By Patrick J. Hurley
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It follows that the permutation of a cut upward does not always reduce cutheight but can increase it. For this reason, we shall explicitly calculate the height of each cut in what follows. As with weakening and contraction, we may assume that there is only one occurrence of the rule of cut, as the last step. 3: The rule of cut, F => D £>, A => C r, A => c -Cut is admissible in G3ip. Proof: The proof is organized as follows: We consider first the case that at least one premiss in a cut is an axiom or conclusion of L_L and show how cut is eliminated.
Proof: The proof is by induction on the height of derivation n. If n = 0, D, D, F =>• C is an axiom or conclusion of L_L and either C is an atom in the antecedent or the antecedent contains _L. In either case, also D, F =>• C is an axiom or conclusion of L_L. Let contraction be admissible up to derivation height n. We have two cases according to whether the contraction formula is not principal or is principal in the last inference step. If the contraction formula D is not principal in the last (one-premiss) rule concluding the premiss of contraction we have £>, P , T => C D, D, T = ^ C 34 STRUCTURAL PROOF THEORY which has a derivation height ^ n, so by inductive hypothesis we obtain \-n D, F r =>• Cr and by applying the last rule h n + i D, F =>• C.
In Chapter 8 we find a somewhat different explanation of cut: It arises, in terms of natural deduction, from non-normal instances of elimination rules. This points to an important analogy between normal derivations in natural deduction and cut-free derivations in sequent calculus, an analogy that will be made precise in Chapter 8. 2, there is in natural deduction a series of concepts from the existence of normal form to strong normalization and uniqueness of normal form. In systems of sequent calculus, it is possible that two derivations F =^ A and A, A =>• C are cut-free, but the derivation F, A =>• C obtained by cut need not, in general, have any such form.