/usr/share/axiom-20170501/src/algebra/EQ.spad is in axiom-source 20170501-3.
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--FOR THE BENEFIT OF LIBAX0 GENERATION
++ Author: Stephen M. Watt, enhancements by Johannes Grabmeier
++ Date Created: April 1985
++ Date Last Updated: June 3, 1991; September 2, 1992
++ Description:
++ Equations as mathematical objects. All properties of the basis domain,
++ for example being an abelian group are carried over the equation domain,
++ by performing the structural operations on the left and on the
++ right hand side.
-- The interpreter translates "=" to "equation". Otherwise, it will
-- find a modemap for "=" in the domain of the arguments.
Equation(S) : SIG == CODE where
S : Type
Ex ==> OutputForm
SIG ==> Type with
"=" : (S, S) -> $
++ a=b creates an equation.
equation : (S, S) -> $
++ equation(a,b) creates an equation.
swap : $ -> $
++ swap(eq) interchanges left and right hand side of equation eq.
lhs : $ -> S
++ lhs(eqn) returns the left hand side of equation eqn.
rhs : $ -> S
++ rhs(eqn) returns the right hand side of equation eqn.
map : (S -> S, $) -> $
++ map(f,eqn) constructs a new equation by applying f to both
++ sides of eqn.
if S has InnerEvalable(Symbol,S) then
InnerEvalable(Symbol,S)
if S has SetCategory then
SetCategory
CoercibleTo Boolean
if S has Evalable(S) then
eval : ($, $) -> $
++ eval(eqn, x=f) replaces x by f in equation eqn.
eval : ($, List $) -> $
++ eval(eqn, [x1=v1, ... xn=vn])
++ replaces xi by vi in equation eqn.
if S has AbelianSemiGroup then
AbelianSemiGroup
"+" : (S, $) -> $
++ x+eqn produces a new equation by adding x to both sides of
++ equation eqn.
"+" : ($, S) -> $
++ eqn+x produces a new equation by adding x to both sides of
++ equation eqn.
if S has AbelianGroup then
AbelianGroup
leftZero : $ -> $
++ leftZero(eq) subtracts the left hand side.
rightZero : $ -> $
++ rightZero(eq) subtracts the right hand side.
"-" : (S, $) -> $
++ x-eqn produces a new equation by subtracting both sides of
++ equation eqn from x.
"-" : ($, S) -> $
++ eqn-x produces a new equation by subtracting x from both sides
++ of the equation eqn.
if S has SemiGroup then
SemiGroup
"*" : (S, $) -> $
++ x*eqn produces a new equation by multiplying both sides of
++ equation eqn by x.
"*" : ($, S) -> $
++ eqn*x produces a new equation by multiplying both sides of
++ equation eqn by x.
if S has Monoid then
Monoid
leftOne : $ -> Union($,"failed")
++ leftOne(eq) divides by the left hand side, if possible.
rightOne : $ -> Union($,"failed")
++ rightOne(eq) divides by the right hand side, if possible.
if S has Group then
Group
leftOne : $ -> Union($,"failed")
++ leftOne(eq) divides by the left hand side.
rightOne : $ -> Union($,"failed")
++ rightOne(eq) divides by the right hand side.
if S has Ring then
Ring
BiModule(S,S)
if S has CommutativeRing then
Module(S)
--Algebra(S)
if S has IntegralDomain then
factorAndSplit : $ -> List $
++ factorAndSplit(eq) make the right hand side 0 and
++ factors the new left hand side. Each factor is equated
++ to 0 and put into the resulting list without repetitions.
if S has PartialDifferentialRing(Symbol) then
PartialDifferentialRing(Symbol)
if S has Field then
VectorSpace(S)
"/" : ($, $) -> $
++ e1/e2 produces a new equation by dividing the left and right
++ hand sides of equations e1 and e2.
inv : $ -> $
++ inv(x) returns the multiplicative inverse of x.
if S has ExpressionSpace then
subst : ($, $) -> $
++ subst(eq1,eq2) substitutes eq2 into both sides of eq1
++ the lhs of eq2 should be a kernel
CODE ==> add
Rep := Record(lhs: S, rhs: S)
eq1,eq2: $
s : S
if S has IntegralDomain then
factorAndSplit eq ==
(S has factor : S -> Factored S) =>
eq0 := rightZero eq
[equation(rcf.factor,0) for rcf in factors factor lhs eq0]
[eq]
l:S = r:S == [l, r]
equation(l, r) == [l, r] -- hack! See comment above.
lhs eqn == eqn.lhs
rhs eqn == eqn.rhs
swap eqn == [rhs eqn, lhs eqn]
map(fn, eqn) == equation(fn(eqn.lhs), fn(eqn.rhs))
if S has InnerEvalable(Symbol,S) then
s:Symbol
ls:List Symbol
x:S
lx:List S
eval(eqn,s,x) == eval(eqn.lhs,s,x) = eval(eqn.rhs,s,x)
eval(eqn,ls,lx) == eval(eqn.lhs,ls,lx) = eval(eqn.rhs,ls,lx)
if S has Evalable(S) then
eval(eqn1:$, eqn2:$):$ ==
eval(eqn1.lhs, eqn2 pretend Equation S) =
eval(eqn1.rhs, eqn2 pretend Equation S)
eval(eqn1:$, leqn2:List $):$ ==
eval(eqn1.lhs, leqn2 pretend List Equation S) =
eval(eqn1.rhs, leqn2 pretend List Equation S)
if S has SetCategory then
eq1 = eq2 == (eq1.lhs = eq2.lhs)@Boolean and
(eq1.rhs = eq2.rhs)@Boolean
coerce(eqn:$):Ex == eqn.lhs::Ex = eqn.rhs::Ex
coerce(eqn:$):Boolean == eqn.lhs = eqn.rhs
if S has AbelianSemiGroup then
eq1 + eq2 == eq1.lhs + eq2.lhs = eq1.rhs + eq2.rhs
s + eq2 == [s,s] + eq2
eq1 + s == eq1 + [s,s]
if S has AbelianGroup then
- eq == (- lhs eq) = (-rhs eq)
s - eq2 == [s,s] - eq2
eq1 - s == eq1 - [s,s]
leftZero eq == 0 = rhs eq - lhs eq
rightZero eq == lhs eq - rhs eq = 0
0 == equation(0$S,0$S)
eq1 - eq2 == eq1.lhs - eq2.lhs = eq1.rhs - eq2.rhs
if S has SemiGroup then
eq1:$ * eq2:$ == eq1.lhs * eq2.lhs = eq1.rhs * eq2.rhs
l:S * eqn:$ == l * eqn.lhs = l * eqn.rhs
eqn:$ * l:S == eqn.lhs * l = eqn.rhs * l
-- We have to be a bit careful here: raising to a +ve integer is OK
-- (since it's the equivalent of repeated multiplication)
-- but other powers may cause contradictions
-- Watch what else you add here! JHD 2/Aug 1990
if S has Monoid then
1 == equation(1$S,1$S)
recip eq ==
(lh := recip lhs eq) case "failed" => "failed"
(rh := recip rhs eq) case "failed" => "failed"
[lh :: S, rh :: S]
leftOne eq ==
(re := recip lhs eq) case "failed" => "failed"
1 = rhs eq * re
rightOne eq ==
(re := recip rhs eq) case "failed" => "failed"
lhs eq * re = 1
if S has Group then
inv eq == [inv lhs eq, inv rhs eq]
leftOne eq == 1 = rhs eq * inv rhs eq
rightOne eq == lhs eq * inv rhs eq = 1
if S has Ring then
characteristic() == characteristic()$S
i:Integer * eq:$ == (i::S) * eq
if S has IntegralDomain then
factorAndSplit eq ==
(S has factor : S -> Factored S) =>
eq0 := rightZero eq
[equation(rcf.factor,0) for rcf in factors factor lhs eq0]
(S has Polynomial Integer) =>
eq0 := rightZero eq
MF ==> MultivariateFactorize(Symbol, IndexedExponents Symbol, _
Integer, Polynomial Integer)
p : Polynomial Integer := (lhs eq0) pretend Polynomial Integer
[equation((rcf.factor) pretend S,0) _
for rcf in factors factor(p)$MF]
[eq]
if S has PartialDifferentialRing(Symbol) then
differentiate(eq:$, sym:Symbol):$ ==
[differentiate(lhs eq, sym), differentiate(rhs eq, sym)]
if S has Field then
dimension() == 2 :: CardinalNumber
eq1:$ / eq2:$ == eq1.lhs / eq2.lhs = eq1.rhs / eq2.rhs
inv eq == [inv lhs eq, inv rhs eq]
if S has ExpressionSpace then
subst(eq1,eq2) ==
eq3 := eq2 pretend Equation S
[subst(lhs eq1,eq3),subst(rhs eq1,eq3)]
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