A Formal Treatment of Bidirectional Typing

There has been much progress in designing bidirectional type systems and associated type synthesis algorithms, but mainly on a case-by-case basis. To remedy the situation, this paper develops a general and formal theory of bidirectional typing for simply typed languages: for every signature that specifies a mode-correct bidirectionally typed language, there exists a proof-relevant type synthesiser which, given an input abstract syntax tree, constructs a typing derivation if any, gives its refutation if not, or reports that the input does not have enough type annotations. Sufficient conditions for deriving a type synthesiser such as soundness, completeness, and mode-correctness are studied universally for all signatures. We propose a preprocessing step called mode decoration, which helps the user to deal with missing type annotations. The entire theory is formally implemented in Agda, so we provide a verified generator of proof-relevant type synthesisers as a by-product of our formalism.