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Proof-theoretical analysis of order relations. (English) Zbl 1062.03055

In a previous work, the first and second author proposed a proof-theoretical treatment of axiomatic systems in a general setting by showing that a wide class of mathematical axioms are convertible into rules of inference added to certain contraction-free sequent calculi so that we can apply several establised methods of proof analysis in sequent calculi.
In this paper, the argument is enriched by some developments on duality of mathematical rules and on Harrop theories, and then applied to a concrete analysis of elementary theories of order relations. The main result obtained by the analysis is a conservativity theorem which gives rise to a proof-theoretical version of Szpilrajn’s theorem on the extension of a partical order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is also shown by giving terminating methods of proof-search. Finally the third author discusses these results by an alternative approach, featuring the notion of entailment relation introduced by D. Scott.

MSC:

03F05 Cut-elimination and normal-form theorems
03F07 Structure of proofs
03B25 Decidability of theories and sets of sentences
06A05 Total orders
06A06 Partial orders, general
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