Proof normalization with nonstandard objects

Proof Normalization Modulo

We define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church’s simple type theory. Deduction modulo [11] is a formulation of first-order logic, that g...

Fragment of Nonstandard Analysis with a Finitary Consistency Proof

We introduce a nonstandard arithmetic NQA− based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by NQA+), with a weakened external open minimization schema. A finitary consistency proof for NQA− formalizable in PRA is presented. We also show interesting facts about the strength of the theories NQA− and NQA+; NQA− is mutually interpretable with I∆0 + EXP, and on th...

Transfer Principle Proof Tactic for Nonstandard Analysis

This paper describes a type constructor for nonstandard extensions of arbitrary types, generalizing the previous formalization of nonstandard analysis in Isabelle by Fleuriot and Paulson [1]. In addition to numeric types, nonstandard extensions of function spaces and set types also prove useful—they provide an elegant way to define internal functions and internal sets. The paper also describes ...

Proof Normalization and Normal Proofs

Affine Normalization of Symmetric Objects

A new method of normalization is used for the construction of the affine moment invariants. The affine transform is decomposed into translation, scaling, stretching, two rotations and mirror reflection. The object is successively normalized to these elementary transforms by means of low order moments. After normalization, other moments of normalized object can be used as affine invariant featur...