Here we give two examples of defining functions by mutual recursion in FriCAS
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.
(1) -> <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 SEC.
(time taken in buildFunctor: 0) Time: 0 SEC.
Cumulative Statistics for Constructor even Time: 0 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 10 OCT 2025 03:54:34 AM):
; wrote /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.fasl ; compilation finished in 0:00:00.008 ------------------------------------------------------------------------ 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)$odd. It depends on EVEN.
)abbrev domain ODD odd
odd(): E == I where
E == with
parity: Integer -> Boolean
I == add
parity(n:Integer) ==
-- output("ODD",
Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7392352806104595030-25px002.spad using old system compiler. ODD abbreviates domain odd ------------------------------------------------------------------------ initializing NRLIB ODD for odd compiling into NRLIB ODD compiling exported parity : Integer -> Boolean Time: 0 SEC.
(time taken in buildFunctor: 0) Time: 0 SEC.
Cumulative Statistics for Constructor odd Time: 0 seconds
finalizing NRLIB ODD Processing odd for Browser database: --->-->odd(constructor): Not documented!!!! --->-->odd((parity ((Boolean) (Integer)))): Not documented!!!! --->-->odd(): Missing Description ; compiling file "/var/aw/var/LatexWiki/ODD.NRLIB/ODD.lsp" (written 10 OCT 2025 03:54:34 AM):
; wrote /var/aw/var/LatexWiki/ODD.NRLIB/ODD.fasl ; compilation finished in 0:00:00.008 ------------------------------------------------------------------------ odd is now explicitly exposed in frame initial odd will be automatically loaded when needed from /var/aw/var/LatexWiki/ODD.NRLIB/ODD
But the bootstrap definition of EVEN is incomplete. It really
depends (recusively) on ODD. So finally we need the full
(re-)definition of parity(n)$even
)abbrev domain EVEN even
even(): E == I where
E == with
parity: Integer -> Boolean
I == add
parity(n) ==
-- output("EVEN",
Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8584362905457126194-25px003.spad using old system compiler. Illegal NRLIB EVENNRLIB 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 SEC.
(time taken in buildFunctor: 0)
;;; *** |even| REDEFINED
;;; *** |even| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor even Time: 0 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 10 OCT 2025 03:54:34 AM):
; wrote /var/aw/var/LatexWiki/EVEN.NRLIB/EVEN.fasl ; compilation finished in 0:00:00.004 ------------------------------------------------------------------------ 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:
parity(10)$even
parity(8)$odd
parity(-1111)$odd
It is possible to write this same recursion as a domain that exports two functions Even and Odd. In this case we do not need to supply any initial bootstrap code because the compiler is able to resolve both functions simultaneously.
)abbrev domain PARITY Parity Parity(): Exports == Implements where Exports == with Even: Integer -> Boolean Odd: Integer -> Boolean Implements == add Odd(n: Integer) == n>0 => Even(n-1) n<0 => Even(n+1) false Even(n: Integer) == n>0 => Odd(n-1) n<0 => Odd(n+1) true
Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2188084076688500475-25px005.spad using old system compiler. PARITY abbreviates domain Parity ------------------------------------------------------------------------ initializing NRLIB PARITY for Parity compiling into NRLIB PARITY compiling exported Odd : Integer -> Boolean Time: 0 SEC.
compiling exported Even : Integer -> Boolean Time: 0 SEC.
(time taken in buildFunctor: 0) Time: 0 SEC.
Cumulative Statistics for Constructor Parity Time: 0 seconds
finalizing NRLIB PARITY Processing Parity for Browser database: --->-->Parity(constructor): Not documented!!!! --->-->Parity((Even ((Boolean) (Integer)))): Not documented!!!! --->-->Parity((Odd ((Boolean) (Integer)))): Not documented!!!! --->-->Parity(): Missing Description ; compiling file "/var/aw/var/LatexWiki/PARITY.NRLIB/PARITY.lsp" (written 10 OCT 2025 03:54:34 AM):
; wrote /var/aw/var/LatexWiki/PARITY.NRLIB/PARITY.fasl ; compilation finished in 0:00:00.008 ------------------------------------------------------------------------ Parity is now explicitly exposed in frame initial Parity will be automatically loaded when needed from /var/aw/var/LatexWiki/PARITY.NRLIB/PARITY
Test
Even(10)$Parity
Odd(8)$Parity
Odd(-1111)$Parity