Higher-order abstract syntax in Coq
The terms of the simply-typed λ-calculus can be used to express the higher-order abstract syntax of objects such as logical formulas, proofs, and programs. Support for the manipulation of such objects is provided in several programming languages (e.g. λProlog, Elf). Such languages also provide embedded implication, a tool which is widely used for expressing hypothetical judgments in natural deduction. In this paper, we show how a restricted form of second-order syntax and embedded implication can be used together with induction in the Coq Proof Development system. We specify typing rules and evaluation for a simple functional language containing only function abstraction and application, and we fully formalize a proof of type soundness in the system. One difficulty we encountered is that expressing the higher-order syntax of an object-language as an inductive type in Coq generates a class of terms that contains more than just those that directly represent objects in the language. We overcome this difficulty by defining a predicate in Coq that holds only for those terms that correspond to programs. We use this predicate to express and prove the adequacy for our syntax.
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