Sound and complete bidirectional typechecking for higher-rank polymorphism with existentials and indexed types

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. CCS Concepts: • Software and its engineering → Data types and structures; • Theory of computation → Type theory;

A vision for online verification-validation

“…The synthesizing judgement form Γ e ⇒ C can be read as, "the typing context Γ and term e synthesize the computation type C ." Specifically, the algorithm computes the type C , when given Γ and e. For the core forms of λ-VMF, the bidirectional rules for values and computations follow the usual patterns found in bidirectional type systems [10,16]. We show several standard-looking rules, sub, lam, app and annot.…”

Bidirectional type checking for relational properties

“…In our prototype implementation of BiRelCost, we use a custom algorithm to eliminate these quantifiers, which works well. An alternative approach, used for example in liquid types [42] and by Dunfield and Krishnaswami for GADTs [23], is to engineer or restrict the type system so that generated constraints do not have existential quantifiers. In the context of bidirectional type checking, this seems fundamentally difficult for quantitative effects like costs.…”

“…However, the idea was popularized only more recently by Pierce and Turner [37]. The technique has shown great applicability-it has been used for dependent types [16], indexed refinement types [43,44], intersection and union types [20,24], higher-rank polymorphism [22,23,35], contextual modal types [36], algebraic effect handlers [29] and gradual typing [41]. Our approach is inspired by many of these papers, in particular DML [43,44], but departs in the technical design of the algorithmic type system due to new challenges offered by relational and modal types, and costs.…”

“…Hence, the broad goal of our work is to investigate issues in implementing a type checker for relational type systems with relational refinements and relational effects. Bidirectional type checking is a natural starting point for the reasons mentioned above, and because it has been used for implementing refinement type systems [20,23,44] and subtyping [37], which are important common features in most of the type systems we are inspired from. However, bidirectional type checking has not been extensively applied to effect systems, although some examples exist [41], and it has not been applied to relational type systems at all.…”

Quantitative program reasoning with graded modal types

“…A more flexible system could be provided by arbitrary rank quantification rather than the ML-style type schemes that Granule features, e.g. via the recent bidirectional results of Dunfield and Krishnaswami [2019]. Even more attractive is a generalisation of our system to full Martin-Löf-style dependent types.…”