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On Aug 12, 2007 6:35 PM Weiss, Juergen wrote:
I found the old document about type equivalence in Scratchpad. It's:
A New Algebra System, May, 29 th 1984, James H Davenport. I have only
found a paper version. Maybe someone has an online version.
It states that:
- Two named types are only equivalent if the names are the same.
- Anonymous types are equivalent when stucturally equivalent
- An anonymous type is never equivalent to a named type.
So following 1:
t1 == List Term
t2 == List Term
x : t1
y : t2
y := x
is not supposed to work,
[t1 and t2 should be treated as different types]
fricas
(1) -> <spad>
fricas
)abbrev package TEST test
test(): with
main:()->Integer
== add
main() ==
t1:IntegerNumberSystem == Integer
t2:IntegerNumberSystem == Integer
x : t1
x := 1$t1
y : t2
y := x</spad>
fricas
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4406049181591081329-25px001.spad
using old system compiler.
TEST abbreviates package test
------------------------------------------------------------------------
initializing NRLIB TEST for test
compiling into NRLIB TEST
compiling exported main : () -> Integer
Semantic Errors:
[1] main: +-> is not a known type
****** comp fails at level 3 with expression: ******
error in function main
(SEQ (DEF (|t1|) ((|IntegerNumberSystem|)) (|Integer|))
(DEF (|t2|) ((|IntegerNumberSystem|)) (|Integer|)) (|:| |x| |t1|)
(|:=| |x| | << | (|Sel| |t1| 1) | >> |) (|:| |y| |t2|)
(|exit| 1 (|:=| |y| |x|)))
****** level 3 ******
x:= (Sel t1 (One))
m:= t1
f:=
((((|x| #) (|t2| #) (|t1| #) (|main| #) ...)))
>> Apparent user error:
NoValueMode
is an unknown modeThere should be a compile error above.
fricas
main()
There are no library operations named main
Use HyperDoc Browse or issue
)what op main
to learn if there is any operation containing " main " in its
name.
Cannot find a no-argument definition or library operation named main
.
but following 2:
x : List Term
y : List Term
y := x
is ok,
spad
)abbrev package TEST test2
test2(): with
main2:()->Integer
== add
main2()==
x : Integer
x := 1
y : Integer
y := x
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3154192227704798176-25px003.spad
using old system compiler.
TEST abbreviates package test2
------------------------------------------------------------------------
initializing NRLIB TEST for test2
compiling into NRLIB TEST
compiling exported main2 : () -> Integer
Time: 0 SEC.
(time taken in buildFunctor: 0)
Time: 0 SEC.
Cumulative Statistics for Constructor test2
Time: 0 seconds
finalizing NRLIB TEST
Processing test2 for Browser database:
--->-->test2(constructor): Not documented!!!!
--->-->test2((main2 ((Integer)))): Not documented!!!!
--->-->test2(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TEST.NRLIB/TEST.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TEST.NRLIB/TEST.fasl
; compilation finished in 0:00:00.024
------------------------------------------------------------------------
test2 is now explicitly exposed in frame initial
test2 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TEST.NRLIB/TESTfricas
main2()
also
spad
)abbrev package TEST test4
test4(): with
main4:()->Integer
== add
main4():Integer ==
t1():IntegerNumberSystem == Integer
x : t1
x := 1$t1
y : t1
y := x
1
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/346730625682591668-25px005.spad
using old system compiler.
TEST abbreviates package test4
------------------------------------------------------------------------
initializing NRLIB TEST for test4
compiling into NRLIB TEST
compiling exported main4 : () -> Integer
Semantic Errors:
[1] main4: +-> is not a known type
****** comp fails at level 3 with expression: ******
error in function main4
(SEQ (DEF (|t1|) ((|IntegerNumberSystem|)) (|Integer|)) (|:| |x| |t1|)
(|:=| |x| | << | (|Sel| |t1| 1) | >> |) (|:| |y| |t1|) (|:=| |y| |x|)
(|exit| 1 1))
****** level 3 ******
x:= (Sel t1 (One))
m:= t1
f:=
((((|x| #) (|t1| #) (|main4| #) (|$DomainsInScope| # # #) ...)))
>> Apparent user error:
NoValueMode
is an unknown modeThis compiles ok but there is a run-time error. Why?
fricas
main4()
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 no-argument definition or library operation named
main4 .
and following 3:
t == List Term
x : List Term
y : t
y := x
is not supposed to work as well
spad
)abbrev package TEST test3
test3(): with
main3:()->Integer
== add
main3() ==
t:IntegerNumberSystem == Integer
x : Integer
x := 1
y : t
y := x
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2232261183027663296-25px007.spad
using old system compiler.
TEST abbreviates package test3
------------------------------------------------------------------------
initializing NRLIB TEST for test3
compiling into NRLIB TEST
compiling exported main3 : () -> Integer
Semantic Errors:
[1] main3: +-> is not a known type
****** comp fails at level 3 with expression: ******
error in function main3
(SEQ (DEF (|t|) ((|IntegerNumberSystem|)) (|Integer|)) (|:| |x| (|Integer|))
(|:=| |x| 1) (|:| |y| |t|) (|exit| 1 | << | (|:=| |y| |x|) | >> |))
****** level 3 ******
x:= (:= y x)
m:= (Integer)
f:=
((((|y| #) (|x| # #) (|t| #) (|main3| #) ...)))
>> Apparent user error:
CANNOT ASSIGN: x
OF MODE: (Integer)
TO: y
OF MODE: tThere should produce a compile error above.
fricas
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 .
All examples are taken from the paper.
I am not sure how much of this design is preserved in the current
system. But without having had an intense look at the examples,
I got the impression, that they follow the rules above.
Regards
Juergen Weiss
The failure above is not convincing, since the spad code did not compile. It's not clear where the failure lies. For example, in Test4, the compiler message says |t1| is not defined. A slight modification causes a different error (for example adding () after t1 in test4):
spad
)abbrev package TESTFIVE test5
test5(): with
main5:()->Integer
== add
main5():Integer ==
t1():IntegerNumberSystem == Integer
x : t1()
x := 1$t1()
y : t1()
y := x
1$Integer
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3611641870102941622-25px009.spad
using old system compiler.
TESTFIVE abbreviates package test5
------------------------------------------------------------------------
initializing NRLIB TESTFIVE for test5
compiling into NRLIB TESTFIVE
compiling exported main5 : () -> Integer
Internal Error
Error while instantiating type t1fricas
main5()
There are no library operations named main5
Use HyperDoc Browse or issue
)what op main5
to learn if there is any operation containing " main5 " in its
name.
Cannot find a no-argument definition or library operation named
main5 .
The better way to test equivalence of domains is to actually construct new domains.
spad
)abbrev domain TEEONE tee1
tee1():IntegerNumberSystem == Integer
)abbrev domain TEETWO tee2
tee2():IntegerNumberSystem == Integer
)abbrev package TESTSIX test6
test6(): with
main6:()->Integer
== add
main6():Integer ==
x : tee1
x := 1$tee1
y : tee1
y := x
1$Integer
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5058510425788394140-25px011.spad
using old system compiler.
TEEONE abbreviates domain tee1
------------------------------------------------------------------------
initializing NRLIB TEEONE for tee1
compiling into NRLIB TEEONE
(time taken in buildFunctor: 3086)
Time: 0 SEC.
Cumulative Statistics for Constructor tee1
Time: 0 seconds
--------------non extending category----------------------
.. tee1 of cat
(|IntegerNumberSystem|) has no
(|LinearlyExplicitOver| (|Integer|)) finalizing NRLIB TEEONE
Processing tee1 for Browser database:
--->-->tee1(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TEEONE.NRLIB/TEEONE.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TEEONE.NRLIB/TEEONE.fasl
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
tee1 is now explicitly exposed in frame initial
tee1 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TEEONE.NRLIB/TEEONE
TEETWO abbreviates domain tee2
------------------------------------------------------------------------
initializing NRLIB TEETWO for tee2
compiling into NRLIB TEETWO
(time taken in buildFunctor: 3976)
Time: 0 SEC.
Cumulative Statistics for Constructor tee2
Time: 0 seconds
--------------non extending category----------------------
.. tee2 of cat
(|IntegerNumberSystem|) has no
(|LinearlyExplicitOver| (|Integer|)) finalizing NRLIB TEETWO
Processing tee2 for Browser database:
--->-->tee2(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TEETWO.NRLIB/TEETWO.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TEETWO.NRLIB/TEETWO.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
tee2 is now explicitly exposed in frame initial
tee2 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TEETWO.NRLIB/TEETWO
TESTSIX abbreviates package test6
------------------------------------------------------------------------
initializing NRLIB TESTSIX for test6
compiling into NRLIB TESTSIX
compiling exported main6 : () -> Integer
Time: 0 SEC.
(time taken in buildFunctor: 0)
Time: 0 SEC.
Cumulative Statistics for Constructor test6
Time: 0 seconds
finalizing NRLIB TESTSIX
Processing test6 for Browser database:
--->-->test6(constructor): Not documented!!!!
--->-->test6((main6 ((Integer)))): Not documented!!!!
--->-->test6(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TESTSIX.NRLIB/TESTSIX.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TESTSIX.NRLIB/TESTSIX.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
test6 is now explicitly exposed in frame initial
test6 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TESTSIX.NRLIB/TESTSIXfricas
main6()
spad
)abbrev package TESTSEV test7
test7(): with
foo7:tee1 -> tee2
main7:(tee1, tee2) -> tee2
== add
foo7(x) == 5$tee2
main7(x,y) == foo7(x) + y
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5193838586659015839-25px013.spad
using old system compiler.
TESTSEV abbreviates package test7
------------------------------------------------------------------------
initializing NRLIB TESTSEV for test7
compiling into NRLIB TESTSEV
compiling exported foo7 : tee1 -> tee2
****** comp fails at level 1 with expression: ******
error in function foo7
((|Sel| (|tee2|) 5))
****** level 1 ******
x:= (Sel (tee2) 5)
m:= (tee2)
f:=
((((|x| # #) (|main7| #) (|foo7| #) (|$DomainsInScope| # # #) ...)))
>> Apparent user error:
unspecified errorWould you have expected a compile error with test7? This shows there is no automatic coercion in the compiler from Integer to tee2.
spad
)abbrev package TESTEGT test8
test8(): with
foo8:tee1 -> tee2
main8:(tee1, tee2) -> tee2
== add
foo8(x) == 1$tee2 + 1$tee2 + 1$tee2 + 1$tee2 + 1$tee2
main8(x,y) == foo8(x) + y
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2052161184486383446-25px014.spad
using old system compiler.
TESTEGT abbreviates package test8
------------------------------------------------------------------------
initializing NRLIB TESTEGT for test8
compiling into NRLIB TESTEGT
compiling exported foo8 : tee1 -> tee2
Time: 0 SEC.
compiling exported main8 : (tee1,tee2) -> tee2
Time: 0 SEC.
(time taken in buildFunctor: 0)
Time: 0 SEC.
Cumulative Statistics for Constructor test8
Time: 0 seconds
finalizing NRLIB TESTEGT
Processing test8 for Browser database:
--->-->test8(constructor): Not documented!!!!
--->-->test8((foo8 ((tee2) (tee1)))): Not documented!!!!
--->-->test8((main8 ((tee2) (tee1) (tee2)))): Not documented!!!!
--->-->test8(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TESTEGT.NRLIB/TESTEGT.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TESTEGT.NRLIB/TESTEGT.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
test8 is now explicitly exposed in frame initial
test8 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TESTEGT.NRLIB/TESTEGTNote that tee2 does have a multiplicative unit without requiring a coercion from Integer!
So test8 compiles okay. Let's see this in action.
fricas
-- expects 6
main8(1$tee1, 1$tee2)
Type: tee2
fricas
-- expects error
main8(1$tee2, 1$tee1)
There are 1 exposed and 0 unexposed library operations named main8
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op main8
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named main8
with argument type(s)
tee2
tee1
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
So even the interpreter does not coerce tee1 into tee2. Guess the next three:
fricas
main8(1$Integer, 1$Integer)
Type: tee2
fricas
main8(2$tee1, 5$tee2)
Type: tee2
fricas
main8(3::tee1, (2$tee1)::tee2)
Cannot convert the value from type tee1 to tee2 .
The compiler would not recognize 5$tee2, but the interpreter has no problem coercing any integer automatically to tee1 or tee2. This shows Integer is very special for the interpreter, and not treated the same as tee1.
Let's see if we change the name of foo8 to convert and make the converted value to be 4 instead of 5.
spad
)abbrev package TESTNINE test9
test9(): with
convert:tee1 -> tee2
main9:(tee1, tee2) -> tee2
== add
convert(x) == 1$tee2 + 1$tee2 + 1$tee2 + 1$tee2
main9(x,y) == convert(x) + y
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8608886520085933680-25px017.spad
using old system compiler.
TESTNINE abbreviates package test9
------------------------------------------------------------------------
initializing NRLIB TESTNINE for test9
compiling into NRLIB TESTNINE
compiling exported convert : tee1 -> tee2
Time: 0 SEC.
compiling exported main9 : (tee1,tee2) -> tee2
Time: 0 SEC.
(time taken in buildFunctor: 0)
Time: 0 SEC.
Cumulative Statistics for Constructor test9
Time: 0 seconds
finalizing NRLIB TESTNINE
Processing test9 for Browser database:
--->-->test9(constructor): Not documented!!!!
--->-->test9((convert ((tee2) (tee1)))): Not documented!!!!
--->-->test9((main9 ((tee2) (tee1) (tee2)))): Not documented!!!!
--->-->test9(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TESTNINE.NRLIB/TESTNINE.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TESTNINE.NRLIB/TESTNINE.fasl
; compilation finished in 0:00:00.008
------------------------------------------------------------------------
test9 is now explicitly exposed in frame initial
test9 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TESTNINE.NRLIB/TESTNINEtest9 also compiles okay. Let's see this in action. Recall that there is a convert: tee1 -> tee2, but no coerce.
fricas
-- expects 5
main9(1$tee1, 1$tee2)
Type: tee2
fricas
-- expects error
main9(1$tee2, 1$tee1)
There are 1 exposed and 0 unexposed library operations named main9
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op main9
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named main9
with argument type(s)
tee2
tee1
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Note convert is not automatically used by the interpreter. Now let's change convert to coerce. Even though there is now a coerce, tee1 is not declared to be CoercibleTo?(tee2) (to do so would have made the two domain constructors different in structure). We use a value 1 this time for coerce.
spad
)abbrev package TESTTEN test10
test10(): with
coerce:tee1 -> tee2
main10:(tee1, tee2) -> tee2
== add
coerce(x) == 1$tee2
main10(x,y) == coerce(x) + y
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7317946861256718720-25px019.spad
using old system compiler.
TESTTEN abbreviates package test10
------------------------------------------------------------------------
initializing NRLIB TESTTEN for test10
compiling into NRLIB TESTTEN
compiling exported coerce : tee1 -> tee2
Time: 0 SEC.
compiling exported main10 : (tee1,tee2) -> tee2
Time: 0 SEC.
(time taken in buildFunctor: 0)
Time: 0 SEC.
Cumulative Statistics for Constructor test10
Time: 0 seconds
finalizing NRLIB TESTTEN
Processing test10 for Browser database:
--->-->test10(constructor): Not documented!!!!
--->-->test10((coerce ((tee2) (tee1)))): Not documented!!!!
--->-->test10((main10 ((tee2) (tee1) (tee2)))): Not documented!!!!
--->-->test10(): Missing Description
; compiling file "/var/aw/var/LatexWiki/TESTTEN.NRLIB/TESTTEN.lsp" (written 20 SEP 2025 01:29:48 AM):
; wrote /var/aw/var/LatexWiki/TESTTEN.NRLIB/TESTTEN.fasl
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
test10 is now explicitly exposed in frame initial
test10 will be automatically loaded when needed from
/var/aw/var/LatexWiki/TESTTEN.NRLIB/TESTTENfricas
-- expects 2
main10(1$tee1, 1$tee2)
Type: tee2
fricas
-- expects error
main10(1$tee2, 1$tee1)
There are 1 exposed and 0 unexposed library operations named main10
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op main10
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
main10 with argument type(s)
tee2
tee1
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Now 1$tee1 is automatically coerced to 1$tee2 even though tee1 does not have CoercibleTo?(tee2), but this is the interpreter!
The next thing to do would be to test equivalence of categories. Some other time.
