3 research outputs found
Schematic Cut elimination and the Ordered Pigeonhole Principle [Extended Version]
In previous work, an attempt was made to apply the schematic CERES method [8]
to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof
encapsulating the infinitary pigeonhole principle) [5]. However the derived
schematic refutation for the characteristic clause set of the proof could not
be expressed in the formal language provided in [8]. Without this formalization
a Herbrand system cannot be algorithmically extracted. In this work, we provide
a restriction of the proof found in [5], the ECA-schema (Eventually Constant
Assertion), or ordered infinitary pigeonhole principle, whose analysis can be
completely carried out in the framework of [8], this is the first time the
framework is used for proof analysis. From the refutation of the clause set and
a substitution schema we construct a Herbrand system.Comment: Submitted to IJCAR 2016. Will be a reference for Appendix material in
that paper. arXiv admin note: substantial text overlap with arXiv:1503.0855
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page