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Hence, veriﬁcation and analysis of complicated control systems with time can be carried out in a consistent formal theory. The authors think that automated veriﬁcation with human assistance of (labeled) @-calculus can be devised relatively easily, since it is based on NK with PA, for which automatic veriﬁcation methods have been studied well. The way that natural numbers are used so as to represent the time-dependent position of trains as axioms in section 3 seems to be a promising solution for the diﬃculty of treating continuous phenomena and discrete changes like decisions, computer operations, etc together, while the external variables with very elementary diﬀerential equations were introduced in [17] and [23] to cope with this problem.

It should be mentioned that we are not bound to a particular logic6 in our deﬁnition of Γ ∗ . We rely only on entailment systems and consider the sequent relation as a subrelation of the entailment relation. e. all statements in Γ ∗ are derivable from Γ in the entailment system. In particular for Γn−1 ⊇ σP and P ϕ we know that Γn entails σ(P ∪ ϕ) and since P ∪ ϕ can be viewed as a partial theory we call this entailment a partial theory inclusion. Let us now assume that we build our transitive closure, now denoted by Γ ∗ , only with total theory inclusions.

24 W. McCune 3. W. McCune. Mace4 Reference Manual and Guide. Tech. Memo ANL/MCS-TM264, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, August 2003. 4. W. McCune. 3 Reference Manual. Tech. Memo ANL/MCS-TM-263, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, August 2003. 5. W. McCune. Prover9. gov/~mccune/prover9/, 2005. 6. W. McCune and L. Henschen. Experiments with semantic paramodulation. J. Automated Reasoning, 1(3):231–261, 1984.