Abstract
The Mizar Verifier uses definitional expansions for controlling proof structures. In this paper we propose another use of definitional expansions--enriching verified inferences by expansions of definitions of formulae included in the inferences and increasing the number of premises accessible by Checker. This introduces more knowledge to the reasoning, which helps to draw more conclusions. Some statistics about influence of such expansions on the Mizar Mathematical Library are presented.
Highlights
In this paper we propose another utilization of definitional expansions by the MIZAR VERIFIER
MIZAR [11, 23] is a proof assistant used for computerized verification of mathematical proofs
MIZAR is a common name for the components of the system: a formal language derived from the mathematical vernacular and dozens of computer programs responsible for formal verification, optimization of proofs [24, 25]
Summary
One of the aspects of the evolution of the language of mathematics is the development of a conceptual apparatus. An important question is which objects or collections of objects are worth to have own names. Definitions of which objects should be introduced to the language. One extreme answer could be that everything should be named. Some researchers treat definitions as an investment in mathematics, while theorems as profits from mathematics. Following this opinion, definitions should be set up carefully. Another question concerns communication between defined objects and their definitions. It is an especially important issue in the case of proof assistants, computer systems for formal proofs management, when their efficiency is desirable. Some related questions are: “How much automation should be implemented?”, and “Which definitions should be expanded automatically to facilitate reasonings, but not to burden a given system too much, making it unusable in practice?”
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