# Proof Normalization for a First-order Formulation of Higher-order Logic

Abstract : We define a notion of cut and a proof reduction process for a class of theories, including all equational theories and a first-order formulation of higher-order logic. Proofs normalize for all equational theories. We show that the proof of the normalization theorem for the usual formulation of higher-order logic can be adapted to prove normalization for its first-order formulation. The «hard part» of the proof, that cannot be carried out in higher-order logic itself (the normalization of the system F-omega) is left unchanged. Thus, from the point of view of proof normalization, defining higher-order logic as a different logic or as a first-order theory does not matter. This result also explains a relation between the normalization of propositions and the normalization of proofs in equational theories and in higher-order logic: normalizing propositions does not eliminate cuts, but it transforms them.
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https://hal.inria.fr/inria-00073306
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 12:29:50 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
Long-term archiving on: : Sunday, April 4, 2010 - 11:41:47 PM

### Identifiers

• HAL Id : inria-00073306, version 1

### Citation

Gilles Dowek. Proof Normalization for a First-order Formulation of Higher-order Logic. [Research Report] RR-3383, INRIA. 1998. ⟨inria-00073306⟩

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