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Proof Generalization in \(\mathrm {LK}\) by Second Order Unifier Minimization

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Abstract

We devise a method for generalizing proofs in Gentzen’s sequent calculus \(\mathrm {LK}\), presented in a typed \(\lambda \)-calculus flavor. A constrained version \(\mathrm {LK}^{{{\mathrm {c}}}}\) of the calculus is introduced, aiming at collecting a second order constraint ensuring that all the inference steps occurring in a proof are syntactically correct. A semantics is provided for \(\mathrm {LK}^{{{\mathrm {c}}}}\), extending the standard semantics of \(\mathrm {LK}\). It is then established that \(\mathrm {LK}\)-proofs correspond to \(\mathrm {LK}^{{{\mathrm {c}}}}\)-proofs with valid constraint thanks to the use of eigenterms replacing \(\mathrm {LK}\)’s eigenvariables. Next, a lifting theorem shows how a valid \(\mathrm {LK}^{{{\mathrm {c}}}}\)-proof can be lifted to a most general proof, yielding a non-trivial constraint together with a solution. An algorithm is then provided that minimizes this solution of the constraint. The result, applied to the most general proof, yields a valid proof that translates to an \(\mathrm {LK}\)-proof more general than the initial one. Finally, clues are given for extending this method to other logics with due care on proof lifting.

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Notes

  1. Formal definitions are given in Sect. 2, see in particular Fig. 1 for the inference rules.

  2. We assume w.l.o.g. that \(D \cap (\mathcal {C}\cup \mathcal {V})= \emptyset \) and that D contains no non-atomic terms in \(\mathcal {T}_D\).

  3. More precisely on a special kind of formulæ called “matrices”.

  4. Note that the \(\mathcal {X}_i\)’s are not part of these meta-variables.

  5. In all rigor the same should be done in [13] for the (\(\Rightarrow \)-L) rule from [25].

  6. These constants cannot be generalized simply because there is no variable of the corresponding types. The equality predicate could be generalized if its specific properties are not used in the proof, i.e., if no paramodulation inference is applied on it. Of course, if no \(\wedge \)-rule is applied on a formula \((\wedge \;t_1\, t_2)\) then it can be generalized by a variable of type \(\mathbf {o}\).

  7. In particular, if \(v\in \{\forall ,\exists \}\) then \(m=1\) and \(z_1\) has type \({\varvec{\i }}^{n+1}\rightarrow \mathbf {o}\). If v is a binary connective then \(m=2\) since t has \(\mathcal {V}\)-type.

  8. Another way to do this is to allow principal formulæ to occur anywhere in the conclusions of the rules. For instance, the (\(\lnot \)-L) rule would be \(\displaystyle \frac{\varGamma ,\varSigma \vdash \varDelta ,\phi }{\varGamma ,\lnot \phi ,\varSigma \vdash \varDelta }.\)

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Acknowledgments

We thank the anonymous referees for their helpful comments and references.

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  1. CNRS/LIG, University of Grenoble, 220, rue de la Chimie, 38400, Saint Martin d’Hères, France

    Thierry Boy de la Tour & Nicolas Peltier

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  1. Thierry Boy de la Tour
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Correspondence to Thierry Boy de la Tour.

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Boy de la Tour, T., Peltier, N. Proof Generalization in \(\mathrm {LK}\) by Second Order Unifier Minimization. J Autom Reasoning 57, 245–280 (2016). https://doi.org/10.1007/s10817-016-9367-3

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