On Duplication in Mathematical Repositories

Abstract

Building a repository of proof-checked mathematical knowledge is without any doubt a lot of work, and besides the actual formalization process there also is the task of maintaining the repository. Thus it seems obvious to keep a repsoitory as small as possible, in particular each piece of mathematical knowledge should be formalized only once. In this paper, however, we claim that it might be reasonable or even necessary to duplicate knowledge in a mathematical repository. We analyze different situations and reasons for doing so and provide a number of examples supporting our thesis.

Introducing Euclidean Relations to Mizar

“…in which the antecedent contains a conjunction of two instances of predicative formulas and the consequent is also the same predicate (with appropriately instantiated variables). As a result, such theorems could be eliminated from the MML in order to avoid redundancy [21].…”

On algebraic hierarchies in mathematical repository of Mizar

“…In practice, this means that notations are generic: There is no need to define, for example, greatest common divisors in Euclidean domains. They can be already introduced in GCD domains (see [20]) and are therefore available in Euclidean domains, once Euclidean domains have been incorporated into the hierarchy. The proofs necessary to built the above-mentioned hierarchy of rings have been carried out in a number of Mizar articles [2], [28], [30], [36].…”

Formal Development of Rough Inclusion Functions

“…where U is a non-empty set called the universe, I : U → ℘U is an uncertainty mapping, and κ : ℘U × ℘U → [0, 1] is a rough inclusion function. The formalization of uncertainty mappings was discussed in [12], and this article tries to define the missing part of the above definition, with future possibility of merging approaches via theory merging mechanism [6], avoiding duplication as much as we can [11].…”