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This is an experiment to see what the differences are between defining NonNegativeInteger (NNI) as a SubDomain of Integer, versus defining it simply based on Integer itself.

The surprizing conclusion is that the latter definition of NNI produces nearly identical code as the first definition except for the now required definitions of coerce:%->Integer and convert:Integer->%, but without the extra CategoryFrame datastructure. This would seem to confirm that the SubDomain construct does not have any great advantages over the more direct definition.

First, NNI the usual way:

spad
)abbrev domain NNI NonNegativeInteger
++ Description: \spadtype{NonNegativeInteger} provides functions for non
++   negative integers.
NonNegativeInteger: Join(OrderedAbelianMonoidSup,Monoid) with
            _quo : (%,%) -> %
              ++ a quo b returns the quotient of \spad{a} and b, forgetting
              ++ the remainder.
            _rem : (%,%) -> %
              ++ a rem b returns the remainder of \spad{a} and b.
            gcd  : (%,%) -> %
              ++ gcd(a,b) computes the greatest common divisor of two
              ++ non negative integers \spad{a} and b.
            divide: (%,%) -> Record(quotient:%,remainder:%)
              ++ divide(a,b) returns a record containing both
              ++ remainder and quotient.
            _exquo: (%,%) -> Union(%,"failed")
              ++ exquo(a,b) returns the quotient of \spad{a} and b, or "failed"
              ++ if b is zero or \spad{a} rem b is zero.
            shift: (%, Integer) -> %
              ++ shift(a,i) shift \spad{a} by i bits.
            random   : % -> %
              ++ random(n) returns a random integer from 0 to \spad{n-1}.
            CommutativeStar
              ++ CommutativeStar means multiplication is commutative : \spad{x*y = y*x}.
== SubDomain(Integer,#1 >= 0) add x,y:% sup(x,y) == MAX(x,y)$Lisp shift(x:%, n:Integer):% == ASH(x,n)$Lisp subtractIfCan(x, y) == c:Integer := (x pretend Integer) - (y pretend Integer) c < 0 => "failed" c pretend %
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4255648489146469329-25px001.spad
      using old system compiler.
   NNI abbreviates domain NonNegativeInteger 
------------------------------------------------------------------------
   initializing NRLIB NNI for NonNegativeInteger 
   compiling into NRLIB NNI 
   compiling exported sup : ($,$) -> $
      NNI;sup;3$;1 is replaced by MAX 
;;; *** |NNI;sup;3$;1| REDEFINED Time: 0.01 SEC.
compiling exported shift : ($,Integer) -> $ NNI;shift;$I$;2 is replaced by ASH
;;; *** |NNI;shift;$I$;2| REDEFINED Time: 0 SEC.
compiling exported subtractIfCan : ($,$) -> Union($,failed) Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |NonNegativeInteger| REDEFINED
;;; *** |NonNegativeInteger| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor NonNegativeInteger Time: 0.01 seconds
--------------non extending category---------------------- .. NonNegativeInteger of cat (|Join| (|OrderedAbelianMonoidSup|) (|Monoid|) (CATEGORY |domain| (SIGNATURE |quo| ($ $ $)) (SIGNATURE |rem| ($ $ $)) (SIGNATURE |gcd| ($ $ $)) (SIGNATURE |divide| ((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $)) (SIGNATURE |exquo| ((|Union| $ "failed") $ $)) (SIGNATURE |shift| ($ $ (|Integer|))) (SIGNATURE |random| ($ $)) (ATTRIBUTE (|CommutativeStar|)))) has no (|IntegerNumberSystem|) finalizing NRLIB NNI Processing NonNegativeInteger for Browser database: --------constructor--------- --------(quo (% % %))--------- --------(rem (% % %))--------- --------(gcd (% % %))--------- --------(divide ((Record (: quotient %) (: remainder %)) % %))--------- --------(exquo ((Union % failed) % %))--------- --------(shift (% % (Integer)))--------- --------(random (% %))--------- --------constructor--------- --->/usr/local/lib/fricas/target/x86_64-unknown-linux/../../src/algebra/NNI.spad-->NonNegativeInteger(): Spurious comments: CommutativeStar means multiplication is commutative : \spad{x*y = y*x}. ; compiling file "/var/aw/var/LatexWiki/NNI.NRLIB/NNI.lsp" (written 31 JUL 2013 04:41:02 PM):
; /var/aw/var/LatexWiki/NNI.NRLIB/NNI.fasl written ; compilation finished in 0:00:00.018 ------------------------------------------------------------------------ NonNegativeInteger is now explicitly exposed in frame initial NonNegativeInteger will be automatically loaded when needed from /var/aw/var/LatexWiki/NNI.NRLIB/NNI

fricas
x1:NNI:=1$NNI

\label{eq1}1(1)
fricas
y1:NNI:=2$NNI

\label{eq2}2(2)
fricas
z1:NNI:=(x1 + y1)$NNI

\label{eq3}3(3)
fricas
z1::INT

\label{eq4}3(4)
Type: Integer
fricas
subtractIfCan(y1,x1)$NNI

\label{eq5}1(5)
Type: Union(NonNegativeInteger,...)
fricas
(4::INT)::NNI
fricas
Compiling function G708 with type Integer -> Boolean

\label{eq6}4(6)
fricas
((-1)::INT)::NNI
Cannot convert from type Integer to NonNegativeInteger for value - 1

Now write NNI without using SubDomain:

spad
)abbrev domain NNI2 NonNegativeInteger2
++ Description: \spadtype{NonNegativeInteger} provides functions for non
++   negative integers.
NonNegativeInteger2: Join(OrderedAbelianMonoidSup,Monoid,CoercibleTo(Integer)) with
            _quo : (%,%) -> %
              ++ a quo b returns the quotient of \spad{a} and b, forgetting
              ++ the remainder.
            _rem : (%,%) -> %
              ++ a rem b returns the remainder of \spad{a} and b.
            gcd  : (%,%) -> %
              ++ gcd(a,b) computes the greatest common divisor of two
              ++ non negative integers \spad{a} and b.
            divide: (%,%) -> Record(quotient:%,remainder:%)
              ++ divide(a,b) returns a record containing both
              ++ remainder and quotient.
            _exquo: (%,%) -> Union(%,"failed")
              ++ exquo(a,b) returns the quotient of \spad{a} and b, or "failed"
              ++ if b is zero or \spad{a} rem b is zero.
            shift: (%, Integer) -> %
              ++ shift(a,i) shift \spad{a} by i bits.
            random   : % -> %
              ++ random(n) returns a random integer from 0 to \spad{n-1}.
            CommutativeStar
              ++ commutative("*") means multiplication is commutative : \spad{x*y = y*x}.
            convert: Integer -> %
== Integer add
sup(x:%,y:%) == MAX(x,y)$Lisp shift(x:%, n:Integer):% == ASH(x,n)$Lisp subtractIfCan(x, y) == c:Integer := (x pretend Integer) - (y pretend Integer) c < 0 => "failed" c pretend % coerce(x:%):Integer == x pretend Integer convert(x:Integer):% == x >= 0 => x pretend % error(["Cannot convert from type Integer to NonNegativeInteger for value", string(x)$String])
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2203784355024913414-25px003.spad
      using old system compiler.
   NNI2 abbreviates domain NonNegativeInteger2 
------------------------------------------------------------------------
   initializing NRLIB NNI2 for NonNegativeInteger2 
   compiling into NRLIB NNI2 
   compiling exported sup : ($,$) -> $
      NNI2;sup;3$;1 is replaced by MAX 
Time: 0 SEC.
compiling exported shift : ($,Integer) -> $ NNI2;shift;$I$;2 is replaced by ASH Time: 0 SEC.
compiling exported subtractIfCan : ($,$) -> Union($,failed) Time: 0 SEC.
compiling exported coerce : $ -> Integer NNI2;coerce;$I;4 is replaced by x Time: 0 SEC.
compiling exported convert : Integer -> $ Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |NonNegativeInteger2| REDEFINED
;;; *** |NonNegativeInteger2| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor NonNegativeInteger2 Time: 0 seconds
--------------non extending category---------------------- .. NonNegativeInteger2 of cat (|Join| (|OrderedAbelianMonoidSup|) (|Monoid|) (|CoercibleTo| (|Integer|)) (CATEGORY |domain| (SIGNATURE |quo| ($ $ $)) (SIGNATURE |rem| ($ $ $)) (SIGNATURE |gcd| ($ $ $)) (SIGNATURE |divide| ((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $)) (SIGNATURE |exquo| ((|Union| $ "failed") $ $)) (SIGNATURE |shift| ($ $ (|Integer|))) (SIGNATURE |random| ($ $)) (ATTRIBUTE (|CommutativeStar|)) (SIGNATURE |convert| ($ (|Integer|))))) has no (|IntegerNumberSystem|) finalizing NRLIB NNI2 Processing NonNegativeInteger2 for Browser database: --------constructor--------- --------(quo (% % %))--------- --------(rem (% % %))--------- --------(gcd (% % %))--------- --------(divide ((Record (: quotient %) (: remainder %)) % %))--------- --------(exquo ((Union % failed) % %))--------- --------(shift (% % (Integer)))--------- --------(random (% %))--------- --->-->NonNegativeInteger2((convert (% (Integer)))): Not documented!!!! --------constructor--------- --->-->NonNegativeInteger2(): Spurious comments: commutative(\spad{"*"}) means multiplication is commutative : \spad{x*y = y*x}. ; compiling file "/var/aw/var/LatexWiki/NNI2.NRLIB/NNI2.lsp" (written 31 JUL 2013 04:41:02 PM):
; /var/aw/var/LatexWiki/NNI2.NRLIB/NNI2.fasl written ; compilation finished in 0:00:00.016 ------------------------------------------------------------------------ NonNegativeInteger2 is now explicitly exposed in frame initial NonNegativeInteger2 will be automatically loaded when needed from /var/aw/var/LatexWiki/NNI2.NRLIB/NNI2

The behaviour is identical to NNI

fricas
x2:NNI2:=1$NNI2

\label{eq7}1(7)
Type: NonNegativeInteger2?
fricas
y2:NNI2:=2$NNI2

\label{eq8}2(8)
Type: NonNegativeInteger2?
fricas
z2:NNI2:=(x2+y2)$NNI2

\label{eq9}3(9)
Type: NonNegativeInteger2?
fricas
z2::INT

\label{eq10}3(10)
Type: Integer
fricas
subtractIfCan(y2,x2)$NNI2

\label{eq11}1(11)
Type: Union(NonNegativeInteger2?,...)
fricas
(4::INT)::NNI2

\label{eq12}4(12)
Type: NonNegativeInteger2?
fricas
((-1)::INT)::NNI2
>> Error detected within library code: Cannot convert from type Integer to NonNegativeInteger for value "-1"

The generated lisp code for NNI is:

fricas
)lisp (system "cat < /var/zope2/var/LatexWiki/NNI.NRLIB/NNI.lsp | tee /tmp/NNI.lsp")
>> System error: The function BOOT::SYSTEM is undefined.

Compared NNI (-) to NNI2 (+):

fricas
)lisp (system "cat < /var/zope2/var/LatexWiki/NNI2.NRLIB/NNI2.lsp | diff -au /tmp/NNI.lsp -")
>> System error: The function BOOT::SYSTEM is undefined.

SubDomain written in Aldor:

aldor
#include "axiom.as"
SubSetDom(X:Type, E:X->Boolean):
with {
  convert: X -> %;
  coerce: % -> X;
} == X add {
  coerce(x:%):X == x pretend X;
  convert(x:X):% == {
    E(x) => x pretend %;
    error("Cannot convert");
  }
}
aldor
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3345325303575885968-25px007.as
      using AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libaxiom.al'.
#2 (Warning) Could not use archive file `libaxiom.al'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 4: 
import from AxiomLib;
............^
[L4 C13] #3 (Error) No meaning for identifier `AxiomLib'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 15: import { true: %, false: % } from Boolean; ..................................^ [L15 C35] #4 (Error) No meaning for identifier `Boolean'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 17: string: Literal -> %; ........................^.......^ [L17 C25] #5 (Error) No meaning for identifier `Literal'. [L17 C33] #6 (Error) There are no suitable meanings for the operator `->'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 18: } from String; .......^ [L18 C8] #8 (Error) No meaning for identifier `String'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3345325303575885968-25px007.as", line 2: SubSetDom(X:Type, E:X->Boolean): ............^.......^^.^ [L2 C13] #9 (Error) No meaning for identifier `Type'. [L2 C21] #10 (Error) There are 0 meanings for `X' in this context. The possible types were: X: Type, a local The context requires an expression of type Tuple(Type). [L2 C22] #12 (Error) There are no suitable meanings for the operator `->'. [L2 C24] #11 (Error) No meaning for identifier `Boolean'. [L2 C24] #13 (Fatal Error) Too many errors (use `-M emax=n' or `-M no-emax' to change the limit).
The )library system command was not called after compilation.

fricas
)sh SubSetDom
The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer .
SubSetDom is not the name of a known type constructor. If you want to see information about any operations named SubSetDom , issue )display operations SubSetDom
fricas
)set functions compile on

Test the new subdomain.

fricas
E:Integer->Boolean
Type: Void
fricas
E(z) == (z >= 0)
Type: Void
fricas
E(3)
fricas
Compiling function E with type Integer -> Boolean

\label{eq13} \mbox{\rm true} (13)
Type: Boolean
fricas
)sh SubSetDom(Integer, E)
The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer .
SubSetDom is not the name of a known type constructor. If you want to see information about any operations named SubSetDom , issue )display operations SubSetDom
Integer is not the name of a known type constructor. If you want to see information about any operations named Integer , issue )display operations Integer
E is not the name of a known type constructor. If you want to see information about any operations named E , issue )display operations E N:SubSetDom(Integer, E)
There are no library operations named SubSetDom Use HyperDoc Browse or issue )what op SubSetDom to learn if there is any operation containing " SubSetDom " in its name.
Cannot find a definition or applicable library operation named SubSetDom with argument type(s) Type FunctionCalled(E)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. N:=1

\label{eq14}1(14)
fricas
N:=2

\label{eq15}2(15)
fricas
N:=-2

\label{eq16}- 2(16)
Type: Integer

Now define NNI the using the new SubSetDom?:

For some unknown reason this fails:

aldor
#include "axiom.as"
#pile
NNI3: Join(OrderedAbelianMonoidSup,Monoid, CoercibleTo(Integer))
   with
     convert: Integer -> %
   == SubSetDom(Integer, (z:Integer):Boolean +-> (z >= 0)) add
sup(x:%,y:%):% == x>y => x y
subtractIfCan(x:%, y:%):Union(ok:%,failed:'failed') == c:Integer == (x pretend Integer) - (y pretend Integer) c < 0 => return [failed] [c pretend %]
aldor
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3694095911335604383-25px010.as
      using AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libaxiom.al'.
#2 (Warning) Could not use archive file `libaxiom.al'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 4: 
import from AxiomLib;
............^
[L4 C13] #3 (Error) No meaning for identifier `AxiomLib'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 15: import { true: %, false: % } from Boolean; ..................................^ [L15 C35] #4 (Error) No meaning for identifier `Boolean'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 17: string: Literal -> %; ........................^.......^ [L17 C25] #5 (Error) No meaning for identifier `Literal'. [L17 C33] #6 (Error) There are no suitable meanings for the operator `->'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 18: } from String; .......^ [L18 C8] #8 (Error) No meaning for identifier `String'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3694095911335604383-25px010.as", line 4: with ...^ [L4 C4] #12 (Error) There are no suitable meanings for the operator `Join'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3694095911335604383-25px010.as", line 5: convert: Integer -> % ..............^.......^ [L5 C15] #9 (Error) No meaning for identifier `Integer'. [L5 C23] #10 (Error) There are no suitable meanings for the operator `->'. [L5 C23] #13 (Fatal Error) Too many errors (use `-M emax=n' or `-M no-emax' to change the limit).
The )library system command was not called after compilation.

But this works:

aldor
#include "axiom.as"
#pile
NNI4: Join(OrderedAbelianMonoidSup,Monoid, CoercibleTo(Integer))
   with
     convert: Integer -> %
-- == SubSetDom(Integer, (z:Integer):Boolean +-> (z >= 0)) add
   == Integer add
coerce(x:%):Integer == x pretend Integer;
convert(x:Integer):% == x>=0 => return x pretend % error "Cannot convert"
sup(x:%,y:%):% == x>y => x y
subtractIfCan(x:%, y:%):Union(ok:%,failed:'failed') == c:Integer == (x pretend Integer) - (y pretend Integer) c < 0 => return [failed] [c pretend %]
aldor
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/84397582143168417-25px011.as
      using AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libaxiom.al'.
#2 (Warning) Could not use archive file `libaxiom.al'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 4: 
import from AxiomLib;
............^
[L4 C13] #3 (Error) No meaning for identifier `AxiomLib'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 15: import { true: %, false: % } from Boolean; ..................................^ [L15 C35] #4 (Error) No meaning for identifier `Boolean'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 17: string: Literal -> %; ........................^.......^ [L17 C25] #5 (Error) No meaning for identifier `Literal'. [L17 C33] #6 (Error) There are no suitable meanings for the operator `->'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 18: } from String; .......^ [L18 C8] #8 (Error) No meaning for identifier `String'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/84397582143168417-25px011.as", line 4: with ...^ [L4 C4] #12 (Error) There are no suitable meanings for the operator `Join'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/84397582143168417-25px011.as", line 5: convert: Integer -> % ..............^.......^ [L5 C15] #9 (Error) No meaning for identifier `Integer'. [L5 C23] #10 (Error) There are no suitable meanings for the operator `->'. [L5 C23] #13 (Fatal Error) Too many errors (use `-M emax=n' or `-M no-emax' to change the limit).
The )library system command was not called after compilation.

fricas
)set message bottomup on
x4:NNI4:=1$NNI4
The function One is not implemented in NIL . y4:NNI4:=2$NNI4
Function Selection for NNI4 Arguments: () -> no function NNI4 found for arguments () There are no library operations named NNI4 Use HyperDoc Browse or issue )what op NNI4 to learn if there is any operation containing " NNI4 " in its name.
Cannot find a no-argument definition or library operation named NNI4 . z4:NNI4:=(x4 + y4)$NNI4
The function + is not implemented in NIL . z4::INT
Cannot convert from type Variable(z4) to Integer for value z4
subtractIfCan(y4,x4)$NNI4
The function subtractIfCan is not implemented in NIL . (4::INT)::NNI4
Cannot convert from type Integer to NIL for value 4
((-1)::INT)::NNI4
Cannot convert from type Integer to NIL for value - 1

aldor
#pile
#include "axiom"
--)abbrev package TEST test
NNI ==> NonNegativeInteger2
test(): with
    main: ()-> NNI
  == add
    n1:NNI:=1
    n2:NNI:=2
    main():NNI == n1-n2
aldor
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5451681632571730514-25px013.as
      using AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
#1 (Warning) Could not use archive file `libaxiom.al'.
#2 (Warning) Could not use archive file `libaxiom.al'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 4: 
import from AxiomLib;
............^
[L4 C13] #3 (Error) No meaning for identifier `AxiomLib'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 15: import { true: %, false: % } from Boolean; ..................................^ [L15 C35] #4 (Error) No meaning for identifier `Boolean'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 17: string: Literal -> %; ........................^.......^ [L17 C25] #5 (Error) No meaning for identifier `Literal'. [L17 C33] #6 (Error) There are no suitable meanings for the operator `->'.
"/usr/local/aldor/linux/1.1.0/include/axiom.as", line 18: } from String; .......^ [L18 C8] #8 (Error) No meaning for identifier `String'.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5451681632571730514-25px013.as", line 4: NNI ==> NonNegativeInteger2 ........^ [L4 C9] #9 (Error) (After Macro Expansion) No meaning for identifier `NonNegativeInteger2'. Expanded expression was: NonNegativeInteger2
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5451681632571730514-25px013.as", line 6: main: ()-> NNI ............^ [L6 C13] #10 (Error) There are no suitable meanings for the operator `->'. [L6 C13] #13 (Fatal Error) Too many errors (use `-M emax=n' or `-M no-emax' to change the limit).
The )library system command was not called after compilation.

fricas
)show TEST
The )show system command is used to display information about types or partial types. For example, )show Integer will show information about Integer .
TEST is not the name of a known type constructor. If you want to see information about any operations named TEST , issue )display operations TEST

fricas
main()
Function Selection for main Arguments: () -> no function main found for arguments () 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 .




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