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 Topics FrontPage SandBox SandBoxInequation <-- You are here. spad)abbrev domain NEQ Inequation ++ Author: Bill Page ++ Based on: Equation by Stephen M. Watt, enhancements by Johannes Grabmeier ++ Date Created: June 2008 ++ Basic Operations: ~= ++ Related Domains: Equation ++ Also See: ++ AMS Classifications: ++ Keywords: inequation ++ Examples: ++ References: ++ Description: ++ Inequations as mathematical objects. All properties of the basis domain, ++ e.g. 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 "inequation". Otherwise, it will -- find a modemap for "~=" in the domain of the arguments. Inequation(S: Type): public == private where public ==> Type with "~=" : (S, S) -> % ++ a~=b creates an inequation. inequation: (S, S) -> % ++ inequation(a,b) creates an inequation. swap: % -> % ++ swap(neq) interchanges left and right hand side of inequation neq. lhs: % -> S ++ lhs(neq) returns the left hand side of inequation neq. rhs: % -> S ++ rhs(neq) returns the right hand side of inequation neq. map: (S -> S, %) -> % ++ map(f,neq) constructs a new inequation by applying f to both ++ sides of neq. (f must be an injection) 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: (%, Equation S) -> % ++ eval(neq, x=f) replaces x by f in inequation neq. eval: (%, List Equation S) -> % ++ eval(neq, [x1=v1, ... xn=vn]) replaces xi by vi in inequation neq. if S has AbelianSemiGroup then "+" : (S, %) -> % ++ x+neq produces a new inequation by adding x to both sides of ++ inequation neq. "+" : (%, S) -> % ++ neq+x produces a new inequation by adding x to both sides of ++ inequation neq. if S has AbelianGroup then "-": % -> % leftZero : % -> % ++ leftZero(neq) subtracts the left hand side. rightZero : % -> % ++ rightZero(neq) subtracts the right hand side. "-": (S, %) -> % ++ x-neq produces a new equation by subtracting both sides of ++ equation neq from x. "-": (%, S) -> % ++ neq-x produces a new equation by subtracting x from both sides of ++ equation neq. if S has Monoid then recip: % -> Union(%,"failed") leftOne : % -> Union(%,"failed") ++ leftOne(neq) divides by the left hand side, if possible. rightOne : % -> Union(%,"failed") ++ rightOne(neq) divides by the right hand side, if possible. if S has Group then leftOne : % -> Union(%,"failed") ++ leftOne(neq) divides by the left hand side. rightOne : % -> Union(%,"failed") ++ rightOne(neq) divides by the right hand side. if S has IntegralDomain then factorAndSplit : % -> List % ++ factorAndSplit(neq) 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 ExpressionSpace then subst: (%, Equation S) -> % ++ subst(neq1,eq2) substitutes eq2 into both sides of neq1 ++ the lhs of eq2 should be a kernel private ==> add Rep := Record(lhs: S, rhs: S) neq1,neq2, neq: % eq2: Equation S s : S if S has IntegralDomain then factorAndSplit neq == (S has factor : S -> Factored S) => neq0 := rightZero neq [inequation(rcf.factor,0) for rcf in factors factor lhs neq0] [neq] l:S ~= r:S == [l, r] inequation(l, r) == [l, r] -- hack! See comment above. lhs neq == neq.lhs rhs neq == neq.rhs swap neq == [rhs neq, lhs neq] map(fn, neq) == inequation(fn(neq.lhs), fn(neq.rhs)) if S has InnerEvalable(Symbol,S) then s:Symbol ls:List Symbol x:S lx:List S eval(neq,s,x) == eval(neq.lhs,s,x) ~= eval(neq.rhs,s,x) eval(neq,ls,lx) == eval(neq.lhs,ls,lx) ~= eval(neq.rhs,ls,lx) if S has Evalable(S) then eval(neq1 : %, eqn2 : Equation S) : % == eval(neq1.lhs, eqn2) ~= eval(neq1.rhs, eqn2) eval(neq1 : %, leqn2 : List Equation S) : % == eval(neq1.lhs, leqn2) ~= eval(neq1.rhs, leqn2) if S has SetCategory then neq1 = neq2 == (neq1.lhs = neq2.lhs)@Boolean and (neq1.rhs = neq2.rhs)@Boolean coerce(neq : %) : OutputForm == blankSeparate([neq.lhs::OutputForm, "~=", neq.rhs::OutputForm])$OutputForm coerce(neq : %) : Boolean == neq.lhs ~= neq.rhs if S has AbelianSemiGroup then s + neq2 == s+neq2.lhs ~= s+neq2.rhs neq1 + s == neq1.lhs+s ~= neq1.rhs+s if S has AbelianGroup then - neq == -neq.lhs ~= -neq.rhs s - neq2 == s-neq2.lhs ~= s-neq2.rhs neq1 - s == neq1.lhs-s ~= neq1.rhs-s leftZero neq == 0 ~= rhs neq - lhs neq rightZero neq == lhs neq - rhs neq ~= 0 if S has Monoid then recip neq == (lh := recip lhs neq) case "failed" => "failed" (rh := recip rhs neq) case "failed" => "failed" [lh :: S, rh :: S] leftOne neq == (re := recip lhs neq) case "failed" => "failed" 1 ~= rhs neq * re rightOne neq == (re := recip rhs neq) case "failed" => "failed" lhs neq * re ~= 1 if S has Group then leftOne neq == 1 ~= rhs neq * inv rhs neq rightOne neq == lhs neq * inv rhs neq ~= 1 if S has IntegralDomain then factorAndSplit neq == (S has factor : S -> Factored S) => neq0 := rightZero neq [inequation(rcf.factor,0) for rcf in factors factor lhs neq0] (S has Polynomial Integer) => neq0 := rightZero neq MF ==> MultivariateFactorize(Symbol, IndexedExponents Symbol, _ Integer, Polynomial Integer) p : Polynomial Integer := (lhs neq0) pretend Polynomial Integer [inequation((rcf.factor) pretend S,0) for rcf in factors factor(p)$MF] [neq] if S has ExpressionSpace then subst(neq1,eq2) == [subst(lhs neq1,eq2),subst(rhs neq1,eq2)] spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5578458406968568069-25px001.spad using old system compiler. NEQ abbreviates domain Inequation ------------------------------------------------------------------------ initializing NRLIB NEQ for Inequation compiling into NRLIB NEQ ****** Domain: S already in scope augmenting S: (IntegralDomain) compiling exported factorAndSplit : $-> List$ augmenting S: (SIGNATURE S factor ((Factored S) S)) Time: 0.03 SEC. compiling exported ~= : (S,S) -> $NEQ;~=;2S$;2 is replaced by CONS Time: 0 SEC. compiling exported inequation : (S,S) -> $NEQ;inequation;2S$;3 is replaced by CONS Time: 0 SEC. compiling exported lhs : $-> S NEQ;lhs;$S;4 is replaced by QCAR Time: 0 SEC. compiling exported rhs : $-> S NEQ;rhs;$S;5 is replaced by QCDR Time: 0 SEC. compiling exported swap : $->$ Time: 0 SEC. compiling exported map : (S -> S,$) ->$ Time: 0 SEC. ****** Domain: S already in scope augmenting S: (InnerEvalable (Symbol) S) compiling exported eval : ($,Symbol,S) ->$ Time: 0 SEC. compiling exported eval : ($,List Symbol,List S) ->$ Time: 0 SEC. ****** Domain: S already in scope augmenting S: (Evalable S) compiling exported eval : ($,Equation S) ->$ Time: 0 SEC. compiling exported eval : ($,List Equation S) ->$ Time: 0 SEC. ****** Domain: S already in scope augmenting S: (SetCategory) compiling exported = : ($,$) -> Boolean Time: 0.01 SEC. compiling exported coerce : $-> OutputForm ****** comp fails at level 3 with expression: ****** error in function coerce ((|Sel| (|OutputForm|) |blankSeparate|) (|construct| (|::| (|neq| |lhs|) (|OutputForm|)) | << ~= >> | (|::| (|neq| |rhs|) (|OutputForm|)))) ****** level 3 ******$x:= ~= $m:= (OutputForm)$f:= ((((|neq| # #) (S # #) (~= #) (= #) ...))) >> Apparent user error: Cannot coerce ~= of mode ~= to mode (OutputForm) spad)abbrev domain INEQ Inequality ++ Author: Bill Page ++ Based on: Equation by Stephen M. Watt, enhancements by Johannes Grabmeier ++ Date Created: June 2008 ++ Basic Operations: < ++ Related Domains: Equation Inequation ++ Also See: ++ AMS Classifications: ++ Keywords: inequality ++ Examples: ++ References: ++ Description: ++ Inequalities as mathematical objects. All properties of the basis domain, ++ e.g. 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 "x < y" to "inequality(x,y)", -- "x > y" to "inequality(y,x)", "x <= y" to "not inequality(y,x)" -- and "x >= y" to "not inequality(x,y)". Inequality(S: Type): public == private where public ==> Type with "<": (S, S) -> % ++ a < b creates an inequality ">=": (S, S) -> % ++ a >= b creates opposite inequality (not a % ++ a < b creates an inequality ge: (S, S) -> % ++ a >= b creates opposite inequality (not a % ++ equality(a,b) creates an inequality. lhs: % -> S ++ lhs(leq) returns the left hand side of inequality leq. rhs: % -> S ++ rhs(leq) returns the right hand side of inequality leq. cmp: % -> String ++ cmp(leq) returns the type of inequality "<", ">=" if S has InnerEvalable(Symbol,S) then InnerEvalable(Symbol,S) if S has OrderedSet then SetCategory CoercibleTo Boolean if S has Evalable(S) then eval: (%, Equation S) -> % ++ eval(leq, x=f) replaces x by f in inequality leq. eval: (%, List Equation S) -> % ++ eval(leq, [x1=v1, ... xn=vn]) replaces xi by vi in inequality leq. coerce : Union(%, Equation S) -> OutputForm if S has AbelianSemiGroup then "+": (S, %) -> % ++ x+leq produces a new inequality by adding x to both sides of ++ inequality leq. "+": (%, S) -> % ++ leq+x produces a new inequality by adding x to both sides of ++ inequality leq. if S has AbelianGroup then "-": % -> % leftZero : % -> % ++ leftZero(leq) subtracts the left hand side. rightZero : % -> % ++ rightZero(leq) subtracts the right hand side. "-": (S, %) -> % ++ x-leq produces a new inquality by subtracting both sides of ++ inequality leq from x. "-": (%, S) -> % ++ leq-x produces a new inequality by subtracting x from both sides of ++ inequality leq. if S has ExpressionSpace then subst: (%, Equation S) -> % ++ subst(leq,eq2) substitutes eq2 into both sides of leq ++ the lhs of eq2 should be a kernel private ==> add Rep := Record(lhs : S, cmp : String, rhs : S) leq1, leq2, leq: % eq2 : Equation S s : S lt(l:S, r:S) == [l, "<", r] l:S < r:S == lt(l,r) inequality(l, r) == lt(l,r) -- hack! See comment above. ge(l:S, r:S) == [l, ">=", r] l:S >= r:S == ge(l,r) lhs leq == leq.lhs rhs leq == leq.rhs cmp leq == leq.cmp if S has InnerEvalable(Symbol,S) then s:Symbol ls:List Symbol x:S lx:List S eval(leq,s,x) == eval(leq.lhs,s,x) < eval(leq.rhs,s,x) eval(leq,ls,lx) == eval(leq.lhs,ls,lx) < eval(leq.rhs,ls,lx) if S has Evalable(S) then eval(leq : %, eqn2 : Equation S) : % == eval(leq.lhs, eqn2) < eval(leq.rhs, eqn2) eval(leq : %, eqn2 : List Equation S) : % == eval(leq.lhs, eqn2) < eval(leq.rhs, eqn2) if S has OrderedSet then leq1 = leq2 == (leq1.lhs = leq2.lhs)@Boolean and (leq1.rhs = leq2.rhs)@Boolean coerce(leq : %) : OutputForm == leq.cmp="<" => blankSeparate([leq.lhs::OutputForm, "<", leq.rhs::OutputForm])$OutputForm blankSeparate([leq.lhs::OutputForm, ">=", leq.rhs::OutputForm])$OutputForm coerce(leq : %) : Boolean == leq.cmp="<" => leq.lhs < leq.rhs (leq.lhs >= leq.rhs)$S if S has AbelianSemiGroup then s + leq2 == s+leq2.lhs < s+leq2.rhs leq1 + s == leq1.lhs+s < leq1.rhs+s if S has AbelianGroup then - leq == (-rhs leq) < (- lhs leq) leftZero leq == 0 < rhs leq - lhs leq rightZero leq == lhs leq - rhs leq < 0 s - leq2 == s-leq2.rhs < s-leq2.lhs leq1 - s == leq1.lhs-s < leq1.rhs-s if S has ExpressionSpace then subst(leq1,eq2) == [subst(lhs leq1,eq2),leq1.cmp,subst(rhs leq1,eq2)] spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8281229116200575938-25px002.spad using old system compiler. INEQ abbreviates domain Inequality ------------------------------------------------------------------------ initializing NRLIB INEQ for Inequality compiling into NRLIB INEQ Local variable Rep type redefined: (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: lhs S) (: cmp (String)) (: rhs S)) S (String) S)) (SIGNATURE ~= ((Boolean) (Record (: lhs S) (: cmp (String)) (: rhs S)) (Record (: lhs S) (: cmp (String)) (: rhs S)))) (SIGNATURE coerce ((OutputForm) (Record (: lhs S) (: cmp (String)) (: rhs S)))) (SIGNATURE elt (S (Record (: lhs S) (: cmp (String)) (: rhs S)) lhs)) (SIGNATURE elt ((String) (Record (: lhs S) (: cmp (String)) (: rhs S)) cmp)) (SIGNATURE elt (S (Record (: lhs S) (: cmp (String)) (: rhs S)) rhs)) (SIGNATURE setelt! (S (Record (: lhs S) (: cmp (String)) (: rhs S)) lhs S)) (SIGNATURE setelt! ((String) (Record (: lhs S) (: cmp (String)) (: rhs S)) cmp (String))) (SIGNATURE setelt! (S (Record (: lhs S) (: cmp (String)) (: rhs S)) rhs S)) (SIGNATURE copy ((Record (: lhs S) (: cmp (String)) (: rhs S)) (Record (: lhs S) (: cmp (String)) (: rhs S)))))) to (Join (SetCategory) (CATEGORY domain (SIGNATURE construct ((Record (: lhs S) (: rhs S)) S S)) (SIGNATURE ~= ((Boolean) (Record (: lhs S) (: rhs S)) (Record (: lhs S) (: rhs S)))) (SIGNATURE coerce ((OutputForm) (Record (: lhs S) (: rhs S)))) (SIGNATURE elt (S (Record (: lhs S) (: rhs S)) lhs)) (SIGNATURE elt (S (Record (: lhs S) (: rhs S)) rhs)) (SIGNATURE setelt! (S (Record (: lhs S) (: rhs S)) lhs S)) (SIGNATURE setelt! (S (Record (: lhs S) (: rhs S)) rhs S)) (SIGNATURE copy ((Record (: lhs S) (: rhs S)) (Record (: lhs S) (: rhs S)))))) compiling exported lt : (S,S) ->$ INEQ;lt;2S$;1 is replaced by VECTORl$ Time: 0 SEC. compiling exported inequality : (S,S) -> $Time: 0 SEC. compiling exported ge : (S,S) ->$ INEQ;ge;2S$;4 is replaced by VECTORl>=r Time: 0 SEC. compiling exported >= : (S,S) ->$ Time: 0 SEC. compiling exported lhs : $-> S INEQ;lhs;$S;6 is replaced by QVELTleq0 Time: 0 SEC. compiling exported rhs : $-> S INEQ;rhs;$S;7 is replaced by QVELTleq2 Time: 0 SEC. compiling exported cmp : $-> String INEQ;cmp;$S;8 is replaced by QVELTleq1 Time: 0 SEC. ****** Domain: S already in scope augmenting S: (InnerEvalable (Symbol) S) compiling exported eval : ($,Symbol,S) ->$ Time: 0 SEC. compiling exported eval : ($,List Symbol,List S) ->$ Time: 0.01 SEC. ****** Domain: S already in scope augmenting S: (Evalable S) compiling exported eval : ($,Equation S) ->$ Time: 0 SEC. compiling exported eval : ($,List Equation S) ->$ Time: 0 SEC. ****** Domain: S already in scope augmenting S: (OrderedSet) compiling exported = : ($,$) -> Boolean Time: 0 SEC. compiling exported coerce : $-> OutputForm ****** comp fails at level 6 with expression: ****** error in function coerce (SEQ (|:=| (|:| #1=#:G704 (|Boolean|)) (= (|leq| |cmp|) "<")) (|exit| 1 (IF #1# ((|Sel| (|OutputForm|) |blankSeparate|) (|construct| (|::| (|leq| |lhs|) (|OutputForm|)) | << < >> | (|::| (|leq| |rhs|) (|OutputForm|)))) ((|Sel| (|OutputForm|) |blankSeparate|) (|construct| (|::| (|leq| |lhs|) (|OutputForm|)) ">=" (|::| (|leq| |rhs|) (|OutputForm|))))))) ****** level 6 ******$x:= < $m:= (OutputForm)$f:= ((((#:G704 # #) (|leq| # #) (S # #) (~= #) ...))) >> Apparent user error: Cannot coerce < of mode < to mode (OutputForm) spad)abbrev package REL Relations ++ Author: Bill Page ++ Date Created: June 2008 ++ Basic Operations: not ++ Related Domains: Equation Inequation Inequality ++ Also See: ++ AMS Classifications: ++ Keywords: negation of relations ++ Examples: ++ References: ++ Description: ++ The Relations package provides the 'not' operation on ++ Inequalities, Inequations and Equations. -- The interpreter translates "x < y" to "inequality(x,y)", and -- normalizes "x > y" to "inequality(y,x)", -- "x <= y" to "not inequality(y,x)" -- and "x >= y" to "not inequality(x,y)". Relations(S: Type): public == private where public ==> Type with _not: Equation(S) -> Inequation(S) _not: Inequation(S) -> Equation(S) _not: Inequality(S) -> Inequality(S) private ==> add _not(leq:Inequality(S)):Inequality(S) == cmp(leq)="<" => ge(lhs(leq),rhs(leq))$Inequality(S) lt(lhs(leq),rhs(leq))$Inequality(S) _not(neq:Inequation(S)):Equation(S) == equation(lhs(neq),rhs(neq)) _not(eq:Equation(S)):Inequation(S) == inequation(lhs(eq),rhs(eq)) spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/2613099951730133301-25px003.spad using old system compiler. REL abbreviates package Relations ------------------------------------------------------------------------ initializing NRLIB REL for Relations compiling into NRLIB REL compiling exported not : Inequality S -> Inequality S Semantic Errors:  Inequality is not a known type Warnings:  not: cmp has no value ****** comp fails at level 4 with expression: ****** error in function not (SEQ (|:=| (|:| #1=#:G726 (|Boolean|)) (= | << | (|cmp| |leq|) | >> | "<")) (|exit| 1 (IF #1# ((|Sel| (|Inequality| S) |ge|) (|lhs| |leq|) (|rhs| |leq|)) ((|Sel| (|Inequality| S) |lt|) (|lhs| |leq|) (|rhs| |leq|))))) ****** level 4 ****** $x:= (cmp leq)$m:= $EmptyMode$f:= ((((#:G726 #) (|leq| # #) (|$DomainsInScope| # # #) (|not| #) ...))) >> Apparent user error: NoValueMode is an unknown mode It works but the LaTeX output does not display fricas)set output tex on  fricas)set output algebra on inequation(a,b) There are no library operations named inequation Use HyperDoc Browse or issue )what op inequation to learn if there is any operation containing " inequation " in its name. Cannot find a definition or applicable library operation named inequation with argument type(s) Variable(a) Variable(b) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. fricas)set output tex on  fricas)set output algebra on w1:=inequality(a,b) There are no library operations named inequality Use HyperDoc Browse or issue )what op inequality to learn if there is any operation containing " inequality " in its name. Cannot find a definition or applicable library operation named inequality with argument type(s) Variable(a) Variable(b) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. test --yixin.cao, Wed, 09 Jul 2008 19:26:24 -0700 replyfricas2+%i < 2*%i +1 There are 3 exposed and 1 unexposed library operations named < having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op < to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named < with argument type(s) Complex(Integer) Complex(Integer) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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