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SandBoxNonZeroInteger
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SandBoxNonZeroIntegerInOpenAxiom
  
last edited 16 years ago by page 

Algebra

SandBoxNonZeroInteger is an attempt to define the domain of Integers without 0 as a SubDomain.

\begin{spad} )abbrev domain NZINT NonZeroInteger NonZeroInteger: Join(OrderedAbelianSemiGroup, Monoid --,CoercibleTo(Integer) ) with commutative("*") gcd: (%,%) -> % ++ gcd(a,b) computes the greatest common divisor of two ++ positive integers \spad{a} and b. _-: % -> % --retract:Integer->% --retractIfCan:Integer->Union(%,"failed") --retract:NonNegativeInteger->% --retractIfCan:NonNegativeInteger->Union(%,"failed") --convert:NonNegativeInteger->% == SubDomain(Integer,not(#1 = 0)) add Rep == Integer x:% y:% z:Integer nz:NonNegativeInteger -x == (-(x pretend Integer)) pretend % x+y == (z:=(x pretend Integer)+(y pretend Integer)) = 0 => error "zero" z pretend %

--coerce(x):Integer == x pretend Integer --retract(z)== z pretend % --convert(nz):% == retract(nz) --retractIfCan(z) == -- zero?(z) => "failed" -- retract(z) --retract(nz)== nz pretend % --retractIfCan(nz) == -- zero?(nz) => "failed" -- retract(nz) \end{spad}

\begin{axiom} )show NZINT \end{axiom}

\begin{axiom} i:NZINT := 1$NZINT i:NZINT := (1::Integer) i:NZINT := 1 \end{axiom} \begin{axiom} j:NZINT := -1 \end{axiom} \begin{axiom} j:NZINT:=-i i+1 j+1 i+j \end{axiom}

\begin{axiom} i-j retractIfCan(0)$NZINT \end{axiom}

Can we define NonZero as a functor? \begin{spad} )abbrev domain NZ NonZero NonZero(T:Ring): Join(OrderedAbelianSemiGroup, Monoid) with Commutative("*") _-: % -> % == SubDomain(T,not(#1 = 0)) add Rep == T x:% y:% z:T -x == (-(x pretend T)) pretend % x+y == (z:=(x pretend T)+(y pretend T)) = 0 => error "zero" z pretend % \end{spad}

\begin{axiom} )show NonZero(Float) \end{axiom}

\begin{axiom} f:NonZero(Float):=1 g:NonZero(Float):=-f f+(1.1::NonZero(Float)) f + -g f+g \end{axiom}

Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3437484353503122027-25px.axm


debugger invoked on a SIMPLE-CONDITION in thread
#<THREAD "main thread" RUNNING {1001C48003}>:
  break


Checking for foreign routines
FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu"
spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so"
foreign routines found
openServer result -2
                       FriCAS Computer Algebra System 
             Version: FriCAS 1.3.10 built with sbcl 2.2.9.debian
                   Timestamp: Wed 10 Jan 02:19:45 CET 2024
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------


(1) -> (1) -> (1) -> (1) -> (1) -> (1) -> <spad>
)abbrev domain NZINT NonZeroInteger
NonZeroInteger: Join(OrderedAbelianSemiGroup, Monoid
                     --,CoercibleTo(Integer)
                    ) with
            commutative("*")
            gcd: (%,%) -> %
              ++ gcd(a,b) computes the greatest common divisor of two
              ++ positive integers \spad{a} and b.
            _-: % -> %
            --retract:Integer->%
            --retractIfCan:Integer->Union(%,"failed")
            --retract:NonNegativeInteger->%
            --retractIfCan:NonNegativeInteger->Union(%,"failed")
            --convert:NonNegativeInteger->%
 == SubDomain(Integer,not(#1 = 0)) add
     Rep == Integer
     x:%
     y:%
     z:Integer
     nz:NonNegativeInteger
     -x == (-(x pretend Integer)) pretend %
     x+y ==
      (z:=(x pretend Integer)+(y pretend Integer)) = 0 => error "zero"
      z pretend %
     --coerce(x):Integer == x pretend Integer
     --retract(z)== z pretend %
     --convert(nz):% == retract(nz)
     --retractIfCan(z) ==
     --  zero?(z) => "failed"
     --  retract(z)
     --retract(nz)== nz pretend %
     --retractIfCan(nz) ==
     --  zero?(nz) => "failed"
     --  retract(nz)</spad>
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5347600158247770088-25px001.spad
      using old system compiler.
   NZINT abbreviates domain NonZeroInteger 
------------------------------------------------------------------------
   initializing NRLIB NZINT for NonZeroInteger 
   compiling into NRLIB NZINT 




Type HELP for debugger help, or (SB-EXT:EXIT) to exit from SBCL.


restarts (invokable by number or by possibly-abbreviated name):
  0: [CONTINUE] Return from BREAK.
  1: [ABORT   ] Exit from the current thread.


(|JoinInner| (#(NIL (((|gcd| #) T (ELT % 6)) ((- #) T (ELT % 6))) (((|commutative| "*") T)) (|Category|) (NIL NIL NIL) NIL)))
   error finding frame source: Bogus form-number: the source file has probably
                               changed too much to cope with.
   source: NIL
0]
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l.137  == SubDomain(Integer,not(#
                                 1 = 0)) add
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l.147 


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l.185  == SubDomain(T,not(#
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                $
l.194 \end{spad}
                \newpage


Overfull \hbox (19.6396pt too wide) in paragraph at lines 181--194
\T1/cmr/m/n/12 )abbrev do-main NZ NonZero NonZero(T:Ring): Join(OrderedAbelianS
emiGroup, Monoid) with Com-mu-ta-tive("*") $[] \OT1/cmr/m/n/12 :== \OML/cmm/m/i
t/12 SubDomain\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 T; not\OT1/cmr/m/n/12 (1 =
[2]


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