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Classically, a regular formula sequent that is not provable has a set-valued countermodel, with empty sorts allowed. Start with the finite diagram of its antecedent, successively adjoin witnesses to the existential consequences of each axiom, and identify elements when equality axioms require it. Every new fact about the original parameters has a finite derivation, so the nonderivable consequent remains false. The filtered union of this witness construction is a model. The conservativity property of this chase construction is studied in Forssell and Lumsdaine's paper.

A regular theory has axioms that are sequents between regular formulas. It has a regular syntactic category and a classifying topos of sheaves for the regular coverage. Its regular formulas enjoy the disjunction property of regular theories when a finite disjunction is allowed as an outer coherent conclusion.