This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. fns1 http://www.openmath.org/cd http://www.openmath.org/cd/fns1.ocd 2014-04-01 2009-04-01 4 0 Author: OpenMath Consortium SourceURL: https://github.com/OpenMath/CDs official This CD is intended to be `compatible' with the corresponding elements in Content MathML. In this CD we give a set of functions concerning functions themselves. Functions can be constructed from expressions via a lambda expression. Also there are basic function functions like compose, etc. restriction application restriction takes two arguments, a function f, and a set S, which should be a subset of domain(f) and returns the function f restricted to S. domainofapplication application Deprecated. This symbol was intended to model MathML domainofapplication but as defined it is a synonym for domain. In MathML3, MathML compatibility is defined to use the new restriction symbol. domain application This symbol denotes the domain of a given function, which is the set of values it is defined over. x is in the domain of f if and only if there exists a y in the range of f and f(x) = y range application This symbol denotes the range of a function, that is a set that the function will map to. The single argument should be the function whos range is being queried. It should be noted that this is not necessarily equal to the image, it is merely required to contain the image. the range of f is a subset of the image of f image application This symbol denotes the image of a given function, which is the set of values the domain of the given function maps to. x in the domain of f implies f(x) is in the image f identity application The identity function, it takes one argument and returns the same value. for all x | identity(x)=x left_inverse application This symbol is used to describe the left inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the left inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this left inverse. right_inverse application This symbol is used to describe the right inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the right inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this right inverse. inverse application This symbol is used to describe the inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the inverse function is ill-defined without further stipulations. No assumptions are made on the semantics of this inverse. (inverse(f))(f(x)) = x if f is injective, that is (for all x,y | f(x) = f(y) implies x=y) implies (inverse(f))(f(z)) = z f(inverse(f(y))=y if f is defined at inverse(f)(y) that is, if there exists an x s.t. inverse(f)(y) = x then this implies f(inverse(f)(y)) = y left_compose application This symbol represents the function which forms the left-composition of its two (function) arguments. for all f,g,x | left_compose(f,g)(x) = f(g(x)) lambda binder This symbol is used to represent anonymous functions as lambda expansions. It is used in a binder that takes two further arguments, the first of which is a list of variables, and the second of which is an expression, and it forms the function which is the lambda extraction of the expression An example to show the connection between curried and uncurried applications of a binary function f (lambda(x,y).(f))(a,b)= (lambda(x).((lambda(y).(f))(b)))(a)