgment. We use refinement types to avoid explicitly passing equality proofs in our term syntax, making our calculus similar to languages such as Haskell and OCaml. We also extend the declarative specification with an explicit rules for deducing when a type is principal, permitting us to give a complete declarative specification for a rich type system with significant type inference. We also give a set of algorithmic typing rules, and prove that it is sound and complete with respect to the declarative system. The proof requires a number of technical innovations, including proving soundness and completeness in a mutually recursive fashion.