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SandBoxMaybe

last edited 11 years ago by Bill Page

Edit detail for SandBoxMaybe revision 23 of 46

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Editor: Bill Page Time: 2008/05/27 13:08:04 GMT-7
Note: exports inferred from RHS

changed:
-x:MyPair(INT):=[-1,-2]
-\end{axiom}
x:MyPair2(INT):=[-1,-2]
\end{axiom}

... --gdr, Mon, 26 May 2008 14:24:29 -0700 reply

spad)abbrev domain MAYBE Maybe
Maybe(T: Type) == Union(T,"failed")

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/3867167196268790065-25px001.spad using 
      Spad compiler.
   MAYBE abbreviates domain Maybe 
------------------------------------------------------------------------
   initializing NRLIB MAYBE for Maybe 
   compiling into NRLIB MAYBE 
(time taken in buildFunctor:  0)
;;;     ***       |Maybe| REDEFINED
;;;     ***       |Maybe| REDEFINED
Time: 0.01 SEC.
   Cumulative Statistics for Constructor Maybe
      Time: 0.01 seconds
   >> System error:
   PROG2 is not of type NUMBER.

need result type --Bill Page, Mon, 26 May 2008 20:25:03 -0700 reply

spad)abbrev domain MAYBE Maybe
Maybe(T: Type):Type == Union(T,"failed")

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/6798525157762352212-25px002.spad using 
      Spad compiler.
   MAYBE abbreviates domain Maybe 
------------------------------------------------------------------------
   initializing NRLIB MAYBE for Maybe 
   compiling into NRLIB MAYBE 
(time taken in buildFunctor:  0)
;;;     ***       |Maybe| REDEFINED
;;;     ***       |Maybe| REDEFINED
Time: 0 SEC.
   Cumulative Statistics for Constructor Maybe
      Time: 0 seconds
--------------non extending category----------------------
.. Maybe #1 of cat 
(|Type|)   has no 
(|UnionCategory| |#1| "failed")    finalizing NRLIB MAYBE 
   Processing Maybe for Browser database:
--->-->Maybe(): Missing Description
------------------------------------------------------------------------
   Maybe is now explicitly exposed in frame initial 
   Maybe will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MAYBE.NRLIB/code.o

axiom)show Maybe
 Maybe T$: Type  is a domain constructor
 Abbreviation for Maybe is MAYBE 
 This constructor is exposed in this frame.
 Issue )edit /var/zope2/var/LatexWiki/6798525157762352212-25px002.spad to see algebra source code for MAYBE 
------------------------------- Operations --------------------------------
                          No exported operations

need result type --gdr, Mon, 26 May 2008 22:03:52 -0700 reply

No, there should not be a need for specifying result type. In fact, if you specify "Type" as result type, you explicitly ask for not having any exported operation. That wasn't the intent. The intent was to get whatever the right hand side exported as exports. Such as in

spad)abbrev domain MYFIELD MyField
MyField() == Fraction Integer

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/2205496008028295448-25px004.spad using 
      Spad compiler.
   MYFIELD abbreviates domain MyField 
------------------------------------------------------------------------
   initializing NRLIB MYFIELD for MyField 
   compiling into NRLIB MYFIELD 
****** Domain: $ already in scope
augmenting $: (CharacteristicNonZero)
augmenting (Integer): (PolynomialFactorizationExplicit)
****** Domain: $ already in scope
augmenting $: (CharacteristicNonZero)
augmenting (Integer): (PolynomialFactorizationExplicit)
augmenting (Integer): (ConvertibleTo (Pattern (Float)))
augmenting (Integer): (Eltable (Integer) (Integer))
augmenting (Integer): (Evalable (Integer))
augmenting (Integer): (InnerEvalable (Symbol) (Integer))
augmenting (Integer): (PartialDifferentialRing (Symbol))
augmenting (Integer): (PatternMatchable (Float))
augmenting (Integer): (PolynomialFactorizationExplicit)
augmenting (Integer): (RetractableTo (Symbol))
(time taken in buildFunctor:  40)
;;;     ***       |MyField| REDEFINED
;;;     ***       |MyField| REDEFINED
Time: 0.79 SEC.
   Cumulative Statistics for Constructor MyField
      Time: 0.79 seconds
   finalizing NRLIB MYFIELD 
   Processing MyField for Browser database:
--->-->MyField(): Missing Description
------------------------------------------------------------------------
   MyField is now explicitly exposed in frame initial 
   MyField will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MYFIELD.NRLIB/code.o

axiom)sh MyField
 MyField  is a domain constructor
 Abbreviation for MyField is MYFIELD 
 This constructor is exposed in this frame.
 Issue )edit /var/zope2/var/LatexWiki/2205496008028295448-25px004.spad to see algebra source code for MYFIELD 
------------------------------- Operations --------------------------------
 ?*? : (%,Integer) -> %                ?*? : (Integer,%) -> %
 ?*? : (Fraction Integer,%) -> %       ?*? : (%,Fraction Integer) -> %
 ?*? : (%,%) -> %                      ?*? : (Integer,%) -> %
 ?*? : (PositiveInteger,%) -> %        ?**? : (%,Integer) -> %
 ?**? : (%,PositiveInteger) -> %       ?+? : (%,%) -> %
 ?-? : (%,%) -> %                      -? : % -> %
 ?/? : (Integer,Integer) -> %          ?/? : (%,%) -> %
 ?=? : (%,%) -> Boolean                1 : () -> %
 0 : () -> %                           ?^? : (%,Integer) -> %
 ?^? : (%,PositiveInteger) -> %        associates? : (%,%) -> Boolean
 coerce : Integer -> %                 coerce : Fraction Integer -> %
 coerce : % -> %                       coerce : Integer -> %
 coerce : % -> OutputForm              denom : % -> Integer
 denominator : % -> %                  factor : % -> Factored %
 gcd : List % -> %                     gcd : (%,%) -> %
 hash : % -> SingleInteger             inv : % -> %
 latex : % -> String                   lcm : List % -> %
 lcm : (%,%) -> %                      numer : % -> Integer
 numerator : % -> %                    one? : % -> Boolean
 prime? : % -> Boolean                 ?quo? : (%,%) -> %
 recip : % -> Union(%,"failed")        ?rem? : (%,%) -> %
 retract : % -> Integer                sample : () -> %
 sizeLess? : (%,%) -> Boolean          squareFree : % -> Factored %
 squareFreePart : % -> %               unit? : % -> Boolean
 unitCanonical : % -> %                zero? : % -> Boolean
 ?~=? : (%,%) -> Boolean              
 ?*? : (NonNegativeInteger,%) -> %
 ?**? : (%,NonNegativeInteger) -> %
 ?<? : (%,%) -> Boolean if Integer has ORDSET
 ?<=? : (%,%) -> Boolean if Integer has ORDSET
 ?>? : (%,%) -> Boolean if Integer has ORDSET
 ?>=? : (%,%) -> Boolean if Integer has ORDSET
 D : (%,(Integer -> Integer)) -> %
 D : (%,(Integer -> Integer),NonNegativeInteger) -> %
 D : (%,List Symbol,List NonNegativeInteger) -> % if Integer has PDRING SYMBOL
 D : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING SYMBOL
 D : (%,List Symbol) -> % if Integer has PDRING SYMBOL
 D : (%,Symbol) -> % if Integer has PDRING SYMBOL
 D : (%,NonNegativeInteger) -> % if Integer has DIFRING
 D : % -> % if Integer has DIFRING
 OMwrite : (OpenMathDevice,%,Boolean) -> Void if Integer has INS and Integer has OM
 OMwrite : (OpenMathDevice,%) -> Void if Integer has INS and Integer has OM
 OMwrite : (%,Boolean) -> String if Integer has INS and Integer has OM
 OMwrite : % -> String if Integer has INS and Integer has OM
 ?^? : (%,NonNegativeInteger) -> %
 abs : % -> % if Integer has OINTDOM
 ceiling : % -> Integer if Integer has INS
 characteristic : () -> NonNegativeInteger
 charthRoot : % -> Union(%,"failed") if $ has CHARNZ and Integer has PFECAT or Integer has CHARNZ
 coerce : Symbol -> % if Integer has RETRACT SYMBOL
 conditionP : Matrix % -> Union(Vector %,"failed") if $ has CHARNZ and Integer has PFECAT
 convert : % -> DoubleFloat if Integer has REAL
 convert : % -> Float if Integer has REAL
 convert : % -> InputForm if Integer has KONVERT INFORM
 convert : % -> Pattern Float if Integer has KONVERT PATTERN FLOAT
 convert : % -> Pattern Integer if Integer has KONVERT PATTERN INT
 differentiate : (%,(Integer -> Integer)) -> %
 differentiate : (%,(Integer -> Integer),NonNegativeInteger) -> %
 differentiate : (%,List Symbol,List NonNegativeInteger) -> % if Integer has PDRING SYMBOL
 differentiate : (%,Symbol,NonNegativeInteger) -> % if Integer has PDRING SYMBOL
 differentiate : (%,List Symbol) -> % if Integer has PDRING SYMBOL
 differentiate : (%,Symbol) -> % if Integer has PDRING SYMBOL
 differentiate : (%,NonNegativeInteger) -> % if Integer has DIFRING
 differentiate : % -> % if Integer has DIFRING
 divide : (%,%) -> Record(quotient: %,remainder: %)
 ?.? : (%,Integer) -> % if Integer has ELTAB(INT,INT)
 euclideanSize : % -> NonNegativeInteger
 eval : (%,Symbol,Integer) -> % if Integer has IEVALAB(SYMBOL,INT)
 eval : (%,List Symbol,List Integer) -> % if Integer has IEVALAB(SYMBOL,INT)
 eval : (%,List Equation Integer) -> % if Integer has EVALAB INT
 eval : (%,Equation Integer) -> % if Integer has EVALAB INT
 eval : (%,Integer,Integer) -> % if Integer has EVALAB INT
 eval : (%,List Integer,List Integer) -> % if Integer has EVALAB INT
 expressIdealMember : (List %,%) -> Union(List %,"failed")
 exquo : (%,%) -> Union(%,"failed")
 extendedEuclidean : (%,%,%) -> Union(Record(coef1: %,coef2: %),"failed")
 extendedEuclidean : (%,%) -> Record(coef1: %,coef2: %,generator: %)
 factorPolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if Integer has PFECAT
 factorSquareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if Integer has PFECAT
 floor : % -> Integer if Integer has INS
 fractionPart : % -> % if Integer has EUCDOM
 gcdPolynomial : (SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
 init : () -> % if Integer has STEP
 map : ((Integer -> Integer),%) -> %
 max : (%,%) -> % if Integer has ORDSET
 min : (%,%) -> % if Integer has ORDSET
 multiEuclidean : (List %,%) -> Union(List %,"failed")
 negative? : % -> Boolean if Integer has OINTDOM
 nextItem : % -> Union(%,"failed") if Integer has STEP
 patternMatch : (%,Pattern Float,PatternMatchResult(Float,%)) -> PatternMatchResult(Float,%) if Integer has PATMAB FLOAT
 patternMatch : (%,Pattern Integer,PatternMatchResult(Integer,%)) -> PatternMatchResult(Integer,%) if Integer has PATMAB INT
 positive? : % -> Boolean if Integer has OINTDOM
 principalIdeal : List % -> Record(coef: List %,generator: %)
 random : () -> % if Integer has INS
 reducedSystem : Matrix % -> Matrix Integer
 reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix Integer,vec: Vector Integer)
 reducedSystem : (Matrix %,Vector %) -> Record(mat: Matrix Integer,vec: Vector Integer) if Integer has LINEXP INT
 reducedSystem : Matrix % -> Matrix Integer if Integer has LINEXP INT
 retract : % -> Integer if Integer has RETRACT INT
 retract : % -> Fraction Integer if Integer has RETRACT INT
 retract : % -> Symbol if Integer has RETRACT SYMBOL
 retractIfCan : % -> Union(Integer,"failed") if Integer has RETRACT INT
 retractIfCan : % -> Union(Fraction Integer,"failed") if Integer has RETRACT INT
 retractIfCan : % -> Union(Symbol,"failed") if Integer has RETRACT SYMBOL
 retractIfCan : % -> Union(Integer,"failed")
 sign : % -> Integer if Integer has OINTDOM
 solveLinearPolynomialEquation : (List SparseUnivariatePolynomial %,SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %,"failed") if Integer has PFECAT
 squareFreePolynomial : SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if Integer has PFECAT
 subtractIfCan : (%,%) -> Union(%,"failed")
 unitNormal : % -> Record(unit: %,canonical: %,associate: %)
 wholePart : % -> Integer if Integer has EUCDOM

macro --Bill Page, Tue, 27 May 2008 07:30:43 -0700 reply

Macro versus domain

axiom
macro Maybe(T) == Union(T,"failed")

Type: Void

axiom)sh Maybe(Integer)
 Union(Integer,"failed") is a domain constructor.
------------------------------- Operations --------------------------------
 ?=? : (%,%) -> Boolean                autoCoerce : % -> failed
 autoCoerce : % -> Integer             autoCoerce : failed -> %
 autoCoerce : Integer -> %             ?case? : (%,failed) -> Boolean
 ?case? : (%,Integer) -> Boolean       coerce : % -> failed
 coerce : % -> Integer                 coerce : % -> OutputForm
 ?~=? : (%,%) -> Boolean

axiom
x:Maybe(Integer)

Type: Void

axiom
x:=-3

(1)

Type: Union(Integer,...)

explicit exports --Bill Page, Tue, 27 May 2008 07:46:20 -0700 reply

spad)abbrev domain MAYBE2 Maybe2
Maybe2(T: Type):with
    _= : (%,%) -> Boolean
    autoCoerce : % -> failed
    autoCoerce : % -> T
    autoCoerce : failed -> %
    autoCoerce : T -> %
    _case: (%,failed) -> Boolean
    _case: (%,T) -> Boolean
    coerce : % -> failed
    coerce : % -> T
    coerce : % -> OutputForm
    coerce : T -> %
  == Union(T,"failed") add
    coerce(x:T):% == autoCoerce(x)

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/7945389118523770429-25px008.spad using 
      Spad compiler.
   MAYBE2 abbreviates domain Maybe2 
------------------------------------------------------------------------
   initializing NRLIB MAYBE2 for Maybe2 
   compiling into NRLIB MAYBE2 
   compiling exported coerce : T$ -> %
Time: 0 SEC.
(time taken in buildFunctor:  0)
;;;     ***       |Maybe2| REDEFINED
;;;     ***       |Maybe2| REDEFINED
Time: 0.01 SEC.
   Cumulative Statistics for Constructor Maybe2
      Time: 0.01 seconds
--------------non extending category----------------------
.. Maybe2 #1 of cat 
(CATEGORY |domain| (SIGNATURE = ((|Boolean|) $ $))
    (SIGNATURE |autoCoerce| (|failed| $))
    (SIGNATURE |autoCoerce| (|#1| $))
    (SIGNATURE |autoCoerce| ($ |failed|))
    (SIGNATURE |autoCoerce| ($ |#1|))
    (SIGNATURE |case| ((|Boolean|) $ |failed|))
    (SIGNATURE |case| ((|Boolean|) $ |#1|))
    (SIGNATURE |coerce| (|failed| $)) (SIGNATURE |coerce| (|#1| $))
    (SIGNATURE |coerce| ((|OutputForm|) $))
    (SIGNATURE |coerce| ($ |#1|)))   has no 
(|UnionCategory| |#1| "failed")    finalizing NRLIB MAYBE2 
   Processing Maybe2 for Browser database:
--->-->Maybe2((= ((Boolean) % %))): Not documented!!!!
--->-->Maybe2((autoCoerce (failed %))): Not documented!!!!
--->-->Maybe2((autoCoerce (T$ %))): Not documented!!!!
--->-->Maybe2((autoCoerce (% failed))): Not documented!!!!
--->-->Maybe2((autoCoerce (% T$))): Not documented!!!!
--->-->Maybe2((case ((Boolean) % failed))): Not documented!!!!
--->-->Maybe2((case ((Boolean) % T$))): Not documented!!!!
--->-->Maybe2((coerce (failed %))): Not documented!!!!
--->-->Maybe2((coerce (T$ %))): Not documented!!!!
--->-->Maybe2((coerce ((OutputForm) %))): Not documented!!!!
--->-->Maybe2((coerce (% T$))): Not documented!!!!
--->-->Maybe2(constructor): Not documented!!!!
--->-->Maybe2(): Missing Description
------------------------------------------------------------------------
   Maybe2 is now explicitly exposed in frame initial 
   Maybe2 will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MAYBE2.NRLIB/code.o

axiom)show Maybe2
 Maybe2 T$: Type  is a domain constructor
 Abbreviation for Maybe2 is MAYBE2 
 This constructor is exposed in this frame.
 Issue )edit /var/zope2/var/LatexWiki/7945389118523770429-25px008.spad to see algebra source code for MAYBE2 
------------------------------- Operations --------------------------------
 ?=? : (%,%) -> Boolean                autoCoerce : % -> failed
 autoCoerce : % -> T$                  autoCoerce : failed -> %
 autoCoerce : T$ -> %                  ?case? : (%,failed) -> Boolean
 ?case? : (%,T$) -> Boolean            coerce : % -> failed
 coerce : % -> T$                      coerce : % -> OutputForm
 coerce : T$ -> %

axiom
y:Maybe2(Integer)

Type: Void

axiomy:=-3
Function:  autoCoerce : Integer -> % is missing from domain: Maybe2 Integer
   Internal Error
   The function autoCoerce with signature $(Integer) is missing from 
      domain Maybe2(Integer)

macro --gdr, Tue, 27 May 2008 08:01:39 -0700 reply

But, the use of macro is a workaround a fundamental problem in the compiler -- which I'm desperately trying to convey.

explicit exports --gdr, Tue, 27 May 2008 08:03:19 -0700 reply

This and the original bug comes from the same source in the compiler.

Re: problem in the compiler --Bill Page, Tue, 27 May 2008 08:06:29 -0700 reply

I agree. I am just trying to understand it.

Why does:

  Maybe(T: Type) == Union(T,"failed")

and:

  Maybe(T: Type):Type == Union(T,"failed")

have such different semantics? Is this a good thing? Maybe there is a need for something that stands for the type of the rhs:

  Maybe(T: Type):rhs == Union(T,"failed")

Re: problem in the compiler --gdr, Tue, 27 May 2008 09:20:19 -0700 reply

Remember that Type is a category that exports no operation. so when you say Maybe(T: Type): Type, you are exlicitly saying that Maybe(T: Type) is a domain constructor that exports no operation. Indeed, if you try )show with OpenAxiom?, you will see that it say, Maybe has no exported operations. Which is right.

However, when one says Maybe(T: Type) == Union(T, "failed"), one is saying that Maybe(T) is exactly like Union(T,"failed"), except for the name. Consequently, Maybe has the exports inferred from the right hand side. This works for all domains, except the `builtin' domains. If you try Record or Enumeration, you should hit a bug in the compiler at some point.

exports inferred from RHS --Bill Page, Tue, 27 May 2008 12:58:47 -0700 reply

spad)abbrev domain MYPAIR MyPair
MyPair(T:Type) == Record(X:T,Y:T)

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/1259072050906152969-25px011.spad using 
      Spad compiler.
   MYPAIR abbreviates domain MyPair 
------------------------------------------------------------------------
   initializing NRLIB MYPAIR for MyPair 
   compiling into NRLIB MYPAIR 
(time taken in buildFunctor:  0)
;;;     ***       |MyPair| REDEFINED
;;;     ***       |MyPair| REDEFINED
Time: 0.01 SEC.
   Cumulative Statistics for Constructor MyPair
      Time: 0.01 seconds
   >> System error:
   SETRECORDELT is not of type NUMBER.

axiom)show MyPair(Integer)
   MyPair is an unknown constructor and so is unavailable. Did you mean
      to use -> but type something different instead?

spad)abbrev domain MYPAIR2 MyPair2
MyPair2(T:Type): with
    _= : (%,%) -> Boolean
    coerce : % -> OutputForm
    copy : % -> %
    _. : (%,X) -> Integer
    _. : (%,Y) -> Integer
    setelt : (%,X,Integer) -> Integer
    setelt : (%,Y,Integer) -> Integer
    _~_= : (%,%) -> Boolean
    construct : (Integer,Integer) -> %
  == Record(X:T,Y:T)

spad

   Compiling OpenAxiom source code from file 
      /var/zope2/var/LatexWiki/5737796087530002187-25px013.spad using 
      Spad compiler.
   MYPAIR2 abbreviates domain MyPair2 
------------------------------------------------------------------------
   initializing NRLIB MYPAIR2 for MyPair2 
   compiling into NRLIB MYPAIR2 
(time taken in buildFunctor:  0)
;;;     ***       |MyPair2| REDEFINED
;;;     ***       |MyPair2| REDEFINED
Time: 0 SEC.
   Cumulative Statistics for Constructor MyPair2
      Time: 0 seconds
--------------non extending category----------------------
.. MyPair2 #1 of cat 
(CATEGORY |domain| (SIGNATURE = ((|Boolean|) $ $))
    (SIGNATURE |coerce| ((|OutputForm|) $)) (SIGNATURE |copy| ($ $))
    (SIGNATURE |.| ((|Integer|) $ X)) (SIGNATURE |.| ((|Integer|) $ Y))
    (SIGNATURE |setelt| ((|Integer|) $ X (|Integer|)))
    (SIGNATURE |setelt| ((|Integer|) $ Y (|Integer|)))
    (SIGNATURE ~= ((|Boolean|) $ $))
    (SIGNATURE |construct| ($ (|Integer|) (|Integer|))))   has no 
(|RecordCategory| (|:| X |#1|) (|:| Y |#1|))    finalizing NRLIB MYPAIR2 
   Processing MyPair2 for Browser database:
--->-->MyPair2((= ((Boolean) % %))): Not documented!!!!
--->-->MyPair2((coerce ((OutputForm) %))): Not documented!!!!
--->-->MyPair2((copy (% %))): Not documented!!!!
--->-->MyPair2((. ((Integer) % X))): Not documented!!!!
--->-->MyPair2((. ((Integer) % Y))): Not documented!!!!
--->-->MyPair2((setelt ((Integer) % X (Integer)))): Not documented!!!!
--->-->MyPair2((setelt ((Integer) % Y (Integer)))): Not documented!!!!
--->-->MyPair2((~= ((Boolean) % %))): Not documented!!!!
--->-->MyPair2((construct (% (Integer) (Integer)))): Not documented!!!!
--->-->MyPair2(constructor): Not documented!!!!
--->-->MyPair2(): Missing Description
------------------------------------------------------------------------
   MyPair2 is now explicitly exposed in frame initial 
   MyPair2 will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MYPAIR2.NRLIB/code.o

axiom)show MyPair2(Integer)
 MyPair2 Integer is a domain constructor.
 Abbreviation for MyPair2 is MYPAIR2 
 This constructor is exposed in this frame.
 Issue )edit /var/zope2/var/LatexWiki/5737796087530002187-25px013.spad to see algebra source code for MYPAIR2 
------------------------------- Operations --------------------------------
 ?.? : (%,X) -> Integer                ?.? : (%,Y) -> Integer
 ?=? : (%,%) -> Boolean                coerce : % -> OutputForm
 copy : % -> %                         setelt : (%,X,Integer) -> Integer
 setelt : (%,Y,Integer) -> Integer     ?~=? : (%,%) -> Boolean
 construct : (Integer,Integer) -> %

axiomx:MyPair2(INT):=[-1,-2]
   You cannot declare x to be of type MyPair2 Integer because either 
      the declared type of x or the type of the value of x is different
      from MyPair2 Integer .