Gradual type theory

Abstract Gradually typed languages are designed to support both dynamically and statically programming styles while preserving the benefits of each. Sound gradually check types at runtime boundary between modules. However, there is much disagreement in gradual typing literature over how enforce complex such as tuples, lists, functions objects. In this paper, we propose a new perspective on design type enforcement: casts exist precisely ensure correctness certain type-based refactorings optimizations. For instance, for simple types, language designer might desire that beta-eta equality valid. We show useful by demonstrating cast semantics can be derived from equality. do providing an axiomatic account program equivalence calculus logic call theory (GTT). Based Levy’s call-by-push-value, GTT allows us axiomatize call-by-value call-by-name languages. then derive behavior corresponding eta principle assumption satisfies property called graduality , also known dynamic guarantee. Since equality, receive contrapositive: any observably different must violate consistency applicability our proving contract-based implementation using lazy gives logical relations model theory, where implies contextual programs. axiomatizes guarantee, establishes central theorem typing. The parameterized so family implementations validate optimization