Reasoning about functional programs by combining interactive and automatic proofs
Andrés Sicard-Ramírez · Americanae (AECID Library) · 2015
We propose a new approach to computer-assisted verification of lazy functional programs where functions can be defined by general recursion. We work in first-order theories of functional programs which are obtained by translating Dybjer's programming logic (Dybjer, P. [1985]. Program Verification in a Logical Theory of Constructions. In: Functional Programming Languages and Computer Architecture. Ed. by Jouannaud, J. P. Vol. 201. Lecture Notes in Computer Science. Springer, pp. 334–349) into a first-order theory, and by extending this programming logic with new (co-)inductive predicates. Rather than building a special purpose system, we formalise our theories in Agda, a proof assistant for dependent type theory which can be used as a generic theorem prover. Agda provides support for interactive reasoning by representing first-order theories using the propositions-as-types principle. Further support is provided by off-the-shelf automatic theorem provers for first-order-logic called by a Haskell program that translates our Agda representations of first-order formulae into the TPTP language understood by the provers. We show some examples where we combine interactive and automatic reasoning, covering both proofs by induction and co-induction. The examples include functions defined by structural recursion, simple general recursion, nested recursion, higher-order recursion, guarded and unguarded co-recursion.