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\documentclass{article}
\usepackage{open-axiom}
\title{src/algebra term.spad}
\begin{document}
\author{Gabriel Dos~Reis}
\maketitle
\begin{abstract}
\end{abstract}
This file contains specifications and implementations for basic
support of term algebras, in the sense of Universal Algebra.
\tableofcontents
\eject
\section{The \code{Arity} domain}
<<domain ARITY Arity>>=
)abbrev domain ARITY Arity
++ Author: Gabriel Dos Reis
++ Date Created: December 04, 2008
++ Date Last Updated: May 09, 2009
++ Description:
++ This domain implements the arity of a function or an operator,
++ e.g. the number of arguments that an operator can take. An
++ arity is either a definition nonnegative integer, and the special
++ value `arbitrary', signifying that an operation can take any
++ number of arguments.
Arity(): Public == Private where
Public == Join(SetCategory, RetractableTo NonNegativeInteger) with
arbitrary: %
++ aribitrary is the arity of a function that accepts any
++ number of arguments.
zero?: % -> Boolean
++ \spad{zero? a} holds if \spad{a} is the arity of niladic function.
one?: % -> Boolean
++ \spad{one? a} holds if \spad{a} is the arity of nullary function.
Private == add
Rep == SingleInteger
arbitrary == per(-1)
zero? a == zero? rep a
one? a == one? rep a
hash x == hash rep x
x = y == rep x = rep y
coerce(x: %): OutputForm ==
x = arbitrary => 'arbitrary::OutputForm
rep(x)::OutputForm
coerce(n: NonNegativeInteger): % ==
max()$SingleInteger < n =>
error "arity is too large for OpenAxiom"
per(n::SingleInteger)
retractIfCan x ==
negative?(n := rep x) => "failed"
n@Integer::NonNegativeInteger
@
\section{Operator Interface: \code{OperatorCategory}}
<<category OPERCAT OperatorCategory>>=
)abbrev category OPERCAT OperatorCategory
++ Author: Gabriel Dos Reis
++ Date Created: December 04, 2008
++ Date Last Updated: December 17, 2008
++ Description:
++ This category specifies the interface for operators used to build
++ terms, in the sense of Universal Algebra. The domain parameter S
++ provides representation for the `external name' of an operator.
OperatorCategory(S: SetCategory): Category ==
SetCategory with
name: % -> S
++ \spad{name(op)} returns the externam name of \spad{op}.
arity: % -> Arity
++ \spad{arity(op)} returns the arity of the operator \spad{op}.
is?: (%,S) -> Boolean
++ \spad{is?(op,n)} holds if the name of the operator \spad{op}
++ is \spad{n}.
add
coerce(op: %): OutputForm == name(op)::OutputForm
is?(op,n) == name op = n
@
\section{Term Algebra Operator}
<<domain TALGOP TermAlgebraOperator>>=
)abbrev domain TALGOP TermAlgebraOperator
++ Author: Gabriel Dos Reis
++ Date Created: April 17, 2010
++ Date Last Modified: April 17, 2010
TermAlgebraOperator(S: SetCategory): Public == Private where
Public == OperatorCategory S with
operator: (S,Arity) -> %
++ \spad{operator(n,a)} returns an operator named \spad{n}
++ and with arity \spad{a}.
Private == add
Rep == Pair(S,Arity)
operator(n,a) == per pair(n,a)
name x == first rep x
arity x == second rep x
@
\section{License}
<<license>>=
--Copyright (C) 2007-2010, Gabriel Dos Reis
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
\section{Putting It Altogether}
<<*>>=
<<license>>
<<domain ARITY Arity>>
<<category OPERCAT OperatorCategory>>
<<domain TALGOP TermAlgebraOperator>>
@
\end{document}
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