solvex 概要
solvex は冪乗根関数で表された方程式,不等式の系を指定された変数についての1次式を原始式とする系に変換する関数です.
例えば,3数の相加平均と相乗平均とが一致する条件を得ようとしても,qex では...
(%i1) qex(a>=0 %and b>=0 %and c>=0 %and (a*b*c)^(1/3)=(a+b+c)/3); Evaluation took 1.0500 seconds (1.4300 elapsed) using 117.327 MB. (%o1) (a >= 0) %and (b >= 0) %and (c >= 0) %and (c^3+3*b*c^2+3*a*c^2+3*b^2*c-21*a*b*c+3*a^2*c+b^3+3*a*b^2+3*a^2*b+a^3 = 0)
一方,solvex は
(%i2) solvex(a>=0 %and b>=0 %and c>=0 %and (a*b*c)^(1/3)=(a+b+c)/3); Evaluation took 8.3100 seconds (16.3700 elapsed) using 761.807 MB. (%o2) [[0 <= a,b = a,c = a]]
と答えます.
solvex の書式は solvex(formula,var1,...,varN)であり,formula を var1,...,varN のうち listofvars(formula) の要素について順に解こうとし,それら以外の変数は定数と見做します.
ただし,solvex(formula) は listofvars(formula) の全ての要素について順に解こうとし,それら以外の変数は定数と見做します.
出力は,bracket dnf,つまり,2重のリストであり,外側のリストのヘッドは %or,内側のリストのヘッドは %and を表わしています.
%or,%and の表示に変換するには,ora コマンドで
(%i3) ora(%); Evaluation took 0.0000 seconds (0.0100 elapsed) using 817.102 KB. (%o3) (0 <= a) %and (b = a) %and (c = a)
元に戻すには,bra コマンドで
(%i4) bra(%); Evaluation took 0.0200 seconds (0.0200 elapsed) using 1.058 MB. (%o4) [[0 <= a,b = a,c = a]]
といった具合です.
また,この書式による入力も可能で
(%i35) solvex([[x^2=a,x>-1]],x); Evaluation took 8.4900 seconds (18.3800 elapsed) using 469.256 MB. (%o35) [[a < 1,x = -sqrt(a)],[x = sqrt(a)]]
と出来,ご覧のように冪乗根を含んだ出力に対応しています.
qex 概要
以下は Maxima 5.39.0 using Lisp CMU Common Lisp 21b (21B Unicode) 上での実行です.
qex は qe コマンドを有理数冪入力対応に拡張したもの(qe extended version)であり,qex([],formula) は qex(formula) のように省略可能です.また,出力は qe と同じく整数係数の多変数多項式による表示です.
平方根.
(%i1) qex(y=sqrt(x)); Evaluation took 0.4700 seconds (0.8600 elapsed) using 28.694 MB. (%o1) (y >= 0) %and (x-y^2 = 0)
複雑な式.
(%i2) qex(y=2^(2/3)/sqrt(2-2^(1/3))); Evaluation took 0.6300 seconds (1.0100 elapsed) using 35.511 MB. (%o2) (y > 0) %and (3*y^6-6*y^4-12*y^2-8 = 0)
不等式.
(%i3) qex(y<sqrt(x)); Evaluation took 0.5900 seconds (0.9900 elapsed) using 40.485 MB. (%o3) (x >= 0) %and ((y < 0) %or (y^2-x < 0))
否定.
(%i4) qex(y#sqrt(x)); Evaluation took 0.4400 seconds (0.8400 elapsed) using 25.491 MB. (%o4) (y < 0) %or (x-y^2 # 0)
絶対値.
(%i5) qex(y=abs(x)); Evaluation took 0.7100 seconds (1.2000 elapsed) using 52.369 MB. (%o5) (y-x >= 0) %and (y+x >= 0) %and ((y-x = 0) %or (y+x = 0))
不等式.
(%i6) qex(x<=abs(x)); Evaluation took 0.4700 seconds (0.8500 elapsed) using 32.271 MB. (%o6) true
元が2つの集合の最大値.
(%i7) qex(x=max2(a,b)); Evaluation took 1.1800 seconds (1.7000 elapsed) using 88.835 MB. (%o7) ((b-x = 0) %or (x-a = 0)) %and (b-x <= 0) %and (x-a >= 0)
3つの場合.
(%i8) qex(x=min2(min2(a,b),c)); Evaluation took 2.4100 seconds (3.5600 elapsed) using 249.620 MB. (%o8) ((b-x = 0) %or (c-x = 0) %or (x-a = 0)) %and (b-x >= 0) %and (c-x >= 0) %and (x-a <= 0)
値域.
(%i9) qex([[E,x]],y=2/sqrt(x)+abs(x)); Evaluation took 1.0100 seconds (1.7000 elapsed) using 69.622 MB. (%o9) y-3 >= 0
値域.
(%i10) qex([[E,x]],y=abs(x)+abs(x-1)+abs(x-2)+abs(x-3)); Evaluation took 3.1500 seconds (25.8000 elapsed) using 237.202 MB. (%o10) y-4 >= 0
値域.
(%i11) qex([[E,x]],y=max2(x,min2(2-x,3-x))); Evaluation took 1.4100 seconds (2.0700 elapsed) using 89.527 MB. (%o11) y-1 >= 0
不等式.
(%i12) qex(abs(a+b)<=abs(a)+abs(b)); Evaluation took 0.9400 seconds (1.4900 elapsed) using 80.652 MB. (%o12) true
不等式.
(%i13) qex(min2(a,b)<=max2(x,a-x)+min2(x,b-x)); Evaluation took 1.3800 seconds (1.7700 elapsed) using 134.412 MB. (%o13) true
相加平均と相乗平均との大小関係.
(%i14) qex([[A,a],[A,b]],(a>=0 %and b>=0) %implies sqrt(a*b)<=(a+b)/n); Evaluation took 2.5200 seconds (3.1200 elapsed) using 252.836 MB. (%o14) (n-2 <= 0) %and (n > 0)
等式となる条件.
(%i15) qex("%and"(a>=0,b>=0,sqrt(a*b)=(a+b)/2)); Evaluation took 0.6700 seconds (1.1000 elapsed) using 57.813 MB. (%o15) (a >= 0) %and (b-a = 0)
一様連続性.
(%i16) qex([[A,e]],(0<e) %implies (qex([[E,d]],(0<d) %and (qex([[A,x],[A,y]],(abs(x-y)<d) %implies (abs(x^(2/3)-y^(2/3))<e)))))); Evaluation took 6.2100 seconds (24.2100 elapsed) using 586.517 MB. (%o16) true
拡張された量化子.
(%i17) qex([[X2,x]],abs(abs(abs(x^2-1)-a)-1)=x+1); Evaluation took 4.8000 seconds (10.2200 elapsed) using 333.388 MB. (%o17) (((a-3 < 0) %and (4*a-5 > 0)) %or ((a-1 < 0) %and (4*a+3 > 0)) %or (a+1 < 0) %or (4*a-13 > 0)) %and (a-3 # 0) %and (a-1 # 0) %and (a+1 # 0) %and (a+3 > 0) %and (4*a-13 # 0) %and (4*a-5 # 0) %and (4*a+3 # 0)
簡約.
(%i18) nnss(%); Evaluation took 4.8300 seconds (8.3800 elapsed) using 186.479 MB. (%o18) ((a-3 < 0) %and (4*a-5 > 0)) %or ((a-1 < 0) %and (4*a+3 > 0)) %or (4*a-13 > 0) %or ((a+1 < 0) %and (a+3 > 0))
整理.(13/4 < a) が最初に来るのは %or が勝手に sort するためです.
(%i19) l2r(%); Evaluation took 0.1400 seconds (0.1400 elapsed) using 4.450 MB. (%o19) (13/4 < a) %or ((-3 < a) %and (a < -1)) %or ((-3/4 < a) %and (a < 1)) %or ((5/4 < a) %and (a < 3))
nns シリーズ更新
新規の関数
・qex
qeの拡張版,有理数冪の入力に対応.
・solvex
多変数の方程式,不等式を原始式とする quantifier free な論理式の solver,有理数冪の入力,出力に対応.
/* 17.01.09.Mon. 07:50:51 */ load(qepmax)$ showtime:on$ display2d:false$ prt(x):=print(x)$ prt(x):=null$ /*********/ /* tools */ /*********/ /* compare length function */ comparelength(f,g):=length(f)<=length(g)$ comparelength2(f,g):=length(g)<=length(f)$ /* cnf dnf */ infix(Or,60,40)$ infix(And,60,40)$ matchdeclare([aa,bb,cc,dd,ee,ff,gg,aa1,bb1,cc1,dd1,ee1,ff1,gg1],true)$ defrule(ru05r, (aa) Or ((bb) And (cc)), ((aa) Or (bb)) And ((aa) Or (cc)))$ defrule(ru08r, ((bb) And (cc)) Or (aa), ((aa) Or (bb)) And ((aa) Or (cc)))$ defrule(ru05a, (aa) And ((bb) Or (cc)), ((aa) And (bb)) Or ((aa) And (cc)))$ defrule(ru08a, ((bb) Or (cc)) And (aa), ((aa) And (bb)) Or ((aa) And (cc)))$ defrule(ru00r, (aa) Or (bb), (aa) %or (bb))$ defrule(ru00a, (aa) And (bb), (aa) %and (bb))$ orand2orand(f):=block([g],g:f, if atom(g) then return(g), if op(g)="%or" then g:xreduce("Or",args(g)), if op(g)="%and" then g:xreduce("And",args(g)),g)$ cnf(f):=applyb2(applyb2(scanmap(orand2orand,f),ru05r,ru08r),ru00r,ru00a)$ dnf(f):=applyb2(applyb2(scanmap(orand2orand,f),ru05a,ru08a),ru00r,ru00a)$ cnfb(f):=applyb2(applyb2(scanmap(orand2orand,f,bottomup),ru05r,ru08r),ru00r,ru00a)$ dnfb(f):=applyb2(applyb2(scanmap(orand2orand,f,bottomup),ru05a,ru08a),ru00r,ru00a)$ /* necessary condition */ defrule(ru00n, (aa) # (bb), "%or"((aa)<(bb),(bb)<(aa)))$ defrule(ru00n, (aa) # (bb), "%not"((aa)=(bb)))$ deli2(f):=(applyb2(f,ru001,ru002,ru003,ru004))$ ncondeq(f):=block([exs,L,exk], if is(ncondtype=off) then return(f), econd:[],L:true, prt("necessary eq. condition"), exs:fullmap(lambda([x],[E,x]), sort(rest(full_listify(powerset(setify(listofvars(f))))),comparelength2)), for exk in exs do (if is(ncondtype=full) then q:qe(exk,(f)) else q:deli2(qe(exk,(f))), prt([L,q]), if is(not(qe([],(L)%implies(q))=true)) then L:(L)%and(q)), prt(L), if is(not(ncondsimp=off)) then L:nnscandc(L), /* if is(qe([],(L)%implies(f))=true) and (tdeg(L)=1) then (prt("equivalent condition"),econd:[dnf(L)],return(dnf(L))) else */ return((L)%and(f)) )$ ncond(f):=block([exs,L,exk],L:true, prt("necessary condition"), exs:fullmap(lambda([x],[E,x]), sort(rest(full_listify(powerset(setify(listofvars(f))))),comparelength2)), for exk in exs do (q:qe(exk,f),prt([L,q]), if not(qe([],(L)%implies(q))=true) then L:(L)%and(nnscan(q))), prt(L), return((L)%and(f)) )$ /***************/ /* nns section */ /***************/ /* non-nonsense */ nns(f):=block([F,L,M], if atom(f) then return(f) elseif op(f)="%and" then (F:sort(args(f),comparelength2),L:setify(F), for g in F do (M:disjoin(g,L),prt(M), if p2emt((substpart("%and",M,0))%implies(g))=true then L:M), return(substpart("%and",L,0))) elseif op(f)="%or" then (F:sort(args(f),comparelength2),L:setify(F), for g in F do (M:disjoin(g,L),prt(M), if p2emt((g)%implies(substpart("%or",M,0)))=true then L:M), return(substpart("%or",L,0))) else f)$ /* non-nonsense scan */ nnscan(f):=(prt("nnscanning..."),scanmap(nns,f))$ nnscanb(f):=(prt("bottomup nnscanning..."),scanmap(nns,f,bottomup))$ /* non-nonsense scan with cnf */ nnscanc(f):=(prt("cnf nnscanning..."),scanmap(nns,cnf(f)))$ nnscancb(f):=(prt("cnf bottomup nnscanning..."),scanmap(nns,cnf(f),bottomup))$ /* non-nonsense scan with dnf */ nnscand(f):=(prt("dnf nnscanning..."),scanmap(nns,dnf(f)))$ nnscandb(f):=(prt("dnf bottomup nnscanning..."),scanmap(nns,dnf(f),bottomup))$ nnscand2(f):=(prt("dnf nnscanning..."),nns(map(nns,dnf(f))))$ /* non-nonsense scan with cnf,dnf */ nnscancd(f):=nnscanc(nnscand(f))$ nnscancdb(f):=nnscancb(nnscandb(f))$ nnscandc(f):=nnscand(nnscanc(f))$ nnscandcb(f):=nnscandb(nnscancb(f))$ /* non-nonsense set */ nnss(f):=block([f1,f2,f3,q,nns4], if mode=d then nns4(g):=nnscand(g) else nns4(g):=nns(g), prt("pre-simplified formula"),f1: dnf (qe ([],f) ) , if atom(f1) then return(f1), prt(length(f1)),prt(f1),prt("necessary condition"), if op(f1)="%or" then (f2:[], for k:1 thru length(f1) do (q:ncond(part(f1,k)),prt(k),prt(q),f2:append(f2,[q])), f2:substpart("%or",f2,0)) else (f2:dnf(ncond(f1)),prt(f2)), prt("simplifing..."), if op(f2)="%or" then (f3:[], for k:1 thru length(f2) do (q:qe([],part(f2,k)),prt(k),prt(q),f3:append(f3,[q])), prt("nnscanning..."),f3:substpart("%or",f3,0),prt(f3),return(nns4(f3))) else q:qe([],f2),prt(q),prt("nnscanning..."),return(nns4(q)) )$ /* nnss with double negation */ nnssdn(f):=qe([],%not(nnss(%not(f))))$ /**************/ /* p2t section */ /**************/ defrule(powertk1, (aa)^(bb), makeprtk(aa,num(bb),denom(bb)))$ defrule(powertk2, abs(aa), makeprtk(aa^2,1,2))$ domain:complex$ defrule(powertk2, abs(aa), (aa^2)^(1/2))$ defrule(powertk3, max2(aa,bb), (aa+bb+abs(aa-bb))/2)$ defrule(powertk4, min2(aa,bb), (aa+bb-abs(aa-bb))/2)$ p2e0(f):=block([L,makeprtk,g,q,rt,gg],L:[], makeprtk(x,a,b):=block([rtk], if b=1 then return(x^a) else (rtk:concat(rt,length(L)+1), if oddp(b) then L:append(L,[(rtk^b=x^a)]) else L:append(L,[(rtk^b=x^a)%and(rtk>=0)]), return(rtk))), g:applyb1(f,powertk4,powertk3,powertk2,powertk1), q:makelist([E,concat(rt,k)],k,length(L)), if L=[] then return(f) else (gg:(substpart("%and",L,0))%and(g), ev(qe(q,gg),noeval)) )$ /* 1-1 type */ p2e1(f):=block([L,M,makeprtk,g,q,rt,gg],M:[],L:[], makeprtk(x,a,b):=block([rtk], if b=1 then return(x^a) elseif is(assoc(x^(a/b),M)#false) then return(concat(rt,assoc(x^(a/b),M))) else (M:append(M,[[x^(a/b),length(L)+1]]),rtk:concat(rt,length(L)+1), if oddp(b) then L:append(L,[(rtk^b=x^a)]) else L:append(L,[(rtk^b=x^a)%and(rtk>=0)]), return(rtk))), g:applyb1(f,powertk4,powertk3,powertk2,powertk1), q:makelist([E,concat(rt,k)],k,length(L)), if L=[] then return(f) else (gg:(substpart("%and",L,0))%and(g), ev(qe(q,gg),noeval)) )$ /* base64 ver */ p2e64(f):=block([sL,L,makeprtk,g,q,gg,rt],sL:[],L:[], makeprtk(x,a,b):=block([strg,rtk], if b=1 then return(x^a) else ( strg:ssubst("equal","=",base64(string(x^(a/b)))), strg:ssubst("plus","+",strg), strg:ssubst("slash","/",strg), rtk:eval_string(strg), if not(member(rtk,sL)) then sL:append(sL,[rtk]), if oddp(b) then L:append(L,[(rtk^b=x^a)]) else L:append(L,[(rtk^b=x^a)%and(rtk>=0)]), return(rtk))), g:applyb1(f,powertk4,powertk3,powertk2,powertk1), q:makelist([E,s],s,sL),/* print([g,sL,q]), if is(is(p2etime>0)=true) then p2etime:p2etime+1 else p2etime:1, */ if L=[] then return(f) else (gg:(substpart("%and",L,0))%and(g),prt(sL), subst(makelist(part(sL,k)=concat(rt,k),k,length(sL)), ev(qe(q,gg),noeval)) ) )$ p2e(f):=if is(p2etype=64) then p2e64(f) elseif is(p2etype=0) then p2e0(f) else p2e1(f)$ /* p2e map */ p2em(f):=block([p2etlocal], if atom(f) then return(f) else p2etlocal(g):=block([], if atom(g) then g else if atom(g) then g elseif op(g)="#" then "%not"(p2e(substpart("=",g,0))) elseif (op(g)="=" or op(g)="<" or op(g)=">" or op(g)="<=" or op(g)=">=") then p2e(g) else g), scanmap(p2etlocal,f) /*qe([],scanmap(p2etlocal,f))*/ )$ /* p2e map tarski */ p2emt(f):=qe([],ev(p2em(f),eval))$ p2t(f):=p2emt(f)$ /*******************/ /* nnsolve section */ /*******************/ /* nnsolvexx */ algexact:true$ /* is imaginary */ imgp(x):=not(is(x>=0) or is(x<0))$ /* del formulas containnig imginary unit */ delimg(f):=block([sel3], if atom(f) then return(f), sel3(g):=if atom(g) then g elseif (op(g)="#" or op(g)="=" or op(g)="<" or op(g)=">" or op(g)="<=" or op(g)=">=") and (ratsimp(imagpart(lhs(g)))=0) and (ratsimp(imagpart(rhs(g)))=0) then ratsimp(realpart(g)) elseif (op(g)="#" or op(g)="=" or op(g)="<" or op(g)=">" or op(g)="<=" or op(g)=">=") and (imgp(lhs(g))=true or imgp(rhs(g))=true) then false else g, scanmap(sel3,f))$ Lintersection(l,m):=listify(intersection(setify(l),setify(m)))$ Lminus(l,m):=listify(setdifference(setify(l),setify(m)))$ /* nnsolve: solver for cnj. of eq. */ /* new ver */ delp(L):=block([LL,ek],LL:L, for ek in L do (if member(rhs(ek),%rnum_list) then LL:subst(rhs(ek)=lhs(ek),LL)), LL )$ /* new ver */ nnsolve(f,[vv]):=block([fs,v,g,gg], linsolvewarn:false, if atom(f) then return(f) elseif op(f)="%and" then fs:args(f) elseif op(f)="=" then fs:[f] else return(f), prt("nnsolving..."),prt(fs),%rnum_list:[], v:Lintersection(vv,listofvars(fs)), if v=[] then v:listofvars(fs), g:delimg(algsys(fs,v)),prt([v,g]), if g=[] then (v:listofvars(fs),g:delimg(algsys(fs,v)),prt([v,g])), if g=[] or setify(g)={[false]} then return(false), solvar:v, return(substpart("%or",map(lambda([l],substpart("%and",l,0)),map(delp,g)),0)) )$ defrule(ru000, (aa) # (bb), true)$ defrule(ru001, (aa) > (bb), true)$ defrule(ru002, (aa) >= (bb), true)$ defrule(ru003, (aa) < (bb), true)$ defrule(ru004, (aa) <= (bb), true)$ defrule(ru006, (aa) = (bb), true)$ /* delete ineq. if f has no eq. then return true */ deli1(f):=(applyb2(f,ru000,ru001,ru002,ru003,ru004))$ deli(f):=block([g], if atom(f) then return(f), g:dnf(f), if op(g)="%or" then substpart("%or",delete(true,map(deli1,args(g))),0) else deli1(g))$ /* delete eq. if f has no ineq. then return true */ dele1(f):=(applyb2(f,ru006))$ dele(f):=block([g], if atom(f) then return(f), g:dnf(f), if op(g)="%or" then substpart("%or",delete(true,map(dele1,args(g))),0) else dele1(g))$ mapply(s,t):=if atom(t) then s(t) elseif op(t)="%or" then map(s,t) else s(t)$ /* solution selector for eq.s and ineq.s */ defrule(ru007, (aa) = (bb), if Lintersection(v,listofvars(expand(aa-bb)))=[] then true else (aa)=(bb))$ defrule(ru008, (aa) = (bb), if Lintersection(v,listofvars(expand(aa-bb)))=[] then (aa)=(bb) else true)$ delec1(f):=(applyb1(f,ru007))$ delec(f):=block([g], if atom(f) then return(f), g:dnf(f), if op(g)="%or" then substpart("%or",delete(true,map(delec1,args(g))),0) else delec1(g))$ delev1(f):=(applyb1(f,ru008))$ delev(f):=block([g], if atom(f) then return(f), g:dnf(f), if op(g)="%or" then substpart("%or",delete(true,map(delev1,args(g))),0) else delev1(g))$ /* new ver for ineq. and conj. of eq. */ solsel(f,[v]):=block([eqss,eqs,ins,sel2,ng], eqss:deli(f), if v=[] or length(listofvars(eqss))=1 then v:listofvars(eqss), eqs:delec(eqss), prt(eqs), if atom(eqs) then return(f), if op(eqs)="%or" then (prt("I can't solve disj. of eq."),return(f)), ins:(dele(f))%and(delev(eqss)), sel2(g):=block([gg], /*(g)%and(qe(map(lambda([x],[E,x]),listofvars(g)),p2et((g)%and(ins))))*/ if /* ins=true then g elseif */ solvar=[] or atom(g) then (g)%and(ins) else (solvar:map(lhs,(if op(g)="%and" then args(g) else [g])), gg:(g)%and(qe(map(lambda([x],[E,x]),solvar),p2t((g)%and(ins)))), ng:qe([],"%not"(qe(map(lambda([x],[E,x]),solvar),p2t(g)))), prt((ng)%and(f)), insolsel:true, gg:(gg)%or(ora(apply(nnsolvexx,flatten([(ng)%and(f),solvar])))), insolsel:false, prt("sel2 returned..."),prt(gg),gg ) ), qq:apply(nnsolve,flatten([eqs,v])),prt(qq), if ins=true then prt("no ins"), qq:mapply(sel2,qq), qq /* for nnsolvexx */ /* nnscan(qq) */ /* for stand alone */ )$ /* nnsolvexx: solver for semi-algebraic system */ nnsolvexx0(f,[v]):=block([g,gg], prt("pre-simplified formula"), g:dnf(qe([],p2t(f))),prt(g), if atom(g) then return(g) else (prt("necessary condition"),g:dnf(ncond(g)),prt(length(g)),prt(g)), if op(g)="%or" then (prt("simplifing..."),gg:dnf(map(ncond,g)),prt(length(gg)),prt(gg),nnscan(map(lambda([t],apply(solsel,flatten([t,v]))),gg))) else nnscan(apply(solsel,flatten([g,v]))) )$ /* total deg */ tdeg0(f):=block([tdegx], hipow(expand(subst(map(lambda([s],s=tdegx),listofvars(f)),f)),tdegx) )$ tdeg(f):=apply(max,map(tdeg0,args(expand( lhs(f)-rhs(f) ))))$ /* new ver */ nnsolvexx(f,[vv]):=block([g,v,gg,q], if not(insolsel=true) then thisin:f, if is(presimp=off) then prt("pre-simplify:off") elseif is(presimp=l) then prt("pre-simplify:lineq with nnscan") else prt("pre-simplify:p2t->qe"), if is(ncondtype=off) then prt("ncond-type:off") elseif is(ncondtype=full) then prt("ncond-type:full formula") else prt("ncond-type:eq. only"), if is(ncondsimp=off) then prt("ncond-post-simplify:off") else prt("ncond-post-simplify:nnscandc"), if is(sineqsimp=nn) then prt("sineq-simplify:nnscan") else prt("sineq-simplify:lineq"), if is(postsimp=off) then prt("post-simplify:off") else prt("post-simplify:nnscanc->nnscand->nnscan"), prt(""),prt("given formula"),prt(f), if is(presimp=off) then g:dnf(f) elseif is(presimp=l) then (prt("pre-simplified dnf"),g:dnf(lineq(f))) else (prt("pre-simplified dnf"),g:dnf(qe([],p2t(f)))),prt(g), prt("var for solve"), v:vv,for xxx in Lminus(vv,listofvars(g)) do v:delete(xxx,v), if v=[] then v:listofvars(f),prt(v), if atom(g) then return(bra(g)) else (if is(ncondsimp=off) then gg:dnf(mapply(ncondeq,g)) else gg:nnscan(dnf(mapply(ncondeq,g))), prt(length(gg)),prt(gg),prt("solving..."), q:scanmap(lambda([t],apply(fineq,flatten([t,last(v)]))), mapply(lambda([t],apply(solsel,flatten([t,v]))),gg) ,bottomup), prt("solved formula"),prt(q), collect(ff):=expand(ff), /* collect(ff):=block([fff,vc],fff:expand(f),vc:listofvars(fff), sum(factor(coeff(fff,vc,d))*x^d,d,0,hipow(fff,vc))), */ if is(postsimp=off) then (thisout:scanmap(collect,l2rx(bra(l2r(q)),last(v))),thisout) else (q:mapply(nnscanc,q),prt("reduced formula"),prt(q), q:mapply(nnscand,q),prt("reduced formula"),prt(q), q:nnscan(q), thisout:scanmap(collect,l2rx(bra(l2r(q)),last(v))),thisout) ) )$ thischeck():=fullall((thisin)%eq(ora(thisout)))$ /* for scanmap */ sineq(ineq,[vv]):=block([p,v,xxx,sineqx,q,r,s,lc,sol,sk,rt,qq,excond], if atom(ineq) then return(ineq), if not(op(ineq)=">" or op(ineq)="<" or op(ineq)=">=" or op(ineq)="<=") then return(ineq), p:ratsimp(lhs(ineq)-rhs(ineq)), if imgp(p)=true then return(ineq), if tdeg(num(factor(p)))<=1 and listofvars(denom(factor(p)))=[] then return(ineq), if length(vv)=1 then (if hipow(p,last(vv))=1 then return(ineq)), v:vv,for xxx in Lminus(vv,listofvars(p)) do v:delete(xxx,v), if v=[] then v:listofvars(p),sineqx:last(v), q:1,r:1, sol:delimg(ratsimp(solve(p,sineqx))), for k:1 thru length(sol) do (sk:part(sol,k),if not(sk=false) then (q:q*(lhs(sk)-rhs(sk))^(part(multiplicities,k)), r:r*(lhs(sk)-concat(rt,k))^(part(multiplicities,k)))), s:factor(quotient(p,q,sineqx)),lc:ratcoef(s,sineqx,hipow(s,sineqx)),lc:lc/abs(lc), prt("factor type"),prt([p,sineqx,r,s,lc]), if s=lc then qq:qe([],qe([],apply(op(ineq),[r*lc,0]))) elseif ((op(ineq)=">") or (op(ineq)="<")) then (prt("nnsolving..."), qq:(qe([],qe([],apply(op(ineq),[r*lc,0]))) %and nnsolvexx0(s#0))) elseif ((op(ineq)=">=") or (op(ineq)="<=")) then (prt("nnsolving..."), qq:(qe([],qe([],apply(op(ineq),[r*lc,0]))) %or nnsolvexx0(s=0))), prt(qq), /* prt(qe([],cnf(qq))),*/ for k:1 thru length(sol) do (sk:part(sol,k),if not(sk=false) then qq:subst(concat(rt,k)=rhs(sk),qq)), if is(sineqsimp=off) then qq:qq elseif is(sineqsimp=ex) then qq:(map(lambda([t],(t)%and(fullex(t))),qq)) elseif is(sineqsimp=l) then qq:lineq(qq) else qq:nnscan(qq), excond:qe([[E,sineqx]],p2t(qq)),prt(excond), if not(is(excond=true)) then (insolsel:true, qq:(qq) %or (ora(nnsolvexx((ineq)%and(%not(excond)),sineqx))), insolsel:false), prt("sineq returned..."),prt(qq), qq )$ /* factor for polynomial inequation */ /* new ver */ fineq(ineq,[vv]):=block([pp,v,xx,yy,dg,lc], if atom(ineq) or listofvars(ineq)=[] then return("%and"(ineq)), if not(op(ineq)=">" or op(ineq)="<" or op(ineq)=">=" or op(ineq)="<=") then return(ineq), pp:ratsimp(lhs(ineq)-rhs(ineq)), if atom(pp) then return(ineq), prt(vv), v:vv,for yy in Lminus(vv,listofvars(pp)) do v:delete(yy,v), if v=[] then v:listofvars(pp),xx:last(v),dg:hipow(pp,xx),prt(xx), if imgp(pp)=true or dg=0 then return(ineq),lc:ratcoef(pp,xx,dg),/*prt(lc),*/ if listofvars(lc)=[] then (prt("case1"),sineq(apply(op(ineq),[pp,0]),xx)) elseif is(fineqfull=on) then ((prt("case2-1"),(nnsolvexx(lc=0) %and fineq(apply(op(ineq),[expand(pp-lc*xx^dg),0]),xx))) %or (prt("case2-2"),(fineq(lc>0) %and (sineq(apply(op(ineq),[pp/lc,0]),xx)))) %or (prt("case2-3"),(fineq(lc<0) %and (sineq(apply(op(ineq),[(-1)*pp/lc,0]),xx))))) else ((prt("case2-1"),((lc=0) %and fineq(apply(op(ineq),[expand(pp-lc*xx^dg),0]),xx))) %or (prt("case2-2"),((lc>0) %and (sineq(apply(op(ineq),[pp/lc,0]),xx)))) %or (prt("case2-3"),((lc<0) %and (sineq(apply(op(ineq),[(-1)*pp/lc,0]),xx))))) )$ /* automatic quantify */ fullex(f):=qe(map(lambda([x],[E,x]),listofvars(f)),p2t(f))$ fullall(f):=qe(map(lambda([x],[A,x]),listofvars(f)),p2t(f))$ /* lineq */ defrule(tolineq1, (aa)^(bb), lineqdec(1,aa^bb,bb))$ defrule(tolineq2, (aa)/(bb), lineqdec(2,aa/bb,bb))$ lineq(f):=block([L,lineqdec,q], if is(lsimp=off) then (prt("lineq off"),return(f)), prt("linear simplifing..."), L:[], lineqdec(m,g,bb):=block([strg], if (m=1 and denom(bb)=1) or (m=2 and listofvars(bb)=[]) then g else ( strg:ssubst("equal","=",base64(string(g))), strg:ssubst("plus","+",strg), strg:ssubst("slash","/",strg), L:append(L,[eval_string(strg)=g]),eval_string(strg))), q:qe([],apply1(f,tolineq1,tolineq2)),L:listify(setify(L)), prt([q,L]), /* nnscan(nnscan(subst(L,q)))*/ /* nnscan(subst(L,q)) */ nnscan(map(lambda([t],(t)%and(fullex(t))),dnf(subst(L,q)))) /* subst(L,q) */ )$ /* l2r */ defrule(l2r1, (aa) > (bb),block([p,l,v],p:expand(bb-aa),l:listofvars(p), if length(l)=1 then (v:first(l), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<v else bb<aa) else bb<aa ))$ defrule(l2r2, (aa) >= (bb),block([p,l,v],p:expand(bb-aa),l:listofvars(p), if length(l)=1 then (v:first(l), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<=-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<=v else bb<=aa) else bb<=aa ))$ defrule(l2r3, (bb) < (aa),block([p,l,v],p:expand(bb-aa),l:listofvars(p), if length(l)=1 then (v:first(l), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<v else bb<aa) else bb<aa ))$ defrule(l2r4, (bb) <= (aa),block([p,l,v],p:expand(bb-aa),l:listofvars(p), if length(l)=1 then (v:first(l), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<=-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<=v else bb<=aa) else bb<=aa ))$ defrule(l2r5, (bb) = (aa),block([p,l,v],p:expand(bb-aa),l:listofvars(p), if length(l)=1 then (v:first(l), if hipow(p,v)=1 then v=-coeff(p,v,0)/coeff(p,v,1) else bb=aa) else bb=aa ))$ defrule(l2r1x, (aa) > (bb),block([p,v],p:expand(bb-aa), if length(vv)=1 then (v:first(vv), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<v else bb<aa) else bb<aa ))$ defrule(l2r2x, (aa) >= (bb),block([p,v],p:expand(bb-aa), if length(vv)=1 then (v:first(vv), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<=-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<=v else bb<=aa) else bb<=aa ))$ defrule(l2r3x, (bb) < (aa),block([p,v],p:expand(bb-aa), if length(vv)=1 then (v:first(vv), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<v else bb<aa) else bb<aa ))$ defrule(l2r4x, (bb) <= (aa),block([p,v],p:expand(bb-aa), if length(vv)=1 then (v:first(vv), if hipow(p,v)=1 and coeff(p,v,1)>0 then v<=-coeff(p,v,0)/coeff(p,v,1) elseif hipow(p,v)=1 and coeff(p,v,1)<0 then -coeff(p,v,0)/coeff(p,v,1)<=v else bb<=aa) else bb<=aa ))$ defrule(l2r5x, (bb) = (aa),block([p,v],p:expand(bb-aa), if length(vv)=1 then (v:first(vv), if hipow(p,v)=1 then v=-coeff(p,v,0)/coeff(p,v,1) else bb=aa) else bb=aa ))$ defrule(l2r6, (bb) < (aa) %and (cc) < (bb), (cc) < (bb) %and (bb) < (aa))$ l2r(f):=block([],apply1(f,l2r1,l2r2,l2r3,l2r4,l2r5))$ l2rx(f,[vv]):=block([],apply1(f,l2r1x,l2r2x,l2r3x,l2r4x,l2r5x,l2r6))$ /* bra, ora */ bra(f):=block([g], if listp(f) then return(f), if atom(f) then return([[f]]), g:dnf(f), if op(g)="%or" then g:args(g) else g:[g], map(lambda([t],if op(t)="%and" then args(t) else [t]),g) )$ ora(f):=block([], if atom(f) then return(f) else substpart("%or",map(lambda([g],substpart("%and",g,0)),f),0) )$ /* lfactor */ lfactor0(ff):=block([f,L,M,cf],L:[],M:[], if not(listp(ff)) then f:bra(ff) else f:ff, cf:delete({},map(lambda([t],apply(intersection,args(t))),sort(rest(listify( setdifference(powerset(setify(map(setify,f))),map(lambda([s],{s}),fullsetify(f))) )),comparelength2))), if cf=[] then return(f), cf:first(cf), for fk in f do if subsetp(cf,setify(fk)) then L:append([substpart("%and",setdifference(setify(fk),cf),0)],L) else M:append([fk],M), append(M,[flatten([substpart("%or",L,0),args(cf)])]) )$ lfactor(ff):=block([f,afr,bfr], if not(listp(ff)) then f:bra(ff) else f:ff, afr:f,bfr:[], for cc:1 unless length(bfr)=length(afr) do (bfr:afr,afr:lfactor0(bfr)), factor(afr))$ /* plineq */ plineq(f,[p]):=sort(append(Lminus(f,part(f,p)),bra(lineq(ora(part(f,p))))))$ /* addex */ addex(L,[v]):=l2r(map( lambda([Lk],append(Lk,first(bra(qe(makelist([E,vk],vk,v),p2t(substpart("%and",Lk,0))))))), L))$ /* qex, solvex */ qex(L,[f]):=if listp(L) then qe(L,apply(p2t,f)) else p2t(L)$ solvex(f,[v]):=apply(nnsolvexx,flatten([ora(bra(f)),v]))$
ncond 動かない式を動かす
以前にも述べましたが,変数の個数を低減させた必要条件を付加する ncond は強力で,nnss,nnsolvexx の求解能力はこの関数によるヒントの賜物とも言えます.
(%i98) f:(a^2+b^2+c^2<=3)%and(a*b*c>=1)$ Evaluation took 0.0100 seconds (0.0200 elapsed) using 538.359 KB. (%i99) qe([],f); Evaluation took 0.3500 seconds (0.5500 elapsed) using 17.918 MB. (%o99) (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (a*b*c-1 = 0) (%i100) nnss(f); Evaluation took 17.1500 seconds (23.0500 elapsed) using 952.370 MB. (%o100) ((a-1 = 0) %and (b-1 = 0) %and (c-1 = 0)) %or ((a-1 = 0) %and (b+1 = 0) %and (c+1 = 0)) %or ((a+1 = 0) %and (b-1 = 0) %and (c+1 = 0)) %or ((a+1 = 0) %and (b+1 = 0) %and (c-1 = 0))
verbose モード.
(%i101) prt(x):=print(x)$ Evaluation took 0.0000 seconds (0.0000 elapsed) using 200 bytes. (%i102) nnss(f); pre-simplified formula 2 (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (a*b*c-1 = 0) necessary condition a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0 a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0 b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0 ((a-1 = 0) %and (a-1 <= 0) %and (a+1 >= 0)) %or ((a-1 <= 0) %and (a+1 = 0) %and (a+1 >= 0)) ((b-1 = 0) %and (b-1 <= 0) %and (b+1 >= 0)) %or ((b-1 <= 0) %and (b+1 = 0) %and (b+1 >= 0)) ((c-1 = 0) %and (c-1 <= 0) %and (c+1 >= 0)) %or ((c-1 <= 0) %and (c+1 = 0) %and (c+1 >= 0)) true 8 ((a-1 = 0) %and (a-1 <= 0) %and (a+1 >= 0) %and (b-1 = 0) %and (b-1 <= 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 = 0) %and (c-1 <= 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 = 0) %and (a-1 <= 0) %and (a+1 >= 0) %and (b-1 = 0) %and (b-1 <= 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 <= 0) %and (c+1 = 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 = 0) %and (a-1 <= 0) %and (a+1 >= 0) %and (b-1 <= 0) %and (b+1 = 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 = 0) %and (c-1 <= 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 = 0) %and (a-1 <= 0) %and (a+1 >= 0) %and (b-1 <= 0) %and (b+1 = 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 <= 0) %and (c+1 = 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 <= 0) %and (a+1 = 0) %and (a+1 >= 0) %and (b-1 = 0) %and (b-1 <= 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 = 0) %and (c-1 <= 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 <= 0) %and (a+1 = 0) %and (a+1 >= 0) %and (b-1 = 0) %and (b-1 <= 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 <= 0) %and (c+1 = 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 <= 0) %and (a+1 = 0) %and (a+1 >= 0) %and (b-1 <= 0) %and (b+1 = 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 = 0) %and (c-1 <= 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) %or ((a-1 <= 0) %and (a+1 = 0) %and (a+1 >= 0) %and (b-1 <= 0) %and (b+1 = 0) %and (b+1 >= 0) %and (a^2*b^4+a^4*b^2-3*a^2*b^2+1 = 0) %and (c-1 <= 0) %and (c+1 = 0) %and (c+1 >= 0) %and (a*b*c-1 = 0) %and (a^2*c^4+a^4*c^2-3*a^2*c^2+1 = 0) %and (b^2*c^4+b^4*c^2-3*b^2*c^2+1 = 0)) simplifing... {(a-1 = 0) %and (b-1 = 0) %and (c-1 = 0), (a-1 = 0) %and (b+1 = 0) %and (c+1 = 0), (a+1 = 0) %and (b-1 = 0) %and (c+1 = 0)} {(a-1 = 0) %and (b-1 = 0) %and (c-1 = 0), (a-1 = 0) %and (b+1 = 0) %and (c+1 = 0), (a+1 = 0) %and (b+1 = 0) %and (c-1 = 0)} {(a-1 = 0) %and (b-1 = 0) %and (c-1 = 0), (a+1 = 0) %and (b-1 = 0) %and (c+1 = 0), (a+1 = 0) %and (b+1 = 0) %and (c-1 = 0)} {(a-1 = 0) %and (b+1 = 0) %and (c+1 = 0), (a+1 = 0) %and (b-1 = 0) %and (c+1 = 0), (a+1 = 0) %and (b+1 = 0) %and (c-1 = 0)} {a-1 = 0,b-1 = 0} {a-1 = 0,c-1 = 0} {b-1 = 0,c-1 = 0} {a-1 = 0,b+1 = 0} {a-1 = 0,c+1 = 0} {b+1 = 0,c+1 = 0} {a+1 = 0,b-1 = 0} {a+1 = 0,c+1 = 0} {b-1 = 0,c+1 = 0} {a+1 = 0,b+1 = 0} {a+1 = 0,c-1 = 0} {b+1 = 0,c-1 = 0} Evaluation took 18.5100 seconds (24.4700 elapsed) using 954.651 MB. (%o102) ((a-1 = 0) %and (b-1 = 0) %and (c-1 = 0)) %or ((a-1 = 0) %and (b+1 = 0) %and (c+1 = 0)) %or ((a+1 = 0) %and (b-1 = 0) %and (c+1 = 0)) %or ((a+1 = 0) %and (b+1 = 0) %and (c-1 = 0))
nnssdn 二重否定の利用
次の不等式を簡約します.
(%i90) f:x^3+y^3+z^3>=3*x*y*z; Evaluation took 0.0000 seconds (0.0000 elapsed) using 1.859 KB. (%o90) z^3+y^3+x^3 >= 3*x*y*z
まずは,qepmax から....
(%i91) qe([],f); Evaluation took 0.0800 seconds (0.2700 elapsed) using 1.674 MB. (%o91) (z+y+x >= 0) %or (z^2-y*z-x*z+y^2-x*y+x^2 = 0)
一方,nnss は....いい感じです.
(%i92) nnss(f); Evaluation took 1.4100 seconds (5.1900 elapsed) using 59.782 MB. (%o92) ((y-x = 0) %and (z-x = 0)) %or (z+y+x >= 0)
では,次はどうでしょう.
(%i93) g:x^3+y^3+z^3>3*x*y*z; Evaluation took 0.0000 seconds (0.0000 elapsed) using 1.859 KB. (%o93) z^3+y^3+x^3 > 3*x*y*z (%i94) qe([],g); Evaluation took 0.0700 seconds (0.2900 elapsed) using 1.510 MB. (%o94) (z+y+x > 0) %and (z^2-y*z-x*z+y^2-x*y+x^2 > 0) (%i95) nnss(g); Evaluation took 1.6900 seconds (3.7200 elapsed) using 92.056 MB. (%o95) (z+y+x > 0) %and (z^2-y*z-x*z+y^2-x*y+x^2 > 0)
共倒れです....ということで,nnssdn の出番です.
(%i96) nnssdn(g); Evaluation took 1.8000 seconds (5.7800 elapsed) using 78.075 MB. (%o96) ((y-x # 0) %or (z-x # 0)) %and (z+y+x > 0)
勿論,万能ではありません.
(%i97) nnssdn(f); Evaluation took 2.5600 seconds (4.8100 elapsed) using 147.524 MB. (%o97) (z+y+x >= 0) %or (z^2-y*z-x*z+y^2-x*y+x^2 = 0)
nnss 見易い式を求めて
次の不等式を簡約します.
(%i86) f:(x^2-1)*(x^2-4)*(x^2-9)<0; Evaluation took 0.0000 seconds (0.0000 elapsed) using 1.344 KB. (%o86) (x^2-9)*(x^2-4)*(x^2-1) < 0
まずは,qepmax + nns です.悪くはありませんが,ちょっと解り難いですね.
(%i87) qe([],f); Evaluation took 0.1000 seconds (0.3300 elapsed) using 2.794 MB. (%o87) (x-3 < 0) %and (x-2 # 0) %and (x-1 # 0) %and (x+1 # 0) %and (x+2 # 0) %and (x+3 > 0) %and ((x-2 > 0) %or (x+2 < 0) %or ((x-1 < 0) %and (x+1 > 0))) (%i88) nns(%); Evaluation took 4.9000 seconds (6.1500 elapsed) using 232.032 MB. (%o88) (x-3 < 0) %and (x+3 > 0) %and ((x-2 > 0) %or (x+2 < 0) %or ((x-1 < 0) %and (x+1 > 0)))
ということで,nnss の出番です.
(%i89) nnss(f); Evaluation took 4.4000 seconds (7.2900 elapsed) using 194.955 MB. (%o89) ((x-3 < 0) %and (x-2 > 0)) %or ((x-1 < 0) %and (x+1 > 0)) %or ((x+2 < 0) %and (x+3 > 0))
dnf から cnf へ
式の簡約は,まず選言形に直し,得られた連言項毎に処理するのが基本です.よって結果は
(%i70) qe([],(a^2=1)%and(b^2=1)); Evaluation took 0.3400 seconds (0.5200 elapsed) using 12.210 MB. (%o70) (a-1 <= 0) %and (a+1 >= 0) %and (((a-1 = 0) %and (b-1 = 0)) %or ((a-1 = 0) %and (b+1 = 0)) %or ((a+1 = 0) %and (b-1 = 0)) %or ((a+1 = 0) %and (b+1 = 0))) %and (b-1 <= 0) %and (b+1 >= 0) (%i71) nns(%); Evaluation took 4.5500 seconds (5.5200 elapsed) using 185.040 MB. (%o71) ((a-1 = 0) %and (b-1 = 0)) %or ((a-1 = 0) %and (b+1 = 0)) %or ((a+1 = 0) %and (b-1 = 0)) %or ((a+1 = 0) %and (b+1 = 0))
のようになります.これは連立方程式の解の表示として標準的ですが,連言形のほうが見易そうです.実際
(%i72) cnf(%); Evaluation took 0.2900 seconds (0.3300 elapsed) using 9.648 MB. (%o72) ((a-1 = 0) %or (a+1 = 0)) %and ((a-1 = 0) %or (a+1 = 0) %or (b-1 = 0)) %and ((a-1 = 0) %or (a+1 = 0) %or (b-1 = 0) %or (b+1 = 0)) %and ((a-1 = 0) %or (a+1 = 0) %or (b+1 = 0)) %and ((a-1 = 0) %or (b-1 = 0) %or (b+1 = 0)) %and ((a+1 = 0) %or (b-1 = 0) %or (b+1 = 0)) %and ((b-1 = 0) %or (b+1 = 0)) (%i73) nns(%); Evaluation took 3.9800 seconds (5.3500 elapsed) using 177.244 MB. (%o73) ((a-1 = 0) %or (a+1 = 0)) %and ((b-1 = 0) %or (b+1 = 0))
といった具合です.勿論,初めからこの結果を目指して
(%i74) map(nnsolve,(a^2=1)%and(b^2=1)); Evaluation took 0.0100 seconds (0.0100 elapsed) using 494.719 KB. (%o74) ((a = -1) %or (a = 1)) %and ((b = -1) %or (b = 1))
とすることも出来ます.