Functions in functional languages have a single elimination form — application — and cannot be compared, hashed, or subjected to other non-application operations. These operations can be approximated via defunctionalization: functions are replaced with first-order data and calls are replaced with invocations of a dispatch function. Operations such as comparison may then be implemented for these first-order data to approximate e.g. deduplication of continuations in algorithms such as unbounded searches. Unfortunately, this encoding is tedious, imposes a maintenance burden, and obfuscates the affected code. We introduce an alternative in intensional functions , a language feature which supports the definition of non-application operations in terms of a function’s definition site and closure-captured values. First-order data operations may be defined on intensional functions without burdensome code transformation. We give an operational semantics and type system and prove their formal properties. We further define intensional monads , whose Kleisli arrows are intensional functions, enabling monadic values to be similarly subjected to additional operations.

In recent years, there has been an increased interest in tools that establish incorrectness rather than correctness of program properties. In this work we build on this approach by developing a novel methodology to prove incorrectness of semantic typing properties of functional programs, extending the incorrectness approach to the model theory of functional program typing. We define a semantic type refuter which refutes semantic typings for a simple functional language. We prove our refuter is co-recursively enumerable, and that it is sound and complete with respect to a semantic typing notion. An initial implementation is described which uses symbolic evaluation to efficiently find type errors over a functional language with a rich type system.