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 1.3.12 compiled at Sat  7 Jun 23:54:49 CEST 2025"

#include "fricas"
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/2635879875939783572-25px002.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.
   Compiling Lisp source code from file 
      ./2635879875939783572-25px002.lsp
   Issuing )library command for 2635879875939783572-25px002
   Reading #P"/var/aw/var/LatexWiki/2635879875939783572-25px002.asy"

   >> System error:
   The value
  |CatA|
is not of type
  LIST

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 "fricas"

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/551345834211954051-25px004.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.
   Compiling Lisp source code from file 
      ./551345834211954051-25px004.lsp
   Issuing )library command for 551345834211954051-25px004
   Reading #P"/var/aw/var/LatexWiki/551345834211954051-25px004.asy"

   >> System error:
   The value
  |CatX|
is not of type
  LIST

Try a little harder to create dynamic types:

#include "fricas"

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/1438580684530002070-25px005.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.
   Compiling Lisp source code from file 
      ./1438580684530002070-25px005.lsp
   Issuing )library command for 1438580684530002070-25px005
   Reading #P"/var/aw/var/LatexWiki/1438580684530002070-25px005.asy"
   CatA is now explicitly exposed in frame initial 

   >> System error:
   The function BOOT::ASHARPMKAUTOLOADFUNCTOR is undefined.

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

   X1 is not a valid type.

Or this way:

#include "fricas"
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/5501580183423487024-25px007.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.
   Compiling Lisp source code from file 
      ./5501580183423487024-25px007.lsp
   Issuing )library command for 5501580183423487024-25px007
   Reading #P"/var/aw/var/LatexWiki/5501580183423487024-25px007.asy"

   >> System error:
   The value
  |CatX|
is not of type
  LIST

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

   Y1 is not a valid type.

Or like this:

#include "fricas"
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/4475696461483594624-25px009.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.
   Compiling Lisp source code from file 
      ./4475696461483594624-25px009.lsp
   Issuing )library command for 4475696461483594624-25px009
   Reading #P"/var/aw/var/LatexWiki/4475696461483594624-25px009.asy"

   >> System error:
   The value
  |CatX|
is not of type
  LIST

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 "fricas"

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/731070323520525114-25px011.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.
   Compiling Lisp source code from file 
      ./731070323520525114-25px011.lsp
   Issuing )library command for 731070323520525114-25px011
   Reading #P"/var/aw/var/LatexWiki/731070323520525114-25px011.asy"
   CatA is already explicitly exposed in frame initial 

   >> System error:
   The function BOOT::ASHARPMKAUTOLOADFUNCTOR is undefined.

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 "fricas"

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/2725589955460952637-25px013.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.
   Compiling Lisp source code from file 
      ./2725589955460952637-25px013.lsp
   Issuing )library command for 2725589955460952637-25px013
   Reading #P"/var/aw/var/LatexWiki/2725589955460952637-25px013.asy"

   >> System error:
   The value
  |CatX|
is not of type
  LIST

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 "fricas"
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/3499552379206056026-25px015.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.
   Compiling Lisp source code from file 
      ./3499552379206056026-25px015.lsp
   Issuing )library command for 3499552379206056026-25px015
   Reading #P"/var/aw/var/LatexWiki/3499552379206056026-25px015.asy"
   CatA is already explicitly exposed in frame initial 

   >> System error:
   The function BOOT::ASHARPMKAUTOLOADFUNCTOR is undefined.

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 "fricas"
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/4520573120428817379-25px017.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.
   Compiling Lisp source code from file 
      ./4520573120428817379-25px017.lsp
   Issuing )library command for 4520573120428817379-25px017
   Reading #P"/var/aw/var/LatexWiki/4520573120428817379-25px017.asy"
   CatA is already explicitly exposed in frame initial 

   >> System error:
   The function BOOT::ASHARPMKAUTOLOADFUNCTOR is undefined.

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 "fricas"
#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/3633650021500177729-25px019.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.
   Compiling Lisp source code from file 
      ./3633650021500177729-25px019.lsp
   Issuing )library command for 3633650021500177729-25px019
   Reading #P"/var/aw/var/LatexWiki/3633650021500177729-25px019.asy"

main5(1)

(1)

main5(2)

(2)

)clear completely

   All user variables and function definitions have been cleared.
   All )browse facility databases have been cleared.
   Internally cached functions and constructors have been cleared.
   )clear completely is finished.

The compiler checks static types.

#include "fricas"
#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/7190335948694378273-25px021.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.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7190335948694378273-25px021.as", line 9: 
  x := String
.......^
[L9 C8] #1 (Error) There is no suitable interpretation for the expression String
  The context requires an expression of type IntegralDomain.
     The possible types of the right hand side (`String') are:
          -- Join(SetCategory, 
                StringCategory with 
      ...

   The )library system command was not called after compilation.

But note that we cannot define domain-valued variables in the FriCAS interpreter.

x:IntegralDomain

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

#include "fricas"
#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/2501096951859201286-25px023.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.
   Compiling Lisp source code from file 
      ./2501096951859201286-25px023.lsp
   Issuing )library command for 2501096951859201286-25px023
   Reading #P"/var/aw/var/LatexWiki/2501096951859201286-25px023.asy"
   MyDom is now explicitly exposed in frame initial

)sh MyDom

 MyDom is a domain constructor.
 Abbreviation for MyDom is MYDOM 
 This constructor is exposed in this frame.
 4 names for 4 operations in this domain.
------------------------------- Operations --------------------------------

 add2 : (MyDom, MyDom) -> MyDom        neg : MyDom -> MyDom
 sigs : () -> List(Category)           sub2 : (MyDom, MyDom) -> MyDom

main8()

   >> System error:
   The value
  NIL
is not of type
  SIMPLE-VECTOR