# A basis in the space of solutions of a convolution equation by Napalkov V. V.

By Napalkov V. V.

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In practice, however, it often runs out of steam without proving a theorem. On success, Prover9 outputs a proof (which is usually not revealing for humans). Mace4 searches for ﬁnite models. It accepts essentially the same input as Prover9, and tries to construct a model of the assumptions that fails the proof goal, that is, a counterexample. Prover9 and Mace4 support inﬁx and postﬁx notation for algebraic operators and precedence declarations. For relation algebra, we use the following code: op(500, op(480, op(300, op(450, op(300, infix, infix, postfix, infix, postfix, "+").

The assumption ﬁle contains the axioms for Kleene algebras with domain, and the deﬁnition of set, transitive closure, precondition, postcondition and invariant, as listed above. The proofs of the ﬁrst and the third goal were instantaneous and very short. The proof of the second goal was slightly harder. It required about 18s and has 129 steps. On Automated Program Construction and Veriﬁcation 31 Termination and Development of the Loop. We now consider termination of the algorithm, and synthesise the body of the loop by considering the proof obligation that the invariant be preserved when executing the loop.

Its backbone is a combination of oﬀ-the-shelf automated theorem proving systems (ATP systems), model generators and computer algebra systems with domain-speciﬁc algebras that are designed and optimised for automation. This combination allows automatic program correctness proofs, but it also supports program development at a more fundamental level through the inference of speciﬁcation statements and algorithmic properties in a game of proof and refutation. While algebraic theories and automation technology can largely be hidden behind an interface, developers can focus on the conceptual level and use simple intuitive relational languages for modelling and reasoning.