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Notes: “mem” in the resolution times column means that the resolution prover stopped due to memory management limitations (here this occurred as a result of unifying huge terms). “mem” in the tableau times column has an analogous meaning, except that on the machine of the authors of the prover those two formulas 4.2 and 4.3 were proved successfully in 43 and 49 seconds. “fail” means that several hours of proof search produced no result and we stopped the search.

Table 1 Notes: “mem” in the resolution times column means that the resolution prover stopped due to memory management limitations (here this occurred as a result of unifying huge terms). “mem” in the tableau times column has an analogous meaning, except that on the machine of the authors of the prover those two formulas 4.2 and 4.3 were proved successfully in 43 and 49 seconds. “fail” means that several hours of proof search produced no result and we stopped the search.

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Other premisses are derived clauses. The rule clauses are analogous to nucleons and the derived clauses are analogous to electrons of the hyperresolution strategy of ordinary classical resolution, see [4]. All the literals to the left of | in a rule clause have to be resolved upon. The literal to the right of | will go to the conclusion. The rule F > is different from the usual rules of classical resolution: two literals to the left of | are resolved upon with two literals in a single non-rule premiss clause. The rules / VY and dF are also nonstandard, since they contain the eigenvariable condition. 3? denotes a set of negative literals. mgu(p1, p\3---3 Pm; P,) denotes the result of unifying the terms f(p1,..., Pm) and f(pi,---,p),). & denotes either a positive literal or no literal at all. L denotes the label introduced by the rule. Each label 2 corresponds to a subformula with a polarity and a leading connective indicated to the left of the rule. VAR(t) denotes that ¢ is a variable, NOOCC (uv, t) denotes that v does not occur in f. Example We continue the example. Recall the formula fF), the labelled form of FP and the corresponding instantiation of GJm. ’conc’ will denote the conclusion of applying the rule clause with the substitution o. The following is the set of instantiated rules from GJm in the clause form:
Other premisses are derived clauses. The rule clauses are analogous to nucleons and the derived clauses are analogous to electrons of the hyperresolution strategy of ordinary classical resolution, see [4]. All the literals to the left of | in a rule clause have to be resolved upon. The literal to the right of | will go to the conclusion. The rule F > is different from the usual rules of classical resolution: two literals to the left of | are resolved upon with two literals in a single non-rule premiss clause. The rules / VY and dF are also nonstandard, since they contain the eigenvariable condition. 3? denotes a set of negative literals. mgu(p1, p\3---3 Pm; P,) denotes the result of unifying the terms f(p1,..., Pm) and f(pi,---,p),). & denotes either a positive literal or no literal at all. L denotes the label introduced by the rule. Each label 2 corresponds to a subformula with a polarity and a leading connective indicated to the left of the rule. VAR(t) denotes that ¢ is a variable, NOOCC (uv, t) denotes that v does not occur in f. Example We continue the example. Recall the formula fF), the labelled form of FP and the corresponding instantiation of GJm. ’conc’ will denote the conclusion of applying the rule clause with the substitution o. The following is the set of instantiated rules from GJm in the clause form:
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Intuitionistic Logicdecision Procedure
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