path: root/src/algebra/op.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra op.spad}
\author{Manuel Bronstein \and Gabriel Dos~Reis}
\maketitle
\begin{abstract}
\end{abstract}
\tableofcontents
\eject

\section{domain BOP BasicOperator}
<<domain BOP BasicOperator>>=
)abbrev domain BOP BasicOperator
++ Basic system operators
++ Author: Manuel Bronstein, Gabriel Dos Reis
++ Date Created: 22 March 1988
++ Date Last Updated: May 09, 2009
++ Description:
++   A basic operator is an object that can be applied to a list of
++   arguments from a set, the result being a kernel over that set.
++ Keywords: operator, kernel.
BasicOperator(): Exports == Implementation where
  O   ==> OutputForm
  P   ==> AssociationList(String, None)
  L   ==> List Record(key:String, entry:None)
  SEX ==> InputForm

  Exports == Join(OrderedSet, OperatorCategory Symbol) with
    properties: $ -> P
      ++ properties(op) returns the list of all the properties
      ++ currently attached to op.
    copy      : $ -> $
      ++ copy(op) returns a copy of op.
    operator  : Symbol -> $
      ++ operator(f) makes f into an operator with arbitrary arity.
    operator  : (Symbol, NonNegativeInteger) -> $
      ++ operator(f, n) makes f into an n-ary operator.
    operator  : (Symbol, Arity) -> $
      ++ \spad{operator(f, a)} makes \spad{f} into an operator
      ++ of arity \spad{a}.
    nullary?  : $ -> Boolean
      ++ nullary?(op) tests if op is nullary.
    unary?    : $ -> Boolean
      ++ unary?(op) tests if op is unary.
    nary?     : $ -> Boolean
      ++ nary?(op) tests if op has arbitrary arity.
    weight    : $ -> NonNegativeInteger
      ++ weight(op) returns the weight attached to op.
    weight    : ($, NonNegativeInteger) -> $
      ++ weight(op, n) attaches the weight n to op.
    equality   : ($, ($, $) -> Boolean) -> $
      ++ equality(op, foo?) attaches foo? as the "%equal?" property
      ++ to op. If op1 and op2 have the same name, and one of them
      ++ has an "%equal?" property f, then \spad{f(op1, op2)} is called to
      ++ decide whether op1 and op2 should be considered equal.
    comparison : ($, ($, $) -> Boolean) -> $
      ++ comparison(op, foo?) attaches foo? as the "%less?" property
      ++ to op. If op1 and op2 have the same name, and one of them
      ++ has a "%less?" property f, then \spad{f(op1, op2)} is called to
      ++ decide whether \spad{op1 < op2}.
    display    : $ -> Maybe(List O -> O)
      ++ display(op) returns the "%display" property of op if
      ++ it has one attached, and \spad{nothing} otherwise.
    display    : ($, List O -> O)      -> $
      ++ display(op, foo) attaches foo as the "%display" property
      ++ of op. If op has a "%display" property f, then \spad{op(a1,...,an)}
      ++ gets converted to OutputForm as \spad{f(a1,...,an)}.
    display    : ($, O -> O)           -> $
      ++ display(op, foo) attaches foo as the "%display" property
      ++ of op. If op has a "%display" property f, then \spad{op(a)}
      ++ gets converted to OutputForm as \spad{f(a)}.
      ++ Argument op must be unary.
    input      : ($, List SEX -> SEX)  -> $
      ++ input(op, foo) attaches foo as the "%input" property
      ++ of op. If op has a "%input" property f, then \spad{op(a1,...,an)}
      ++ gets converted to InputForm as \spad{f(a1,...,an)}.
    input      : $ -> Maybe(List SEX -> SEX)
      ++ input(op) returns the "%input" property of op if
      ++ it has one attached, \spad{nothing} otherwise.
    has?       : (%, Identifier) -> Boolean
      ++ \spad{has?(op,p)} tests if property \spad{s} is attached to \spad{op}.
    assert     : (%, Identifier) -> $
      ++ \spad{assert(op, p)} attaches property \spad{p} to \spad{op}.
      ++ Argument op is modified "in place", i.e. no copy is made.
    deleteProperty!: ($, String) -> $
      ++ deleteProperty!(op, s) unattaches property s from op.
      ++ Argument op is modified "in place", i.e. no copy is made.
    deleteProperty!: ($, Identifier) -> $
      ++ \spad{deleteProperty!(op, p)} unattaches property \spad{p} from
      ++ \spad{op}. Argument \spad{op} is modified "in place", 
      ++ i.e. no copy is made.
    property      : ($, String) -> Union(None, "failed")
      ++ property(op, s) returns the value of property s if
      ++ it is attached to op, and "failed" otherwise.
    property : (%, Identifier) -> Maybe None
      ++ \spad{property(op, p)} returns the value of property \spad{p} if
      ++ it is attached to \spad{op}, otherwise \spad{nothing}.
    setProperty   : ($, String, None) -> $
      ++ setProperty(op, s, v) attaches property s to op,
      ++ and sets its value to v.
      ++ Argument op is modified "in place", i.e. no copy is made.
    setProperty   : ($, Identifier, None) -> $
      ++ \spad{setProperty(op, p, v)} attaches property \spad{p} to \spad{op},
      ++ and sets its value to \spad{v}.
      ++ Argument \spad{op} is modified "in place", i.e. no copy is made.
    setProperties : ($, P) -> $
      ++ setProperties(op, l) sets the property list of op to l.
      ++ Argument op is modified "in place", i.e. no copy is made.

  Implementation ==> add
    -- some internal properties
    macro LESS? == '%less?
    macro EQUAL? == '%equal?
    macro WEIGHT == '%weight
    macro DISPLAY == '%display
    macro SEXPR == '%input
    import Arity
    import String
    -- if narg < 0 then the operator ahs variable arity.
    Rep == Record(opname:Symbol, narg: Arity, props:P)

    is?(op, s)           == name(op) = s
    name op              == rep(op).opname
    properties op        == rep(op).props
    setProperties(op, l) == 
      rep(op).props := l
      op
    operator(s: Symbol)  == per [s, arbitrary(), table()]
    operator(s: Symbol, n: NonNegativeInteger) == per [s, n::Arity, table()]
    operator(s: Symbol, a: Arity) == per [s, a, table()]
    property(op: %, name: String)   == search(name, rep(op).props)
    property(op: %, p: Identifier) == 
      case search(string p, rep(op).props) is
        val@None => just val
        otherwise => nothing
    assert(op: %, p: Identifier) == setProperty(op, p, NIL$Lisp)
    has?(op: %, name: Identifier) == key?(string name, rep(op).props)
    weight(op, n)        == setProperty(op, WEIGHT, n pretend None)
    nullary? op          == zero? rep(op).narg
    unary? op            == one? rep(op).narg
    nary? op             == arbitrary() = rep(op).narg
    equality(op, func)   == setProperty(op, EQUAL?, func pretend None)
    comparison(op, func) == setProperty(op, LESS?, func pretend None)
    display(op:$, f:O -> O)        == display(op, f first #1)
    deleteProperty!(op: %, name: String) == (remove!(name, properties op); op)
    deleteProperty!(op: %, p: Identifier) == 
      remove!(string p, properties op)
      op
    setProperty(op: %, name: String, valu: None) == 
      rep(op).props.name := valu
      op
    setProperty(op: %, p: Identifier, valu: None) == 
      rep(op).props.(string p) := valu
      op
    coerce(op:$):OutputForm        == name(op)::OutputForm
    input(op:$, f:List SEX -> SEX) == setProperty(op, SEXPR, f pretend None)
    display(op:$, f:List O -> O)   == setProperty(op, DISPLAY, f pretend None)

    display op ==
      property(op, DISPLAY) pretend Maybe(List O -> O)

    input op ==
      property(op, SEXPR) pretend Maybe(List SEX -> SEX)

    arity op == rep(op).narg

    copy op ==
      per [name op, rep(op).narg,
            table([[r.key, r.entry] for r in entries(properties op)@L]$L)]

-- property EQUAL? contains a function f: (BOP, BOP) -> Boolean
-- such that f(o1, o2) is true iff o1 = o2
    op1 = op2 ==
      name(op1) ~= name(op2) => false
      rep(op1).narg ~= rep(op2).narg => false
      brace(keys properties op1)~=$Set(String) brace(keys properties op2) => false
      (func := property(op1, EQUAL?)) case None =>
                   ((func@None) pretend (($, $) -> Boolean)) (op1, op2)
      true

-- property WEIGHT allows one to change the ordering around
-- by default, every operator has weigth 1
    weight op ==
      (w := property(op, WEIGHT)) case nothing => 1
      (w@None) pretend NonNegativeInteger

-- property LESS? contains a function f: (BOP, BOP) -> Boolean
-- such that f(o1, o2) is true iff o1 < o2
    op1 < op2 ==
      (w1 := weight op1) ~= (w2 := weight op2) => w1 < w2
      rep(op1).narg ~= rep(op2).narg => 
        -- FIXME: Horrible.
        (rep(op1).narg pretend SingleInteger) < (rep(op2).narg pretend SingleInteger)
      name(op1) ~= name(op2) => name(op1) < name(op2)
      n1 := #(k1 := brace(keys(properties op1))$Set(String))
      n2 := #(k2 := brace(keys(properties op2))$Set(String))
      n1 ~= n2 => n1 < n2
      not zero?(n1 := #(d1 := difference(k1, k2))) =>
        n1 ~= (n2 := #(d2 := difference(k2, k1))) => n1 < n2
        inspect(d1) < inspect(d2)
      (func := property(op1, LESS?)) case None =>
                   ((func@None) pretend (($, $) -> Boolean)) (op1, op2)
      (func := property(op1, EQUAL?)) case None =>
              not(((func@None) pretend (($, $) -> Boolean)) (op1, op2))
      false

@
\section{package BOP1 BasicOperatorFunctions1}
<<package BOP1 BasicOperatorFunctions1>>=
)abbrev package BOP1 BasicOperatorFunctions1
++ Tools to set/get common properties of operators
++ Author: Manuel Bronstein
++ Date Created: 28 Mar 1988
++ Date Last Updated: 15 May 1990
++ Description:
++   This package exports functions to set some commonly used properties
++   of operators, including properties which contain functions.
++ Keywords: operator.
BasicOperatorFunctions1(A:SetCategory): Exports == Implementation where
  OP   ==> BasicOperator

  Exports ==> with
    evaluate        : (OP, List A)      -> Union(A, "failed")
      ++ evaluate(op, [a1,...,an]) checks if op has an "%eval"
      ++ property f. If it has, then \spad{f(a1,...,an)} is returned, and
      ++ "failed" otherwise.
    evaluate        : (OP, List A -> A) -> OP
      ++ evaluate(op, foo) attaches foo as the "%eval" property
      ++ of op. If op has an "%eval" property f, then applying op
      ++ to \spad{(a1,...,an)} returns the result of \spad{f(a1,...,an)}.
    evaluate        : (OP, A -> A)      -> OP
      ++ evaluate(op, foo) attaches foo as the "%eval" property
      ++ of op. If op has an "%eval" property f, then applying op
      ++ to a returns the result of \spad{f(a)}. Argument op must be unary.
    evaluate        : OP                -> Union(List A -> A, "failed")
      ++ evaluate(op) returns the value of the "%eval" property of
      ++ op if it has one, and "failed" otherwise.
    derivative      : (OP, List (List A -> A)) -> OP
      ++ derivative(op, [foo1,...,foon]) attaches [foo1,...,foon] as
      ++ the "%diff" property of op. If op has an "%diff" property
      ++ \spad{[f1,...,fn]} then applying a derivation D to \spad{op(a1,...,an)}
      ++ returns \spad{f1(a1,...,an) * D(a1) + ... + fn(a1,...,an) * D(an)}.
    derivative      : (OP, A -> A) -> OP
      ++ derivative(op, foo) attaches foo as the "%diff" property
      ++ of op. If op has an "%diff" property f, then applying a
      ++ derivation D to op(a) returns \spad{f(a) * D(a)}. Argument op must be unary.
    derivative      : OP -> Union(List(List A -> A), "failed")
      ++ derivative(op) returns the value of the "%diff" property of
      ++ op if it has one, and "failed" otherwise.
    constantOperator: A -> OP
      ++ constantOperator(a) returns a nullary operator op
      ++ such that \spad{op()} always evaluate to \spad{a}.
    constantOpIfCan : OP -> Union(A, "failed")
      ++ constantOpIfCan(op) returns \spad{a} if op is the constant
      ++ nullary operator always returning \spad{a}, "failed" otherwise.

  Implementation ==> add
    macro EVAL == '%eval
    macro CONST == '%constant
    macro DIFF == '%diff
    evaluate(op:OP, func:A -> A) == evaluate(op, func first #1)

    evaluate op ==
      (func := property(op, EVAL)) case nothing => "failed"
      (func@None) pretend (List A -> A)

    evaluate(op:OP, args:List A) ==
      (func := property(op, EVAL)) case nothing => "failed"
      ((func@None) pretend (List A -> A)) args

    evaluate(op:OP, func:List A -> A) ==
      setProperty(op, EVAL, func pretend None)

    derivative op ==
      (func := property(op, DIFF)) case nothing => "failed"
      ((func@None) pretend List(List A -> A))

    derivative(op:OP, grad:List(List A -> A)) ==
      setProperty(op, DIFF, grad pretend None)

    derivative(op:OP, f:A -> A) ==
      unary? op or nary? op =>
        derivative(op, [f first #1]$List(List A -> A))
      error "Operator is not unary"

    cdisp(a: OutputForm, l: List OutputForm): OutputForm == a

    csex(a: InputForm, l: List InputForm): InputForm  == a

    eqconst?(a: OP, b: OP): Boolean ==
      (va := property(a, CONST)) case nothing => not has?(b, CONST)
      ((vb := property(b, CONST)) case None) and
	 ((va@None) pretend A) = ((vb@None) pretend A)

    ltconst?(a: OP, b: OP): Boolean ==
      (va := property(a, CONST)) case nothing => has?(b, CONST)
      ((vb := property(b, CONST)) case None) and
	 before?((va@None) pretend A, (vb@None) pretend A)

    opconst:OP :=
      comparison(equality(operator('constant, 0), eqconst?),
							     ltconst?)

    constOp(a: A): OP ==
      setProperty(display(copy opconst, cdisp(a::OutputForm, #1)),
						CONST, a pretend None)

    constantOpIfCan op ==
      is?(op,'constant) and
	((u := property(op, CONST)) case None) => (u@None) pretend A
      "failed"

    if A has ConvertibleTo InputForm then
      constantOperator a == input(constOp a, csex(convert a, #1))
    else
      constantOperator a == constOp a

@
\section{package COMMONOP CommonOperators}
<<package COMMONOP CommonOperators>>=
)abbrev package COMMONOP CommonOperators
++ Provides commonly used operators
++ Author: Manuel Bronstein
++ Date Created: 25 Mar 1988
++ Date Last Updated: 2 December 1994
++ Description:
++ This package exports the elementary operators, with some semantics
++ already attached to them. The semantics that is attached here is not
++ dependent on the set in which the operators will be applied.
++ Keywords: operator.
CommonOperators(): Exports == Implementation where
  OP  ==> BasicOperator
  O   ==> OutputForm

  Exports ==> with
    operator: Symbol -> OP
        ++ operator(s) returns an operator with name s, with the
        ++ appropriate semantics if s is known. If s is not known,
        ++ the result has no semantics.

  Implementation ==> add
    macro POWER == '%power
    macro ALGOP == '%alg
    macro EVEN  == 'even
    macro ODD   == 'odd
    DUMMYVAR ==> "%dummyVar"
    dpi        : List O -> O
    dgamma     : List O -> O
    dquote     : List O -> O
    dexp       : O -> O
    dfact      : O -> O
    startUp    : Boolean -> Void
    setDummyVar: (OP, NonNegativeInteger) -> OP

    brandNew?:Reference(Boolean) := ref true

    opalg   := operator('rootOf, 2)$OP
    oproot  := operator('nthRoot, 2)
    oppi    := operator('pi, 0)
    oplog   := operator('log, 1)
    opexp   := operator('exp, 1)
    opabs   := operator('abs, 1)
    opsin   := operator('sin, 1)
    opcos   := operator('cos, 1)
    optan   := operator('tan, 1)
    opcot   := operator('cot, 1)
    opsec   := operator('sec, 1)
    opcsc   := operator('csc, 1)
    opasin  := operator('asin, 1)
    opacos  := operator('acos, 1)
    opatan  := operator('atan, 1)
    opacot  := operator('acot, 1)
    opasec  := operator('asec, 1)
    opacsc  := operator('acsc, 1)
    opsinh  := operator('sinh, 1)
    opcosh  := operator('cosh, 1)
    optanh  := operator('tanh, 1)
    opcoth  := operator('coth, 1)
    opsech  := operator('sech, 1)
    opcsch  := operator('csch, 1)
    opasinh := operator('asinh, 1)
    opacosh := operator('acosh, 1)
    opatanh := operator('atanh, 1)
    opacoth := operator('acoth, 1)
    opasech := operator('asech, 1)
    opacsch := operator('acsch, 1)
    opbox   := operator('%box)$OP
    oppren  := operator('%paren)$OP
    opquote := operator('applyQuote)$OP
    opdiff  := operator('%diff, 3)
    opsi    := operator('Si, 1)
    opci    := operator('Ci, 1)
    opei    := operator('Ei, 1)
    opli    := operator('li, 1)
    operf   := operator('erf, 1)
    opli2   := operator('dilog, 1)
    opGamma     := operator('Gamma, 1)
    opGamma2    := operator('Gamma2, 2)
    opBeta      := operator('Beta, 2)
    opdigamma   := operator('digamma, 1)
    oppolygamma := operator('polygamma, 2)
    opBesselJ   := operator('besselJ, 2)
    opBesselY   := operator('besselY, 2)
    opBesselI   := operator('besselI, 2)
    opBesselK   := operator('besselK, 2)
    opAiryAi    := operator('airyAi,  1)
    opAiryBi    := operator('airyBi , 1)
    opint   := operator('integral, 3)
    opdint  := operator('%defint, 5)
    opfact  := operator('factorial, 1)
    opperm  := operator('permutation, 2)
    opbinom := operator('binomial, 2)
    oppow   := operator(POWER, 2)
    opsum   := operator('summation, 3)
    opdsum  := operator('%defsum, 5)
    opprod  := operator('product, 3)
    opdprod := operator('%defprod, 5)

    algop   := [oproot, opalg]$List(OP)
    rtrigop := [opsin, opcos, optan, opcot, opsec, opcsc,
                         opasin, opacos, opatan, opacot, opasec, opacsc]
    htrigop := [opsinh, opcosh, optanh, opcoth, opsech, opcsch,
                   opasinh, opacosh, opatanh, opacoth, opasech, opacsch]
    trigop  := concat(rtrigop, htrigop)
    elemop  := concat(trigop, [oppi, oplog, opexp])
    primop  := [opei, opli, opsi, opci, operf, opli2, opint, opdint]
    combop  := [opfact, opperm, opbinom, oppow,
                                         opsum, opdsum, opprod, opdprod]
    specop  := [opGamma, opGamma2, opBeta, opdigamma, oppolygamma, opabs,
                opBesselJ, opBesselY, opBesselI, opBesselK]
    anyop   := [oppren, opdiff, opbox, opquote]
    allop   := concat(concat(concat(concat(concat(
                            algop,elemop),primop),combop),specop),anyop)

-- odd and even operators, must be maintained current!
    evenop := [opcos, opsec, opcosh, opsech, opabs]
    oddop  := [opsin, opcsc, optan, opcot, opasin, opacsc, opatan,
               opsinh, opcsch, optanh, opcoth, opasinh, opacsch,opatanh,opacoth,
                opsi, operf]

-- operators whose second argument is a dummy variable
    dummyvarop1 := [opdiff,opalg, opint, opsum, opprod]
-- operators whose second and third arguments are dummy variables
    dummyvarop2 := [opdint, opdsum, opdprod]

    operator s ==
      if (deref brandNew?) then startUp false
      for op in allop repeat
        is?(op, s) => return copy op
      operator(s)$OP

    dpi l    == '%pi::O
    dfact x  == postfix("!"::Symbol::O, (not %pair?(x)$Foreign(Builtin) => x; paren x))
    dquote l == prefix(quote(first(l)::O), rest l)
    dgamma l == prefix(hconcat("|"::Symbol::O, overbar(" "::Symbol::O)), l)
    setDummyVar(op, n) == setProperty(op, DUMMYVAR, n pretend None)

    dexp x ==
      e := '%e::O
      x = 1::O => e
      e ** x

    startUp b ==
      setref(brandNew?,b)
      display(oppren, paren)
      display(opbox, commaSeparate)
      display(oppi, dpi)
      display(opexp, dexp)
      display(opGamma, dgamma)
      display(opGamma2, dgamma)
      display(opfact, dfact)
      display(opquote, dquote)
      display(opperm, supersub('A::O, #1))
      display(opbinom, binomial(first #1, second #1))
      display(oppow, first(#1) ** second(#1))
      display(opsum,   sum(first #1, second #1, third #1))
      display(opprod, prod(first #1, second #1, third #1))
      display(opint, int(first #1 * hconcat('d::O, second #1),
                                                   empty(), third #1))
      input(oppren, convert concat(convert("("::Symbol)@InputForm,
                            concat(#1, convert(")"::Symbol)@InputForm)))
      input(oppow, convert concat(convert("**"::Symbol)@InputForm, #1))
      input(oproot,
            convert [convert("**"::Symbol)@InputForm, first #1, 1 / second #1])
      for op in algop   repeat assert(op, ALGOP)
      for op in rtrigop repeat assert(op, 'rtrig)
      for op in htrigop repeat assert(op, 'htrig)
      for op in trigop  repeat assert(op, 'trig)
      for op in elemop  repeat assert(op, 'elem)
      for op in primop  repeat assert(op, 'prim)
      for op in combop  repeat assert(op, 'comb)
      for op in specop  repeat assert(op, 'special)
      for op in anyop   repeat assert(op, 'any)
      for op in evenop  repeat assert(op, EVEN)
      for op in oddop   repeat assert(op, ODD)
      for op in dummyvarop1 repeat setDummyVar(op, 1)
      for op in dummyvarop2 repeat setDummyVar(op, 2)
      assert(oppren, 'linear)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2009, 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.
@
<<*>>=
<<license>>

-- SPAD files for the functional world should be compiled in the
-- following order:
--
--   OP  kl  expr function

<<domain BOP BasicOperator>>
<<package BOP1 BasicOperatorFunctions1>>
<<package COMMONOP CommonOperators>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}