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The Sum domain constructor is intended to be the Categorical Dual of the Product domain constructor

fricas
(1) -> <spad>
fricas
)abbrev domain SUM Sum
++ Description:
++ This domain implements direct union
Sum (A:SetCategory,B:SetCategory) : C == T
 where
  C == SetCategory  with
       if A has Finite and B has Finite then Finite
       if A has Monoid and B has Monoid then Monoid
       if A has AbelianMonoid and B has AbelianMonoid then AbelianMonoid
       if A has CancellationAbelianMonoid and
          B has CancellationAbelianMonoid then CancellationAbelianMonoid
       if A has Group  and B has Group  then  Group
       if A has AbelianGroup and B has AbelianGroup then  AbelianGroup
       if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup
                                             then OrderedAbelianMonoidSup
       if A has OrderedSet and B has OrderedSet then  OrderedSet
selectsum : % -> Union(acomp:A,bcomp:B) ++ selectsum(x) \undocumented in1 : A -> % ++ makefirst(a) \undocumented in2 : B -> % ++ makesecond(b) \undocumented
T == add
--representations Rep ==> Union(acomp:A, bcomp:B) rep x ==> (x@%) pretend Rep per x ==> (x@Rep) pretend %
import Rep
--declarations x,y: % i: NonNegativeInteger p: NonNegativeInteger a: A b: B c: PositiveInteger d: Integer
--define coerce(x:%):OutputForm == rep(x) case acomp => sub(coerce(rep(x).acomp),"1") sub(coerce(rep(x).bcomp),"2") x=y == rep(x)=rep(y) selectsum(x) == rep(x) in1(a) == per [a] in2(b) == per [b]
if A has Monoid and B has Monoid then -- represent unit of Sum(A,B) as 1$A (We could use either 1$A or 1$B) 1 == per [1$A] x * y == rep(x) case acomp and rep(y) case acomp => per [rep(x).acomp * rep(y).acomp] rep(x) case bcomp and rep(y) case bcomp => per [rep(x).bcomp * rep(y).bcomp] -- unit of Sum(A,B)=1$A is unit for B x=1 => y y=1 => x error "not same type" x ^ p == rep(x) case acomp => per [rep(x).acomp ^ p] per [rep(x).bcomp ^ p]
if A has Finite and B has Finite then size == size$A + size$B index(n) == n > size$B => per [index((n::Integer - size$B)::PositiveInteger)$A] per [index(n)$B] random() == random()$Boolean => per [random()$A] per [random()$B] lookup(x) == rep(x) case acomp => (lookup(rep(x).acomp)$A::NonNegativeInteger + size$B)::PositiveInteger lookup(rep(x).bcomp)$B hash(x) == rep(x) case acomp => hash(rep(x).acomp)$A + size$B::SingleInteger hash(rep(x).bcomp)$B
if A has Group and B has Group then inv(x) == rep(x) case acomp => per [inv(rep(x).acomp)] per [inv(rep(x).bcomp)]
if A has AbelianMonoid and B has AbelianMonoid then -- represent zero of Sum(A,B) as 0$A (We could use either 0$A or 0$B) 0 == per [0$A] x + y == rep(x) case acomp and rep(y) case acomp => per [rep(x).acomp + rep(y).acomp] rep(x) case bcomp and rep(y) case bcomp => per [rep(x).bcomp + rep(y).bcomp] -- zero of Sum(A,B)=0$A is zero for B x=0 => y y=0 => x error "not same type" c * x == rep(x) case acomp => per [c * rep(x).acomp] per [c* rep(x).bcomp]
if A has AbelianGroup and B has AbelianGroup then -x == rep(x) case acomp => per [- rep(x).acomp] per [- rep(x).bcomp] (x - y):% == rep(x) case acomp and rep(y) case acomp => per [rep(x).acomp - rep(y).acomp] rep(x) case bcomp and rep(y) case bcomp => per [rep(x).bcomp - rep(y).bcomp] -- zero of Sum(A,B)=0$A is zero for B x=0 => -y y=0 => x error "not same type"
d * x == rep(x) case acomp => per [d * rep(x).acomp] per [d* rep(x).bcomp]
if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then sup(x,y) == rep(x) case acomp and rep(y) case acomp => per [sup(rep(x).acomp,rep(y).acomp)] rep(x) case bcomp and rep(y) case bcomp => per [sup(rep(x).bcomp,rep(y).bcomp)] rep(x) case acomp and rep(y) case bcomp => y x
if A has OrderedSet and B has OrderedSet then x < y == rep(x) case acomp and rep(y) case acomp => rep(x).acomp < rep(y).acomp rep(x) case bcomp and rep(y) case bcomp => rep(x).bcomp < rep(y).bcomp rep(x) case acomp and rep(y) case bcomp</spad>
fricas
Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6614319580939889182-25px001.spad
      using old system compiler.
   SUM abbreviates domain Sum 
------------------------------------------------------------------------
   initializing NRLIB SUM for Sum 
   compiling into NRLIB SUM 
   processing macro definition Rep ==> Union(acomp: A,bcomp: B) 
   processing macro definition rep x ==> pretend(@(x,%),Union(acomp: A,bcomp: B)) 
   processing macro definition per x ==> pretend(@(x,Union(acomp: A,bcomp: B)),%) 
   importing Union(acomp: A,bcomp: B)
   compiling exported coerce : % -> OutputForm
****** comp fails at level 5 with expression: ******
error in function coerce 
(SEQ (|:=| (|:| #1=#:G0 (|Boolean|)) (|case| (|pretend| (@ |x| %) (|Union| (|:| |acomp| A) (|:| |bcomp| B))) |acomp|)) (|exit| 1 (IF #1# (|sub| (|coerce| ((|pretend| (@ |x| %) (|Union| (|:| |acomp| A) (|:| |bcomp| B))) |acomp|)) | << 1 >> |) (|sub| (|coerce| ((|pretend| (@ |x| %) (|Union| (|:| |acomp| A) (|:| |bcomp| B))) |bcomp|)) "2")))) ****** level 5 ****** $x:= 1 $m:= (OutputForm) $f:= ((((#:G0 # #) (|x| # #) (|d| #) (|c| #) ...)))
>> Apparent user error: Cannot coerce 1 of mode 1 to mode (OutputForm)

fricas
size()$Sum(PF 7,PF 13)
Sum is an unknown constructor and so is unavailable. Did you mean to use -> but type something different instead?

The CoTuple? domain constructor is intended to be the Categorical Dual of the Tuple domain constructor

spad
)abbrev domain COT CoTuple
++ This domain is intended to be the categorical dual of a
++ Tuple (comma-delimited homogeneous sequence of values).
++ As such it implements an n-ary disjoint union.
CoTuple(S:Type): CoercibleTo(S) with
    inj: (NonNegativeInteger,S) -> %
      ++ inject(x,n) returns x as an element of the n-th
      ++ component of the CoTuple. CoTuples are 0-based
    ind: % -> NonNegativeInteger
      ++ index(x) returns the component number of x in the CoTuple
    if S has Monoid then Monoid
    if S has AbelianMonoid then AbelianMonoid
    if S has CancellationAbelianMonoid then CancellationAbelianMonoid
    if S has Group then Group
    if S has AbelianGroup then  AbelianGroup
    if S has OrderedAbelianMonoidSup then OrderedAbelianMonoidSup
    if S has SetCategory then SetCategory
    if S has OrderedSet then OrderedSet
  == add
    Rep ==> Record(ind : NonNegativeInteger, elt : S)
    rep x ==> (x@%) pretend Rep
    per x ==> (x@Rep) pretend %
    --declarations
    x,y: %
    s: S
    i: NonNegativeInteger
    p: NonNegativeInteger
    c: PositiveInteger
    d: Integer
coerce(x:%):S == rep(x).elt inj(i,s) == per [i,s] ind(x) == rep(x).ind
if S has SetCategory then x = y == (rep(x).ind = rep(y).ind) and (rep(x).elt = rep(y).elt) coerce(x : %): OutputForm == sub(coerce(rep(x).elt),coerce(rep(x).ind)) if S has Monoid then -- represent unit of CoTuple(S) as 1$S of component 0 1 == per [0,1] x * y == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt * rep(y).elt] x = 1 => y y = 1 => x error "not same type" x ^ p == per [rep(x).ind, rep(x).elt ^ p]
if S has Group then inv(x) == per [rep(x).ind, inv(rep(x).elt)]
if S has AbelianMonoid then -- represent zero of Sum(A,B) as 0$S 0 == per [0,0] x + y == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt + rep(y).elt] x = 0 => y y = 0 => x error "not same type" c * x == per [rep(x).ind, c * rep(x).elt]
if S has AbelianGroup then - x == per [rep(x).ind, -rep(x).elt] (x - y):% == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).elt - rep(y).elt] x = 0 => -y y = 0 => x error "not same type"
d * x == per [rep(x).ind, d* rep(x).elt]
if S has OrderedAbelianMonoidSup then sup(x,y) == rep(x).ind = rep(y).ind => per [rep(x).ind, sup(rep(x).elt,rep(y).elt)] rep(x).ind < rep(y).ind => y x if S has OrderedSet then x < y == rep(x).ind = rep(y).ind => rep(x).elt < rep(y).elt rep(x).ind < rep(y).ind
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2750494589739067594-25px003.spad
      using old system compiler.
   COT abbreviates domain CoTuple 
------------------------------------------------------------------------
   initializing NRLIB COT for CoTuple 
   compiling into NRLIB COT 
   processing macro definition Rep ==> Record(ind: NonNegativeInteger,elt: S) 
   processing macro definition rep x ==> pretend(@(x,%),Record(ind: NonNegativeInteger,elt: S)) 
   processing macro definition per x ==> pretend(@(x,Record(ind: NonNegativeInteger,elt: S)),%) 
   compiling exported coerce : % -> S
      COT;coerce;%S;1 is replaced by QCDR 
Time: 0 SEC.
compiling exported inj : (NonNegativeInteger,S) -> % COT;inj;NniS%;2 is replaced by CONS Time: 0 SEC.
compiling exported ind : % -> NonNegativeInteger COT;ind;%Nni;3 is replaced by QCAR Time: 0 SEC.
****** Domain: S already in scope augmenting S: (SetCategory) compiling exported = : (%,%) -> Boolean Time: 0 SEC.
compiling exported coerce : % -> OutputForm Time: 0 SEC.
****** Domain: S already in scope augmenting S: (Monoid) compiling exported One : () -> % Time: 0 SEC.
compiling exported * : (%,%) -> % Time: 0 SEC.
compiling exported ^ : (%,NonNegativeInteger) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (Group) compiling exported inv : % -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (AbelianMonoid) compiling exported Zero : () -> % Time: 0 SEC.
compiling exported + : (%,%) -> % Time: 0 SEC.
compiling exported * : (PositiveInteger,%) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (AbelianGroup) compiling exported - : % -> % Time: 0 SEC.
compiling exported - : (%,%) -> % Time: 0 SEC.
compiling exported * : (Integer,%) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (OrderedAbelianMonoidSup) compiling exported sup : (%,%) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (OrderedSet) compiling exported < : (%,%) -> Boolean Time: 0 SEC.
****** Domain: S already in scope augmenting S: (AbelianGroup) ****** Domain: S already in scope augmenting S: (AbelianMonoid) ****** Domain: S already in scope augmenting S: (CancellationAbelianMonoid) ****** Domain: S already in scope augmenting S: (Group) ****** Domain: S already in scope augmenting S: (Monoid) ****** Domain: S already in scope augmenting S: (OrderedAbelianMonoidSup) ****** Domain: S already in scope augmenting S: (OrderedSet) ****** Domain: S already in scope augmenting S: (SetCategory) (time taken in buildFunctor: 171)
;;; *** |CoTuple| REDEFINED
;;; *** |CoTuple| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor CoTuple Time: 0.02 seconds
finalizing NRLIB COT Processing CoTuple for Browser database: --------constructor--------- --------(inj (% (NonNegativeInteger) S))--------- --->-->CoTuple((inj (% (NonNegativeInteger) S))): Improper first word in comments: inject "inject(\\spad{x},{}\\spad{n}) returns \\spad{x} as an element of the \\spad{n}-th component of the CoTuple. CoTuples are 0-based" --------(ind ((NonNegativeInteger) %))--------- --->-->CoTuple((ind ((NonNegativeInteger) %))): Improper first word in comments: index "index(\\spad{x}) returns the component number of \\spad{x} in the CoTuple" ; compiling file "/var/aw/var/LatexWiki/COT.NRLIB/COT.lsp" (written 01 DEC 2024 09:07:35 AM):
; wrote /var/aw/var/LatexWiki/COT.NRLIB/COT.fasl ; compilation finished in 0:00:00.024 ------------------------------------------------------------------------ CoTuple is now explicitly exposed in frame initial CoTuple will be automatically loaded when needed from /var/aw/var/LatexWiki/COT.NRLIB/COT

fricas
inj(1,10)

\label{eq1}{10}_{1}(1)
Type: CoTuple?(PositiveInteger?)
fricas
inj(1,10) < inj(2,10)

\label{eq2} \mbox{\rm true} (2)
Type: Boolean
fricas
sup(inj(1,99), inj(2,10))$CoTuple(NNI)

\label{eq3}{10}_{2}(3)
Type: CoTuple?(NonNegativeInteger?)

The DirectSum? domain constructor implements an associative (flat) DirectSum? domain that is the dual to DirectProduct?.

spad
)abbrev domain DIRSUM DirectSum
++ This domain is intended to be the categorical dual of
++ DirectProduct (comma-delimited homogeneous sequence of
++ values). As such it implements an n-ary disjoint union.
DirectSum(S:Type): Type with
    coerce: S -> %
      ++ any type is a 1-ary union
    inj: (NonNegativeInteger,NonNegativeInteger,%) -> %
      ++ inject(i,n,x) injects the m-CoTuple element x as the
      ++ (m + i)-th component of the (n+m)-CoTuple.
    ind: % -> NonNegativeInteger
      ++ index(x) returns the component number of x in the CoTuple
    len: % -> NonNegativeInteger
      ++ len(x) returns the number of components
    if S has Monoid then Monoid
    if S has AbelianMonoid then AbelianMonoid
    if S has CancellationAbelianMonoid then CancellationAbelianMonoid
    if S has Group then Group
    if S has AbelianGroup then  AbelianGroup
    if S has OrderedAbelianMonoidSup then OrderedAbelianMonoidSup
    if S has SetCategory then SetCategory
    if S has OrderedSet then OrderedSet
  == add
    Rep ==> Record(ind : NonNegativeInteger, len : NonNegativeInteger, elt : S)
    rep x ==> (x@%) pretend Rep
    per x ==> (x@Rep) pretend %
    --declarations
    x,y: %
    s: S
    i: NonNegativeInteger
    p: NonNegativeInteger
    c: PositiveInteger
    d: Integer
coerce(s:S):% == per [0,0,s] inj(i,n,x) == i < n => per [rep(x).len+i,rep(x).len+n,rep(x).elt] error "index out of bounds" ind(x) == rep(x).ind len(x) == rep(x).len
if S has SetCategory then x = y == (rep(x).ind = rep(y).ind) and (rep(x).elt = rep(y).elt) coerce(x : %): OutputForm == sub(coerce(rep(x).elt),coerce(rep(x).ind)) if S has Monoid then -- represent unit of CoTuple(S) as 1$S of component 0 1 == per [0,0,1] x * y == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).len, rep(x).elt * rep(y).elt] x = 1 => y y = 1 => x error "not same type" x ^ p == per [rep(x).ind, rep(x).len, rep(x).elt ^ p]
if S has Group then inv(x) == per [rep(x).ind, rep(x).len, inv(rep(x).elt)]
if S has AbelianMonoid then -- represent zero of Sum(A,B) as 0$S 0 == per [0,0,0] x + y == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).len, rep(x).elt + rep(y).elt] x = 0 => y y = 0 => x error "not same type" c * x == per [rep(x).ind, rep(x).len, c * rep(x).elt]
if S has AbelianGroup then - x == per [rep(x).ind, rep(x).len, -rep(x).elt] (x - y):% == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).len, rep(x).elt - rep(y).elt] x = 0 => -y y = 0 => x error "not same type"
d * x == per [rep(x).ind, rep(x).len, d* rep(x).elt]
if S has OrderedAbelianMonoidSup then sup(x,y) == rep(x).ind = rep(y).ind => per [rep(x).ind, rep(x).len, sup(rep(x).elt,rep(y).elt)] rep(x).ind < rep(y).ind => y x if S has OrderedSet then x < y == rep(x).ind = rep(y).ind => rep(x).elt < rep(y).elt rep(x).ind < rep(y).ind
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5742852776520408994-25px005.spad
      using old system compiler.
   DIRSUM abbreviates domain DirectSum 
------------------------------------------------------------------------
   initializing NRLIB DIRSUM for DirectSum 
   compiling into NRLIB DIRSUM 
   processing macro definition Rep ==> Record(ind: NonNegativeInteger,len: NonNegativeInteger,elt: S) 
   processing macro definition rep x ==> pretend(@(x,%),Record(ind: NonNegativeInteger,len: NonNegativeInteger,elt: S)) 
   processing macro definition per x ==> pretend(@(x,Record(ind: NonNegativeInteger,len: NonNegativeInteger,elt: S)),%) 
   compiling exported coerce : S -> %
      DIRSUM;coerce;S%;1 is replaced by VECTOR00s 
Time: 0 SEC.
compiling exported inj : (NonNegativeInteger,NonNegativeInteger,%) -> % Time: 0 SEC.
compiling exported ind : % -> NonNegativeInteger DIRSUM;ind;%Nni;3 is replaced by QVELTx0 Time: 0 SEC.
compiling exported len : % -> NonNegativeInteger DIRSUM;len;%Nni;4 is replaced by QVELTx1 Time: 0 SEC.
****** Domain: S already in scope augmenting S: (SetCategory) compiling exported = : (%,%) -> Boolean Time: 0 SEC.
compiling exported coerce : % -> OutputForm Time: 0 SEC.
****** Domain: S already in scope augmenting S: (Monoid) compiling exported One : () -> % Time: 0 SEC.
compiling exported * : (%,%) -> % Time: 0 SEC.
compiling exported ^ : (%,NonNegativeInteger) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (Group) compiling exported inv : % -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (AbelianMonoid) compiling exported Zero : () -> % Time: 0 SEC.
compiling exported + : (%,%) -> % Time: 0 SEC.
compiling exported * : (PositiveInteger,%) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (AbelianGroup) compiling exported - : % -> % Time: 0 SEC.
compiling exported - : (%,%) -> % Time: 0 SEC.
compiling exported * : (Integer,%) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (OrderedAbelianMonoidSup) compiling exported sup : (%,%) -> % Time: 0 SEC.
****** Domain: S already in scope augmenting S: (OrderedSet) compiling exported < : (%,%) -> Boolean Time: 0 SEC.
****** Domain: S already in scope augmenting S: (AbelianGroup) ****** Domain: S already in scope augmenting S: (AbelianMonoid) ****** Domain: S already in scope augmenting S: (CancellationAbelianMonoid) ****** Domain: S already in scope augmenting S: (Group) ****** Domain: S already in scope augmenting S: (Monoid) ****** Domain: S already in scope augmenting S: (OrderedAbelianMonoidSup) ****** Domain: S already in scope augmenting S: (OrderedSet) ****** Domain: S already in scope augmenting S: (SetCategory) (time taken in buildFunctor: 0)
;;; *** |DirectSum| REDEFINED
;;; *** |DirectSum| REDEFINED Time: 0 SEC.
Cumulative Statistics for Constructor DirectSum Time: 0.02 seconds
finalizing NRLIB DIRSUM Processing DirectSum for Browser database: --------constructor--------- --------(coerce (% S))--------- --->-->DirectSum((coerce (% S))): Improper first word in comments: any "any type is a 1-ary union" --------(inj (% (NonNegativeInteger) (NonNegativeInteger) %))--------- --->-->DirectSum((inj (% (NonNegativeInteger) (NonNegativeInteger) %))): Improper first word in comments: inject "inject(\\spad{i},{}\\spad{n},{}\\spad{x}) injects the \\spad{m}-CoTuple element \\spad{x} as the (\\spad{m} + \\spad{i})\\spad{-}th component of the (\\spad{n+m})-CoTuple." --------(ind ((NonNegativeInteger) %))--------- --->-->DirectSum((ind ((NonNegativeInteger) %))): Improper first word in comments: index "index(\\spad{x}) returns the component number of \\spad{x} in the CoTuple" --------(len ((NonNegativeInteger) %))--------- ; compiling file "/var/aw/var/LatexWiki/DIRSUM.NRLIB/DIRSUM.lsp" (written 01 DEC 2024 09:07:35 AM):
; wrote /var/aw/var/LatexWiki/DIRSUM.NRLIB/DIRSUM.fasl ; compilation finished in 0:00:00.032 ------------------------------------------------------------------------ DirectSum is now explicitly exposed in frame initial DirectSum will be automatically loaded when needed from /var/aw/var/LatexWiki/DIRSUM.NRLIB/DIRSUM

fricas
inj(0,2,10)

\label{eq4}{10}_{0}(4)
Type: DirectSum?(Integer)
fricas
inj(0,2,10) < inj(1,2,10)

\label{eq5} \mbox{\rm true} (5)
Type: Boolean
fricas
sup(inj(0,2,99), inj(1,2,10))$DirectSum(NNI)

\label{eq6}{10}_{1}(6)
Type: DirectSum?(NonNegativeInteger?)
fricas
a0:=inj(0,2,'a)

\label{eq7}a_{0}(7)
Type: DirectSum?(Polynomial(Integer))
fricas
len(a0)

\label{eq8}2(8)
Type: PositiveInteger?
fricas
b1:=inj(1,2,'b)

\label{eq9}b_{1}(9)
Type: DirectSum?(Polynomial(Integer))
fricas
len(b1)

\label{eq10}2(10)
Type: PositiveInteger?
fricas
a2:=inj(0,2,inj(0,2,'aa))

\label{eq11}aa_{2}(11)
Type: DirectSum?(Polynomial(Integer))
fricas
len(a2)

\label{eq12}4(12)
Type: PositiveInteger?
fricas
a3:=inj(1,2,inj(0,2,'ba))

\label{eq13}ba_{3}(13)
Type: DirectSum?(Polynomial(Integer))
fricas
len(a3)

\label{eq14}4(14)
Type: PositiveInteger?
fricas
b2:=inj(0,2,inj(1,2,'ab))

\label{eq15}ab_{2}(15)
Type: DirectSum?(Polynomial(Integer))
fricas
len(b2)

\label{eq16}4(16)
Type: PositiveInteger?
fricas
b3:=inj(1,2,inj(1,2,'bb))

\label{eq17}bb_{3}(17)
Type: DirectSum?(Polynomial(Integer))
fricas
len(b3)

\label{eq18}4(18)
Type: PositiveInteger?
fricas
inj(2,3,b3)

\label{eq19}bb_{6}(19)
Type: DirectSum?(Polynomial(Integer))




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