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

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