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This paper obtains a completeness result for inequational reasoning withapplicative terms without variables in a setting where the intended semanticmodels are the full structures, the full type hierarchies over preorders forthe base types. The syntax allows for the specification that a given symbol beinterpreted as a monotone function, or an antitone function, or both. There isa natural set of five rules for inequational reasoning. One can add variablesand also add a substitution rule, but we observe that this logic would beincomplete for full structures. This is why the completeness result in thispaper pertains to terms without variables. Since the completeness is alreadyknown for the class of general (Henkin) structures, we are interested in fullstructures. We present a completeness theorem. Our result is not optimalbecause we restrict to base preorders which have a weak completeness property:every pair of elements has an upper bound and a lower bound. To compensate weadd several rules to the logic. We also present extensions and variations ofour completeness result.

A Completeness Result for Inequational Reasoning in a Full Higher-Order Setting