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Here we give two examples of defining functions by mutual recursion in Axiom

Recursion between Separate Domains

Mutual recursion between domains is allowed. However, compiling such domains is a little tricky. Fist one need to compile files in "bootstrap mode". That is one needs to give:

)boot $bootStrapMode := true

Then compile domains first time using )compile. Next, compile again in normal way. That is give

)boot $bootStrapMode := false

and angain use )compile.

Note, that goal of "bootstrap mode" compilation is to break cycles, so for example if there are two mutially dependent domains, then it is enough to compile in "bootstrap mode" just one.

This process can not be used on the wiki, because wiki software automatically issues compilation commands. Instead one can use manual workaround which we describe below.

First we show how to define two separate domains EVEN and ODD both of which have an attribute represented by the polymorthic function parity.

We must start with a "bootstrap" definition of parity(n)$even. All that is really needed here is a package that exports a function named parity with the right signature. This particular function will never be called and will be re-defined later. It's only purpose is as a placeholder to allow the later definition of ODD.

spad
)abbrev domain EVEN even
even(): E == I where
  E == with
    parity: Integer -> Boolean
  I == add
    parity(n) == true
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad
      using old system compiler.
   EVEN abbreviates domain even 
------------------------------------------------------------------------
   initializing NRLIB EVEN for even 
   compiling into NRLIB EVEN 
   compiling exported parity : Integer -> Boolean
      EVEN;parity;IB;1 is replaced by QUOTET 
Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |even| REDEFINED
;;; *** |even| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor even Time: 0.01 seconds
finalizing NRLIB EVEN Processing even for Browser database: --->-->even(constructor): Not documented!!!! --->-->even((parity ((Boolean) (Integer)))): Not documented!!!! --->-->even(): Missing Description ; compiling file "/var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.lsp" (written 07 MAY 2016 12:50:34 PM):
; /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.fasl written ; compilation finished in 0:00:00.015 ------------------------------------------------------------------------ even is now explicitly exposed in frame initial even will be automatically loaded when needed from /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN

Now we can define parity(n)$odd. It depends on EVEN.

spad
)abbrev domain ODD odd
odd(): E == I where
  E == with
    parity: Integer -> Boolean
  I == add
    parity(n:Integer) ==
--    output("ODD",n::OutputForm)$OutputPackage
      (n>0) => parity(n-1)$even
      (n<0) => parity(n+1)$even
      false
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7392352806104595030-25px002.spad
      using old system compiler.
   ODD abbreviates domain odd 
------------------------------------------------------------------------
   initializing NRLIB ODD for odd 
   compiling into NRLIB ODD 
   compiling exported parity : Integer -> Boolean
Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |odd| REDEFINED
;;; *** |odd| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor odd Time: 0.01 seconds
finalizing NRLIB ODD Processing odd for Browser database: --->-->odd(constructor): Not documented!!!! --->-->odd((parity ((Boolean) (Integer)))): Not documented!!!! --->-->odd(): Missing Description ; compiling file "/var/aw/var/LatexWiki/ODD.NRLIB/ODD.lsp" (written 07 MAY 2016 12:50:34 PM):
; /var/aw/var/LatexWiki/ODD.NRLIB/ODD.fasl written ; compilation finished in 0:00:00.015 ------------------------------------------------------------------------ odd is now explicitly exposed in frame initial odd will be automatically loaded when needed from /var/aw/var/LatexWiki/ODD.NRLIB/ODD

But the bootstrap definition of EVEN is incomplete. It really depends (recusively) on ODD. So finally we need the full (re-)definition of parity(n)$even

spad
)abbrev domain EVEN even
even(): E == I where
  E == with
    parity: Integer -> Boolean
  I == add
    parity(n) ==
--    output("EVEN",n::OutputForm)$OutputPackage
      n>0 => parity(n-1)$odd
      n<0 => parity(n+1)$odd
      true
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8584362905457126194-25px003.spad
      using old system compiler.
   Illegal NRLIB 
   EVEN.NRLIB claims that its constructor name is the domain even but 
      even is already known to be the for package EVEN .
   EVEN abbreviates domain even 
------------------------------------------------------------------------
   initializing NRLIB EVEN for even 
   compiling into NRLIB EVEN 
   compiling exported parity : Integer -> Boolean
Time: 0.01 SEC.
(time taken in buildFunctor: 0)
;;; *** |even| REDEFINED
;;; *** |even| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor even Time: 0.01 seconds
finalizing NRLIB EVEN Processing even for Browser database: --->/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad-->even(constructor): Not documented!!!! --->/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad-->even((parity ((Boolean) (Integer)))): Not documented!!!! --->/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3342984326569348053-25px001.spad-->even(): Missing Description ; compiling file "/var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.lsp" (written 07 MAY 2016 12:50:34 PM):
; /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.fasl written ; compilation finished in 0:00:00.010 ------------------------------------------------------------------------ even is already explicitly exposed in frame initial even will be automatically loaded when needed from /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN

Now we can test the new function:

fricas
parity(10)$even

\label{eq1} \mbox{\rm true} (1)
Type: Boolean
fricas
parity(8)$odd

\label{eq2} \mbox{\rm false} (2)
Type: Boolean
fricas
parity(-1111)$odd

\label{eq3} \mbox{\rm true} (3)
Type: Boolean

Recursion within a Single Domain

It is possible to write this same recursion as a domain that exports two functions Even and Odd. In this case we do not need to supply any initial bootstrap code because the compiler is able to resolve both functions simultaneously.

spad
)abbrev domain PARITY Parity
Parity(): Exports == Implements where
  Exports == with
    Even: Integer -> Boolean
    Odd: Integer -> Boolean
  Implements == add
    Odd(n: Integer) ==
      n>0 => Even(n-1)
      n<0 => Even(n+1)
      false
    Even(n: Integer) ==
      n>0 => Odd(n-1)
      n<0 => Odd(n+1)
      true
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2188084076688500475-25px005.spad
      using old system compiler.
   PARITY abbreviates domain Parity 
------------------------------------------------------------------------
   initializing NRLIB PARITY for Parity 
   compiling into NRLIB PARITY 
   compiling exported Odd : Integer -> Boolean
Time: 0 SEC.
compiling exported Even : Integer -> Boolean Time: 0 SEC.
(time taken in buildFunctor: 10)
;;; *** |Parity| REDEFINED
;;; *** |Parity| REDEFINED Time: 0.01 SEC.
Cumulative Statistics for Constructor Parity Time: 0.01 seconds
finalizing NRLIB PARITY Processing Parity for Browser database: --->-->Parity(constructor): Not documented!!!! --->-->Parity((Even ((Boolean) (Integer)))): Not documented!!!! --->-->Parity((Odd ((Boolean) (Integer)))): Not documented!!!! --->-->Parity(): Missing Description ; compiling file "/var/aw/var/LatexWiki/PARITY.NRLIB/PARITY.lsp" (written 07 MAY 2016 12:50:34 PM):
; /var/aw/var/LatexWiki/PARITY.NRLIB/PARITY.fasl written ; compilation finished in 0:00:00.011 ------------------------------------------------------------------------ Parity is now explicitly exposed in frame initial Parity will be automatically loaded when needed from /var/aw/var/LatexWiki/PARITY.NRLIB/PARITY

Test

fricas
Even(10)$Parity

\label{eq4} \mbox{\rm true} (4)
Type: Boolean
fricas
Odd(8)$Parity

\label{eq5} \mbox{\rm false} (5)
Type: Boolean
fricas
Odd(-1111)$Parity

\label{eq6} \mbox{\rm true} (6)
Type: Boolean




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