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

Abstract

We devise a method for generalizing proofs in Gentzen's sequent calculus $$\mathrm {LK}$$LK, presented in a typed $$\lambda $$ź-calculus flavor. A constrained version $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc 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 $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc, extending the standard semantics of $$\mathrm {LK}$$LK. It is then established that $$\mathrm {LK}$$LK-proofs correspond to $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc-proofs with valid constraint thanks to the use of eigenterms replacing $$\mathrm {LK}$$LK's eigenvariables. Next, a lifting theorem shows how a valid $$\mathrm {LK}^{{{\mathrm {c}}}}$$LKc-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 $$\mathrm {LK}$$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.