Editing Integer (section)

=== Other approaches ===

In theoretical computer science, other approaches for the construction of integers are used by [[Automated theorem proving|automated theorem provers]] and [[Rewriting|term rewrite engines]]. Integers are represented as [[Term algebra|algebraic terms]] built using a few basic operations (e.g.,  '''zero''', '''succ''', '''pred''') and using [[natural number]]s, which are assumed to be already constructed (using the [[Peano axioms|Peano approach]]).

There exist at least ten such constructions of signed integers.<ref>{{cite conference |last=Garavel |first=Hubert |title=On the Most Suitable Axiomatization of Signed Integers |conference=Post-proceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016) |year=2017 |publisher=Springer |series=Lecture Notes in Computer Science |volume=10644 |pages=120–134 |doi=10.1007/978-3-319-72044-9_9 |isbn=978-3-319-72043-2 |url=https://hal.inria.fr/hal-01667321 |access-date=2018-01-25 |archive-url=https://web.archive.org/web/20180126125528/https://hal.inria.fr/hal-01667321 |archive-date=2018-01-26 |url-status=live }}</ref> These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2), and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.

The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation '''pair'''<math>(x,y)</math> that takes as arguments two natural numbers <math>x</math> and <math>y</math>, and returns an integer (equal to <math>x-y</math>). This operation is not free since the integer 0 can be written '''pair'''(0,0), or '''pair'''(1,1), or '''pair'''(2,2), etc.. This technique of construction is used by the [[proof assistant]] [[Isabelle (proof assistant)|Isabelle]]; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.