Edit detail for SandBox Aldor Semantics revision 4 of 5

The following Aldor code was given as an example by Ralf Hemmecke and added to the wiki by Martin Rubey. See the thread:

http://lists.gnu.org/archive/html/axiom-developer/2006-07/msg00046.html

(1) -> )version

"FriCAS 1.4.0 compiled at Fri 16 Oct 16:10:51 CEST 2026"

#include "axiom"
define CatA: Category == with { }
define CatB: Category == with { }
define SomeCat: Category == with { CatA; CatB; }
Dom: SomeCat == Integer add;
A == Dom;
B: CatA == Dom;
H: CatA == Dom add;
main1():List Record(expression:String,result:Boolean) == [
  ["A has CatA",A has CatA],
  ["A has CatB",A has CatB],
  ["A has SomeCat",A has SomeCat],
  ["B has CatA",B has CatA],
  ["B has CatB",B has CatB],
  ["B has SomeCat",B has SomeCat],
  ["H has CatA",H has CatA],
  ["H has CatB",H has CatB],
  ["H has SomeCat",H has SomeCat]];

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3788028526442996283-25px002.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3788028526442996283-25px002.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

You get...

main1()

   There are no library operations named main1 
      Use HyperDoc Browse or issue
                               )what op main1
      to learn if there is any operation containing " main1 " in its 
      name.

   Cannot find a no-argument definition or library operation named 
      main1 .

In particular, that "B has CatB?" is a bit surprising, isn't it?

Bill Page replied:

  I think you are dealing here with two separate but related issues:
  1) *static* typing, and 2) inheritance rules. All types in Aldor
  are static (or at least *nearly static*) meaning that they must be
  resolved during the compilation phase.

#include "axiom"

define CatA: Category == with;
define CatB: Category == with;
define CatX: Category == with {CatA; CatB;}

A: Join(CatX, CatA) == add;
B: Join(CatX, CatB) == add;
import from Integer;
X: CatX == if odd? random(10) then A else B;

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/934600477956551444-25px004.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/934600477956551444-25px004.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

Try a little harder to create dynamic types:

#include "axiom"

import from Integer;

define CatA: Category == with {n:Integer};
define CatB: Category == with {n:Integer};
define CatX: Category == with {n:Integer};

A: Join(CatX, CatA) == add { n:Integer==1 };
B: Join(CatX, CatB) == add { n:Integer==2 };
X1: CatX == if odd? random(10) then (A add) else (B add);
X2: CatX == if even? random(10) then (A add) else (B add);

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4046171615798297799-25px005.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4046171615798297799-25px005.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

for i in 1..10 repeat output [n()$X1,n()$X2]

   X1 is not a valid type.

Or this way:

#include "axiom"
import from Integer;

define CatA: Category == with {n:Integer};
define CatB: Category == with {n:Integer};
define CatX: Category == with {CatA; CatB; n:Integer};

A: Join(CatX, CatA) == add { n:Integer==1 };
B: Join(CatX, CatB) == add { n:Integer==2 };
Y1: CatX == if odd? random(10) then A else B;
Y2: CatX == if even? random(10) then A else B;

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3471379723332304579-25px007.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3471379723332304579-25px007.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

for i in 1..10 repeat output [n()$Y1,n()$Y2]

   Y1 is not a valid type.

Or like this:

#include "axiom"
import from Integer;
define CatX: Category == with {foo: () -> Integer}
A: CatX == add {foo(): Integer == 0;}
B: CatX == add {foo(): Integer == 1;}
Z: CatX == if odd? random(10) then A else B;

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5702025508533234167-25px009.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5702025508533234167-25px009.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

for i in 1..10 repeat output foo()$Z

   Z is not a valid type.

Ralf Hemmecke asked: Why does the compiler reject the program without the "add" in line (*)?

#include "axiom"

define CatA: Category == with;
define CatB: Category == with;
define CatX: Category == with;
A: CatX with { CatA } == add;
B: CatX with { CatB } == add;
X: CatX == if true then (A add) else (B add); -- (*)

main2():List Record(expression:String,result:Boolean) == [
  ["X has CatA",X has CatA],
  ["X has CatB",X has CatB],
  ["X has CatX",X has CatX]];

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7712043450922662837-25px011.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7712043450922662837-25px011.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

main2()

   There are no library operations named main2 
      Use HyperDoc Browse or issue
                               )what op main2
      to learn if there is any operation containing " main2 " in its 
      name.

   Cannot find a no-argument definition or library operation named 
      main2 .

Christian Aistleitner provided this answer:

I'd consider that a bug in comparison of exports. Replacing your (*) line by:

   X: CatX == if true then (A@CatX) else (B@CatX);

gives a working program.

#include "axiom"

define CatA: Category == with;
define CatB: Category == with;
define CatX: Category == with;
A: CatX with { CatA } == add;
B: CatX with { CatB } == add;
X: CatX == if true then A@CatX else B@CatX; -- (*)

main3():List Record(expression:String,result:Boolean) == [
  ["X has CatA",X has CatA],
  ["X has CatB",X has CatB],
  ["X has CatX",X has CatX]];

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1208324321343406004-25px013.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1208324321343406004-25px013.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

main3()

   There are no library operations named main3 
      Use HyperDoc Browse or issue
                               )what op main3
      to learn if there is any operation containing " main3 " in its 
      name.

   Cannot find a no-argument definition or library operation named 
      main3 .

So the problem (wild guess) is that the compiler Has problems with seeing that the if statement gives CatX? in both branches of the if statement. Mainly because the types of A and B aro not equal. However, you can hint the compiler. My code is telling him "The if part gives CatX? and the else part gives CatX?". Then the compiler can infer, that the whole "if" statement gives CatX?. And it is at least the type of X (which is CatX?). So it matches.

#include "axiom"
import from Integer;
define CatA: Category == with;
define CatB: Category == with;
define CatX: Category == with;
A: Join(CatX, CatA) == add;
B: Join(CatX, CatB) == add;

MyPkg(X: CatX): with {isA?: () -> Boolean} == add {
        isA?(): Boolean == X has CatA;
}
main4(n:Integer): Boolean == {
        X: CatX == if zero? n then (A@CatX) else (B@CatX);
        isA?()$MyPkg(X);
}

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6804229532583544981-25px015.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6804229532583544981-25px015.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

main4(0)

   There are no library operations named main4 
      Use HyperDoc Browse or issue
                               )what op main4
      to learn if there is any operation containing " main4 " in its 
      name.

   Cannot find a definition or applicable library operation named main4
      with argument type(s) 
                             NonNegativeInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

In the following code we have the correspondence:

    A      <--> B
    String <--> with
    "x"    <--> String
    "y"    <--> Integer

#include "axiom"
define CatA(s: String): Category == with;
A(s: String): CatA(s) == add;

define CatB(s: with): Category == with;
B(s: with): CatB(s) == add;

rhxmain(): List Record(s: String, b: Boolean) == [
    ["A x has CatA x", (A("x") has CatA("x"))],
    ["A y has CatA x", (A("y") has CatA("x"))],
    ["B String has CatB String", (B(String)  has CatB(String))],
    ["B Integer has CatB String",(B(Integer) has CatB(String))]
];

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7724360379013079758-25px017.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7724360379013079758-25px017.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

The interesting part is that the truth value of the second and fourth list elements do not agree.

rhxmain()

   There are no library operations named rhxmain 
      Use HyperDoc Browse or issue
                              )what op rhxmain
      to learn if there is any operation containing " rhxmain " in its 
      name.

   Cannot find a no-argument definition or library operation named 
      rhxmain .

An example of a domain-valued variable

#include "axiom"
#pile

main5(n:Integer):List Boolean ==
  local x:IntegralDomain
  local y:Category

  if n=1 then
    y := Field
  else
    y := Ring

  x := Integer
  test1:Boolean := x has y
  x := Fraction Integer
  test2:Boolean := x has y
  [test1,test2]

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1586618870818532942-25px019.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1586618870818532942-25px019.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

main5(1)

   There are no library operations named main5 
      Use HyperDoc Browse or issue
                               )what op main5
      to learn if there is any operation containing " main5 " in its 
      name.

   Cannot find a definition or applicable library operation named main5
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

#include "axiom"
#pile

main6():List Boolean ==
  local x:IntegralDomain

  x := Integer
  test1:Boolean := x has Field
  x := String
  test2:Boolean := x has Field
  [test1,test2]

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/395909315557564304-25px021.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/395909315557564304-25px021.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

x:IntegralDomain

   IntegralDomain is a category, not a domain, and declarations require
      domains.

#include "axiom"
#pile

MyDom: with
  sigs:List Category
  add2:(MyDom,MyDom) -> MyDom
  sub2:(MyDom,MyDom) -> MyDom
  neg: MyDom -> MyDom
 == add
  import from Integer
  Rep == Integer

  sigs:List Category ==
    [with {add2:(MyDom,MyDom)->MyDom},
     with {sub2:(MyDom,MyDom)->MyDom},
     with {neg:MyDom->MyDom}]

  add2(x:%,y:%):% == per(rep(x) + rep(y))
  sub2(x:%,y:%):% == per(rep(x) - rep(y))
  neg(x:%):% == per(-rep(x))

main8():List Boolean ==
  import from ListFunctions2(Category,Boolean)

  map((i:Category):Boolean+->(MyDom has i), sigs$MyDom)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7325620604487185069-25px023.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      . Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libfricas.al'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7325620604487185069-25px023.as", line 1: 
#include "axiom"
^
[L1 C1] #2 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

)sh MyDom

   The )show system command is used to display information about types 
      or partial types. For example, )show Integer will show 
      information about Integer .

      MyDom is not the name of a known type constructor. If you want to
      see information about any operations named MyDom , issue
                          )display operations MyDom 
main8()

   There are no library operations named main8 
      Use HyperDoc Browse or issue
                               )what op main8
      to learn if there is any operation containing " main8 " in its 
      name.

   Cannot find a no-argument definition or library operation named 
      main8 .