実行例（２）
SyNRAC の岩根秀直さん（富士通研究所）のサイト
https://github.com/hiwane/qe_problems/blob/master/problems/exam/manual-fof/tsukuba2010-Ri-1-m.mpl
他の QE ツールの出力との比較
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2019-13.pdf Appendix
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([ex],[a,b,x],0,0,r11,0 ),
qe( bfpcad(ext( '( -1 <= x and x <= 3 and b = x^3/3 - a*x^2/2 and a > 0 ) ))) );
Evaluation took 1.7400 seconds (1.7400 elapsed) using 188.173 MB.
(%o1) [[root(a,1) < a and a < root(a-2,1),
root(6*b+3*a+2,1) <= b and b <= root(2*b+9*a-18,1),true],
[a = root(a-2,1),root(6*b+a^3,1) <= b and b <= root(2*b+9*a-18,1),
true],
[root(a-2,1) < a and a <= root(a-3,1),
root(6*b+a^3,1) <= b and b <= root(b,1),true],
[root(a-3,1) < a,root(2*b+9*a-18,1) <= b and b <= root(b,1),true]]
(%i2) (qvpeds ([ex],[a,b,x],1,0,r11,0 ),
qe( bfpcad(ext( '( -1 <= x and x <= 3 and b = x^3/3 - a*x^2/2 and a > 0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [16,30,1.0b-3,1.0b-16]
x+1 >= 0 and x-3 <= 0 and equal((2*x^3-3*a*x^2-6*b)/6,0) and a > 0
["and",equal((2*x^3-3*a*x^2-6*b)/6,0),x-3 <= 0,x+1 >= 0,a > 0]
[[x-3],[x+1],[2*x^3-3*a*x^2-6*b]]
[[b],[2*b+9*a-18],[6*b+3*a+2],[6*b+a^3]]
[[a-3,a-2,a,a+1,a+6,3*a-7,3*a+2],[b,2*b+9*a-18,6*b+3*a+2,6*b+a^3],
[x-3,x+1,2*x^3-3*a*x^2-6*b]]
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -1.333333333333333b0 -1.333333333333333b0
dist: 0.0b0
common-roots: -1.5b0 -1.5b0
dist: 0.0b0
common-roots: -4.5b0 -4.5b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 9.999999999999999b-1-3.469517903370656b-17*%i
3.469446951953614b-17*%i+1.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-3.0b0*x^2+1.0b0]
dist: 1.309231643767475b-16
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-3.0b0*x^2]
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 6.938893903907228b-17*%i+2.0b0 2.0b0-6.939035806741312b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-6*x^2+8]
dist: 2.61846328753495b-16
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-6*x^2]
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 2.166666666666667b0-4.163490115220211b-17*%i
4.163336342344337b-17*%i+2.166666666666667b0
[2*x^3-3*a*x^2-6*b,2*x^3-6.5b0*x^2+1.01712962962963b1]
dist: 1.861907491918325b-16
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
common-roots: 3.0b0 3.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-6.5b0*x^2]
dist: 0.0b0
1 multi-roots: 2.333333333333333b0-5.551280766043446b-17*%i
5.551115123125783b-17*%i+2.333333333333334b0
[2*x^3-3*a*x^2-6*b,2*x^3-7*x^2+1.27037037037037b1]
dist: 1.171009175116753b-15
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-7*x^2]
dist: 0.0b0
1 multi-roots: 1.387778780781446b-17*%i+2.666666666666667b0
2.666666666666667b0-1.387968122423659b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-8.0b0*x^2+1.896296296296296b1]
dist: 1.144396291987333b-16
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-8.0b0*x^2]
dist: 0.0b0
1 multi-roots: 3.0b0-4.163336342344337b-17*%i 4.163123343336605b-17*%i+3.0b0
[2*x^3-3*a*x^2-6*b,2*x^3-9*x^2+27]
dist: 2.371429941323172b-16
common-roots: 3.0b0 3.0b0
dist: 0.0b0
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-9*x^2]
dist: 0.0b0
1 multi-roots: 5.550866638069772b-17*%i+3.5b0 3.5b0-5.551115123125783b-17*%i
[2*x^3-3*a*x^2-6*b,2*x^3-1.05b1*x^2+4.2875b1]
dist: 5.661043993863231b-16
common-roots: 3.0b0 3.0b0
dist: 5.551115123125783b-17
common-roots: -1.0b0 -1.0b0
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [2*x^3-3*a*x^2-6*b,2*x^3-1.05b1*x^2]
dist: 0.0b0
[T,F]: [39,253]
Evaluation took 2.5500 seconds (2.5600 elapsed) using 169.662 MB.
(%o2) [[root(a,1) < a and a < root(a-2,1),
root(6*b+3*a+2,1) <= b and b <= root(2*b+9*a-18,1),true],
[a = root(a-2,1),root(6*b+a^3,1) <= b and b <= root(2*b+9*a-18,1),
true],
[root(a-2,1) < a and a <= root(a-3,1),
root(6*b+a^3,1) <= b and b <= root(b,1),true],
[root(a-3,1) < a,root(2*b+9*a-18,1) <= b and b <= root(b,1),true]]
projection set に対する数値解の誤差（その２）
．．．とは言え，sep の評価なしでは，いくら R を小さくしても，前回の冒頭で述べたように，近接根に対応した数値解をカウントしてしまう可能性を排除出来ないかのように見えます．
しかし，我々が扱うのは一般の系ではなく，CAD の projection set であり，特に pscs を含む Collins-Hong タイプの projection（定義は，例えば，D.Wilson. Advances in Cylindrical Algebraic Decomposition の 2.3.1 Collins’Algorithm，2.4.1 Alternative Projection Operators をご参照ください）を利用すれば，sep の情報なしに，重根に対応した数値解全体のみを正確にカウントすることが出来ます．以下，そのことをお話しましょう．
まず，pscs（principal subresultant coefficient sequence）とは，多項式 f(x), g(x) に対して，それらの Sylvester 行列
syl(f1,f2,v):=block([d1:hipow(f1,v),d2:hipow(f2,v),c1,c2,z0,i,k],
c1:makelist(coeff(f1,v,i),i,d1,0,-1),
c2:makelist(coeff(f2,v,i),i,d2,0,-1),z0:makelist(0,d1+d2),
[makelist(firstn(append(firstn(z0,k-1),c1,z0),d1+d2),k,d2),
makelist(firstn(append(firstn(z0,k-1),c2,z0),d1+d2),k,d1)])$
の一部を除いた行列の行列式の列（普通の resultant はその初項です）
pscs(f01,f02,v):=block(
[uratmx:ratmx,s1,s2,d,d1,d2,f1:expand(f01),f2:expand(f02),j],
ratmx:on,[s1,s2]:syl(f1,f2,v),d1:hipow(f1,v),d2:hipow(f2,v),
d:makelist(determinant(apply(matrix,
append(
map(lambda([e],firstn(e,d1+d2-2*j)),
firstn(s1,d2-j)),
map(lambda([e],firstn(e,d1+d2-2*j)),
firstn(s2,d1-j))))),j,0,
min(d1,d2)-1),ratmx:uratmx,d)$
のことで
( pscs の初項から続く 0 の個数 ) = ( gcd( f(x), g(x) ) の次数 )
という性質（Polynomial greatest common divisor - Wikipedia）を持ちます．例えば，上の（素朴な）コードを読み込んで，実行すると
(%i24) pscs( (x-1) * (x-2) , (x-3) * (x-4), x ); (%o24) [12,-4] (%i25) pscs( (x-1) * (x-2)^3 * (x-3)^2 , (x-2)^2 * (x-3)^3 * (x-4), x ); (%o25) [0,0,0,0,12,-4] (%i26) factor(pscs( (x-1) * (x-a)^3 * (x-b)^2 , (x-a)^2 * (x-b)^3 * (x-4), x )); (%o26) [0,0,0,0,-3*(a-4)*(b-1)*(b-a),-(b-a+3)]
といった具合になります．
さて，上の性質を g(x) = f'(x) の場合に用いると
gcd( f(x), f'(x) ) の次数
の値が pscs から特定できる訳ですが，この値が f(x) の
(重根の個数) - (相異なる重根の個数)
であることは，積の微分公式により簡単に判ります．例で確認してみると
(%i31) pscs( f : (x-1) * (x-2)^3 * (x-3)^5 , diff(f,x) , x ); (%o31) [0,0,0,0,0,0,-60,-38]
のように 8 - 2 = 6 となっています．
ところが，この差は前回の冒頭で述べた方法における，重根に対応する数値解の検出カウントに他なりません．例えば，上の重根 2, 2, 2, 3, 3, 3, 3, 3 に対応した数値解 s1, s2, s3, t1, t2, t3, t4, t5 は
2*R >= | s1 - s2 |, 2*R >= | s2 - s3 |,
2*R >= | t1 - t2 |, 2*R >= | t2 - t3 |, 2*R >= | t3 - t4 |, 2*R >= | t4 - t5 |
として，合計 (3-1) + (5-1) = 6 回カウントされ，もし，検出カウントがこれを超えるなら，そこには近接根に対応する数値解のカウントが含まれているので，R を小さくして retry すれば，有限回で近接根に対応する数値解を排除でき，同時に，カウントの長さを見れば，各重根の重複度（各クラスタに属する数値解の個数）も判る，と言う仕組みです．
近接根の識別
さて，r, R を先の条件を満たすようにとるとして，そも，R をどのように定めるのか？という問題があります．
確かに，数値解の誤差 R の系において，互いの距離が 2*R 以下である数値解を全て集めれば，重根に対する数値解は必ずそれに属します．しかし，互いの距離が 4*R 以下である厳密相異根（近接根）が存在すると，それらに対する数値解もメンバーになるかも知れません．
ここでもし，系における厳密相異根間の距離の最小値（根差）sep の値が判るならば，sep > 4*R を満たすように R をとっておけば，相異根 r1, r2 に対する数値解 s1, s2 間の距離は
| s2 - s1 | = | (r2 - r1) - (s1 - r1) + (s2 - r2) | >= sep - R - R > 2*R
のように 2*R より大きくなるので，上記のように集めた数値解全体が，重根に対する数値解全体になります．
では如何にして sep の値を得るかですが，それには resultant を繰り返し用いて１変数化する方法があります．maxima では
(%i7) factor(eliminate([2*c+1,4096*c^3+27*b^4,8*a*c-9*b^2-2*a^3,x^4+a*x^2+b*x+c],[c,b,a]));
Evaluation took 0.0300 seconds (0.0400 elapsed) using 2.309 MB.
(%o7) [33554432*x^8*(2*x^4-27)*(6*x^4-1)^3
*(186624*x^32+2985984*x^28+40621824*x^24+12082176*x^20
+41597280*x^16+64694016*x^12+262151536*x^8
-644544*x^4+729)]
のように必要条件を定める多項式が得られ，この実根
-1.91683..., -0.63894..., 0, 0.63894..., 1.91683...
から，sep >= 0.63894...（実際には c は 0 でないので，1.91683...-0.63894...）と判るので，
R < 0.63894.../4 = 0.15973...
のようにとっておけば，上記の系の近接根に対する数値解間の距離はカウントされず，原理的には，これをすべての多項式の組合せに対して実行し，最小の sep に対して， R を定めればよいことになります．
．．．が，一般に，resultant の係数は爆発し，モニター画面が埋まります．
projection set に対する数値解の誤差（その１）
誤差 r の数値解を係数に代入した方程式系の数値解の誤差 R との関係を考えると
・数値解の代入による係数の摂動は Taylor の定理により r 程度．
・係数の摂動 r に対する厳密 n 重根の摂動は同じく r^{1/n} 程度．
なので，例えば，前回の
4096*c^3+27*b^4（4次）, 8*a*c-9*b^2-2*a^3（3次）, x^4+a*x^2+b*x+c（4次）
の場合，次数の最大値 4 により，r を最悪でも R^{4} 程度まで小さくすればよく，先の例のように %ez : 1.0b-5 とするならば，fpprec は 25 程度にすればよいことになります．
重根に対する数値解の誤差
先の出力の dist は，同じ値に対すると見做した数値解間の複素平面上での距離ですが，単根の場合の（絶対）誤差 10^{-30} に対して，例えば，下の部分では 10^{-11} まで膨らんでいます．
1 multi-roots:
(-6.2414826969557199979633340867b-13*%i)-6.38943104254668599248510111018b-1
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.45826385597869537841530414623b-11
2 multi-roots:
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
7.58333029391685279498189770221b-12*%i-6.38943104242614942415515511763b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.4582638571553690760678343437b-11
これは，一般に n 重根に対する数値解の誤差は，単根に対する数値解の誤差の 1/n 乗程度となる性質によるもので，上の場合は，3 重根なので指数が -30 の 1/3 程度となっている訳です．
この例の出所をもう少し詳しく述べると，上記は，下位の lifting で projection factor から得た
c = root(2*c+1, 1), b = root(4096*c^3+27*b^4, 1), a = root(8*a*c-9*b^2-2*a^3, 1)
の数値解を x^4+a*x^2+b*x+c = 0 に代入した
x^4-2.44948974278317809819728407471b0*x^2-2.08677944009771642211264733221b0*x-5.0b-1 = 0
を解いたもので，厳密解
c = - 1/2 = - 0.5, b = - (4*2^(1/4))/(3^(3/4)) = 2.08677944009771642211264733221..., a = - 6^(1/2) = 2.449489742783178098197284074705...
に対して，数値解は
c = - 5.0b-1, b = - 2.08677944009771642211264733221b0, a = - 2.44948974278317809819728407471b0
なので，c,b,a までの精度は保たれており，最後の
x^4+a*x^2+b*x+c = (x-3^(3/4)/2^(1/4))*(x+1/6^(1/4))^3 = 0
の 3 重根
-1/6^(1/4) = - 0.6389431042462724758553493051605564336...
に対してのみ，対応する数値解が
(-6.2414826969557199979633340867b-13*%i)-6.38943104254668599248510111018b-1, (-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1, 7.58333029391685279498189770221b-12*%i-6.38943104242614942415515511763b-1
のようにばらけますが，相加平均をとると
(-3.21582999003662290927010077574b-32*%i)-6.38943104246272475855349305161b-1
となり，この実部として，30 桁精度のサンプルが得られます．
QE on maxima
以下は，現在作成中の QE ツールの出力例です．
・有効桁数が fpprec の多倍長浮動小数点数を使用．
・２つの数値解間の距離が %ez より小ならばそれらは同じ値の根（重根や共通根）に対応し，%ez 以上ならばそれらは異なる値の根に対応する数値解と見做しています．
Maxima 5.39.0 http://maxima.sourceforge.net
using Lisp CMU Common Lisp 21b (21B Unicode)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) (qvpeds ([all], [c,b,a,x],0,h1,r11,0 ),qe( bfpcad(ext( '( x^4+a*x^2+b*x+c>=0 ) ))) );
Evaluation took 11.3900 seconds (11.4100 elapsed) using 1307.972 MB.
(%o1) [[c = root(c,1),b = root(126976*c^3-135*b^4,1),
root(8*a*c-9*b^2-2*a^3,1) <= a,true],
[root(c,1) < c,b < root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true],
[root(c,1) < c,b = root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,1)
<= a,true],
[root(c,1) < c,root(b,1) < b,
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true]]
(%i2) (qvpeds ([all], [c,b,a,x],1,h1,r11,0 ),qe( bfpcad(ext( '( x^4+a*x^2+b*x+c>=0 ) ))) );
[fpprec,fpprintprec,%ez,ratepsilon]: [30,30,1.0b-5,1.0b-30]
x^4+a*x^2+b*x+c >= 0
x^4+a*x^2+b*x+c >= 0
[[c],
[b,256*c^3-27*b^4,1024*c^3-2187*b^4,1024*c^3+3*b^4,4096*c^3+27*b^4,
126976*c^3-135*b^4],
[a,8*a*c-9*b^2-2*a^3,
256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2],
[x^4+a*x^2+b*x+c]]
1 multi-roots: 0.0b0 0.0b0 [256*c^3-27*b^4,-27*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [256*c^3-27*b^4,-27*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [256*c^3-27*b^4,-27*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [1024*c^3-2187*b^4,-2187*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [1024*c^3-2187*b^4,-2187*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [1024*c^3-2187*b^4,-2187*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [1024*c^3+3*b^4,3*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [1024*c^3+3*b^4,3*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [1024*c^3+3*b^4,3*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [4096*c^3+27*b^4,27*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [4096*c^3+27*b^4,27*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [4096*c^3+27*b^4,27*b^4]
dist: 0.0b0
1 multi-roots: 0.0b0 0.0b0 [126976*c^3-135*b^4,-135*b^4]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [126976*c^3-135*b^4,-135*b^4]
dist: 0.0b0
3 multi-roots: 0.0b0 0.0b0 [126976*c^3-135*b^4,-135*b^4]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
4.52295331325015833831979417428b-16*%i-2.44948974278317847258028706083b0
(-4.52295331325015213319900815541b-16*%i)-2.44948974278317772381428108859b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-5.0b-1*(16*a^4+6.27069374152493593138504723125b2*a))
-1.74185937264581553649584645312b1*a^3-3.2b1*a^2-5.44b2]
dist: 1.17428054512251951790867706837b-15
common-roots: -2.44948974278317809819728407471b0
-2.44948974278317809819728407471b0
dist: 1.57772181044202361082345713057b-30
1 multi-roots: -1.41421356237309504880168872421b0*%i
(-1.41421356237309504880168872421b0*%i)-3.86941038573562221171322357299b-43
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-8.0b0*a^4)-3.2b1*a^2-3.2b1]
dist: 7.88860905221011805411728660181b-31
2 multi-roots:
1.41421356237309504880261489839b0*%i-2.95418118567534149019656617711b-22
1.41421356237309504880076255003b0*%i+2.9541811856753414902004355875b-22
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-8.0b0*a^4)-3.2b1*a^2-3.2b1]
dist: 1.94429469761416284867540568328b-21
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
4.52295330147550776509999452971b-16*%i-2.44948974278317847258028887801b0
(-4.52295330147551542569579253843b-16*%i)-2.44948974278317772381427927141b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
(-5.0b-1*(16*a^4+6.27069374152493593138504723125b2*a))
-1.74185937264581553649584645313b1*a^3-3.2b1*a^2
-5.44000000000000000000000000001b2]
dist: 1.17428054562583903487602624948b-15
common-roots: -2.44948974278317809819728407471b0
-2.44948974278317809819728407471b0
dist: 1.57772181044202361082345713057b-30
1 multi-roots: 0.0b0 0.0b0 [8*a*c-9*b^2-2*a^3,-2*a^3]
dist: 0.0b0
2 multi-roots: 0.0b0 0.0b0 [8*a*c-9*b^2-2*a^3,-2*a^3]
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
8.16496580927726032731894282802b-1-1.6183536594179216924381257046b-22*%i
1.61835365941792169243742042834b-22*%i+8.16496580927726032732961767002b-1
[8*a*c-9*b^2-2*a^3,(-2*a^3)+4.0b0*a-2.17732421580726942061980806641b0]
dist: 1.11547535241290012202233114443b-21
1 multi-roots:
(-8.47286141428164487440940737737b-20*%i)-1.41421356237309504909493338483b0
8.47286141428164487616706655174b-20*%i-1.41421356237309504850844406359b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
8.0b0*a^4-3.2b1*a^2+3.2b1]
dist: 6.10479709847422546500953724134b-19
2 multi-roots:
1.41421356237309504879618188426b0-1.96859282073234240509586747592b-21*%i
1.96859282073234238751927573217b-21*%i+1.41421356237309504880719556416b0
[256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,
8.0b0*a^4-3.2b1*a^2+3.2b1]
dist: 1.16962633227785224686827731017b-20
common-roots: 1.41421356237309504880168872421b0
1.41421356237309504880168872421b0
dist: 0.0b0
common-roots: 0.0b0 0.0b0
dist: 0.0b0
common-roots: -1.41421356237309504880168872421b0
-1.41421356237309504880168872421b0
dist: 0.0b0
1 multi-roots:
8.1649658092772603273200008634b-1-1.30109252983440372998384918995b-22*%i
1.30109252983440372998339457472b-22*%i+8.16496580927726032732855963464b-1
[8*a*c-9*b^2-2*a^3,(-2*a^3)+4.0b0*a-2.17732421580726942061980806641b0]
dist: 8.94560966410596198552565674457b-22
common-roots: 0.0b0 0.0b0
dist: 0.0b0
1 multi-roots:
(-1.16997742006955532323735088816b-17*%i)-3.35546475144558657978639517931b-1
1.16997742006955532102907767889b-17*%i-3.35546475144558652208708812765b-1
[x^4+a*x^2+b*x+c,
x^4-4.77860936379455500840332276085b0*x^2
-3.05577241698508990342139722064b0*x-5.0b-1]
dist: 2.41004349697564216587243138313b-17
1 multi-roots:
(-2.24047760152893241737421755167b-20*%i)-1.0109852688012821415609910299b0
2.24047760152893269859199915596b-20*%i-1.01098526880128214151321562239b0
[x^4+a*x^2+b*x+c,
x^4-3.55546677120777806279111679242b0*x^2
-3.05577241698508990342139722064b0*x-5.0b-1]
dist: 6.55010344684744407831148213933b-20
1 multi-roots:
2.22935715805258081302014670225b-18*%i-4.14333797761439630281210468432b-1
(-2.22935715805258088784912140497b-18*%i)-4.14333797761439627962993519833b-1
[x^4+a*x^2+b*x+c,
x^4-3.42753994633320075795235578053b0*x^2
-2.55577241698508990342139722064b0*x-5.0b-1]
dist: 5.02536199426577147169718392348b-18
1 multi-roots:
(-7.63751536911083876261312227633b-19*%i)-8.96295167765296091409417104731b-1
7.63751536911083876548501856105b-19*%i-8.96295167765296090286367856344b-1
[x^4+a*x^2+b*x+c,
x^4-3.03243266180629550631765220844b0*x^2
-2.55577241698508990342139722064b0*x-5.0b-1]
dist: 1.89591805066530344113644713169b-18
1 multi-roots:
4.44329956060120776495651206012b-21*%i-4.7445898461131539492366468499b-1
(-4.44329956060120776491378978904b-21*%i)-4.74458984611315394915833267933b-1
[x^4+a*x^2+b*x+c,
x^4-2.89645721365288302103527609247b0*x^2
-2.32127592854140316276702227643b0*x-5.0b-1]
dist: 1.18449456333131606287348159343b-20
1 multi-roots:
(-5.13518395612155029572934523107b-18*%i)-8.19617912427072101369893178291b-1
5.13518395612155030519077584595b-18*%i-8.19617912427072096157128200993b-1
[x^4+a*x^2+b*x+c,
x^4-2.75961902983845013509216731558b0*x^2
-2.32127592854140316276702227643b0*x-5.0b-1]
dist: 1.15175247237145207854523711501b-17
1 multi-roots:
(-6.2414826969557199979633340867b-13*%i)-6.38943104254668599248510111018b-1
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.45826385597869537841530414623b-11
2 multi-roots:
(-6.95918202422128079528203919324b-12*%i)-6.38943104241533885902022292701b-1
7.58333029391685279498189770221b-12*%i-6.38943104242614942415515511763b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
-2.08677944009771642211264733221b0*x-5.0b-1]
dist: 1.4582638571553690760678343437b-11
1 multi-roots:
6.38943104246144034405208996102b-1-4.422502465774425554602186706b-14*%i
1.33349305972666630723236835579b-13*%i+6.38943104246298402619456223129b-1
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
+2.08677944009771642211264733221b0*x-5.0b-1]
dist: 2.3529170930722626582908906827b-13
2 multi-roots:
1.33349305972666630723236835579b-13*%i+6.38943104246298402619456223129b-1
6.38943104246374990541382696251b-1-8.91242813149223756054934246976b-14*%i
[x^4+a*x^2+b*x+c,
x^4-2.44948974278317809819728407471b0*x^2
+2.08677944009771642211264733221b0*x-5.0b-1]
dist: 2.35287498234870156742355194154b-13
1 multi-roots:
3.72610157565100454685245607331b-20*%i+4.74458984611315394913552119431b-1
4.74458984611315394925945833492b-1-3.72610157565100454690915365651b-20*%i
[x^4+a*x^2+b*x+c,
x^4-2.89645721365288302103527609247b0*x^2
+2.32127592854140316276702227643b0*x-5.0b-1]
dist: 7.55455976814101740503286439967b-20
1 multi-roots:
8.19617912427072098761958869152b-1-3.13943116569295322775812726995b-21*%i
3.13943116569295322775468371035b-21*%i+8.19617912427072098765062510133b-1
[x^4+a*x^2+b*x+c,
x^4-2.75961902983845013509216731558b0*x^2
+2.32127592854140316276702227643b0*x-5.0b-1]
dist: 7.00404879414403669951788837319b-21
1 multi-roots:
2.46936845160507392345254636383b-17*%i+4.14333797761439612007308924107b-1
4.14333797761439646236895064158b-1-2.46936845160507398188288178659b-17*%i
[x^4+a*x^2+b*x+c,
x^4-3.42753994633320075795235578053b0*x^2
+2.55577241698508990342139722064b0*x-5.0b-1]
dist: 6.00897394505265450935583153395b-17
1 multi-roots:
8.96295167765296090779340009951b-1-8.69963119046618331638821004739b-20*%i
8.69963119046618331598771326227b-20*%i+8.96295167765296090916444951125b-1
[x^4+a*x^2+b*x+c,
x^4-3.03243266180629550631765220844b0*x^2
+2.55577241698508990342139722064b0*x-5.0b-1]
dist: 2.21520197802208732404660033187b-19
1 multi-roots:
3.35546475144558654573716194067b-1-3.2321873145433364152600510801b-19*%i
3.23218731454333641635946277158b-19*%i+3.35546475144558655613632136628b-1
[x^4+a*x^2+b*x+c,
x^4-4.77860936379455500840332276085b0*x^2
+3.05577241698508990342139722064b0*x-5.0b-1]
dist: 1.2244617433979626836819761986b-18
1 multi-roots:
1.01098526880128213782781893505b0-2.07015345747187669505738755424b-18*%i
2.07015345747187669030859350914b-18*%i+1.01098526880128214524638771724b0
[x^4+a*x^2+b*x+c,
x^4-3.55546677120777806279111679242b0*x^2
+3.05577241698508990342139722064b0*x-5.0b-1]
dist: 8.49572269592501254899699170041b-18
1 multi-roots:
1.09731695855008129432850901121b-16*%i-6.29960524947436974960114115801b-1
(-1.09731695855008175020983658478b-16*%i)-6.29960524947436189807096491478b-1
[x^4+a*x^2+b*x+c,x^4-1.19055078897614960606377922946b0*x^2-5.0b-1*x]
dist: 8.15248085790726489126442096385b-16
1 multi-roots: 0.0b0 0.0b0 [x^4+a*x^2+b*x+c,x^4-5.0b-1*x^2]
dist: 0.0b0
1 multi-roots:
6.29960524947436582381894689913b-1-5.1915207100556332839659272546b-22*%i
5.19152071005563328395652911378b-22*%i+6.29960524947436582385315917366b-1
[x^4+a*x^2+b*x+c,x^4-1.19055078897614960606377922946b0*x^2+5.0b-1*x]
dist: 3.57531436008650390541071687971b-21
1 multi-roots:
(-8.67770249132643708238919796611b-23*%i)-1.31549862931202979917891670099b0
8.67770249132643708239032323801b-23*%i-1.31549862931202979917841983362b0
[x^4+a*x^2+b*x+c,
x^4-4.90268221512206394838567282523b0*x^2
-3.79286913672985525962281904563b0*x+5.0b-1]
dist: 5.26306169342786570228478616546b-22
1 multi-roots:
2.61198261984724332021478496099b-1-3.73328701456410231125396194552b-23*%i
3.73328701456410231125394316086b-23*%i+2.61198261984724332021514776701b-1
[x^4+a*x^2+b*x+c,
x^4+7.12406849810570318368614385351b0*x^2
-3.79286913672985525962281904563b0*x+5.0b-1]
dist: 8.30135823846948556696946435113b-23
1 multi-roots:
(-8.35793034501348062623930605099b-22*%i)-1.26839633131681513709955065996b0
8.35793034501348062625014797815b-22*%i-1.26839633131681513709478200116b0
[x^4+a*x^2+b*x+c,
x^4-4.51570275957450820467053128978b0*x^2
-3.29286913672985525962281904563b0*x+5.0b-1]
dist: 5.05314820570176695705561472995b-21
1 multi-roots:
3.49188104507532910404015377995b-23*%i+2.98842309414307218065285325202b-1
2.98842309414307218065613316054b-1-3.49188104507532910403992032144b-23*%i
[x^4+a*x^2+b*x+c,
x^4+5.33076228416614959738464811917b0*x^2
-3.29286913672985525962281904563b0*x+5.0b-1]
dist: 3.35343542885119089772093274405b-22
1 multi-roots:
(-5.10955124220135577792387388171b-22*%i)-1.1495332358829680004419098722b0
5.10955124220135577792837906846b-22*%i-1.14953323588296800043902649925b0
[x^4+a*x^2+b*x+c,
x^4-3.58590105561473348013717487389b0*x^2
-2.16812942838935673533957135587b0*x+5.0b-1]
dist: 3.05910773620696643491835445611b-21
1 multi-roots:
4.29760429000264513227806165013b-1-4.9825698827127380563723750383b-17*%i
4.9825698827127377393878597465b-17*%i+4.29760429000264633921325792835b-1
[x^4+a*x^2+b*x+c,
x^4+2.15309806062091933677807113863b0*x^2
-2.16812942838935673533957135587b0*x+5.0b-1]
dist: 1.56516218758945019086233255227b-16
1 multi-roots:
(-1.09877730976256428518192469349b-21*%i)-1.00617122801969079931775349078b0
1.09877730976254185519229728272b-21*%i-1.00617122801969079931174334307b0
[x^4+a*x^2+b*x+c,
x^4-2.54325618867219212419273854599b0*x^2
-1.04338972004885821105632366611b0*x+5.0b-1]
dist: 6.39930635240535273049586094033b-21
1 multi-roots:
6.38943104246272475524586123887b-1-7.84439082707517235056253391037b-20*%i
7.84439082708190432497441062349b-20*%i+6.38943104246272476186112486434b-1
[x^4+a*x^2+b*x+c,x^4-1.04338972004885821105632366611b0*x+5.0b-1]
dist: 6.79875661664383746823339492112b-19
1 multi-roots:
(-4.8300774831537619911512151596b-22*%i)-9.65994523816328465899519279333b-1
4.8300774707834158989233615011b-22*%i-9.65994523816328465899094849168b-1
[x^4+a*x^2+b*x+c,
x^4-2.26361409430139643717899746966b0*x^2
-7.67624175513389459396781893817b-1*x+5.0b-1]
dist: 1.05514307182643029247458493579b-21
1 multi-roots:
1.119137819361247989040796719b-15*%i+6.94963770046457076218883279372b-1
6.94963770046457864444214852857b-1-1.11913781936124711403020335568b-15*%i
[x^4+a*x^2+b*x+c,
x^4-4.13672885217763849890412496005b-1*x^2
-7.67624175513389459396781893817b-1*x+5.0b-1]
dist: 2.37301011549313127333803802423b-15
1 multi-roots:
6.33988972265475161381443097681b-22*%i-9.23416931585132456001273979169b-1
(-6.33988972265475161353799507713b-22*%i)-9.23416931585132455992195460933b-1
[x^4+a*x^2+b*x+c,
x^4-1.9717229852393250817227538327b0*x^2
-4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 9.16663850192252560176095861665b-21
1 multi-roots:
7.49558352273654755133389531399b-1-9.2980021082756567072476690926b-18*%i
9.29800210827546330169426966815b-18*%i+7.49558352273654772974086850138b-1
[x^4+a*x^2+b*x+c,
x^4-7.95576487395121227355033572426b-1*x^2
-4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 2.57701737215882295851533416416b-17
1 multi-roots:
1.03401156272799309188393045594b-19*%i-8.83251837810700396804617510575b-1
(-1.03401156272799304752829508618b-19*%i)-8.83251837810700395251773718947b-1
[x^4+a*x^2+b*x+c,
x^4-1.69948573504594766924564690096b0*x^2
-2.45929315488960353868620060764b-1*x+5.0b-1]
dist: 1.56655387320991285239749179312b-18
1 multi-roots:
7.96303936790265302555727618361b-1-3.40419552049028707429902475843b-21*%i
3.40419552049944803968351960399b-21*%i+7.96303936790265302568553128937b-1
[x^4+a*x^2+b*x+c,
x^4-1.1137806681726386454165880796b0*x^2
-2.45929315488960353868620060764b-1*x+5.0b-1]
dist: 1.45206029523254927419705052618b-20
1 multi-roots:
(-5.70609945371498130990553057283b-21*%i)-8.40896415253714543050874201664b-1
5.70609945371498131018153821252b-21*%i-8.40896415253714543011376750803b-1
[x^4+a*x^2+b*x+c,x^4-1.41421356237309504880168872421b0*x^2+5.0b-1]
dist: 4.11130989883287052898186383085b-20
2 multi-roots:
8.40896415253714543012849282445b-1-6.53339775168399828589095914175b-21*%i
6.53339775168399828561495150206b-21*%i+8.40896415253714543049401670022b-1
[x^4+a*x^2+b*x+c,x^4-1.41421356237309504880168872421b0*x^2+5.0b-1]
dist: 3.88177560186052570543478329165b-20
1 multi-roots:
2.15893555484106906165912452485b-16*%i+8.83251837810700183846822208909b-1
8.83251837810700608209569020613b-1-2.15893555484106797595047500567b-16*%i
[x^4+a*x^2+b*x+c,
x^4-1.69948573504594766924564690097b0*x^2
+2.45929315488960353868620060765b-1*x+5.0b-1]
dist: 6.05412132418777109501344790091b-16
1 multi-roots:
6.37787066397529175282449379547b-20*%i-7.96303936790265302727748521798b-1
(-6.37787066396800919273345251121b-20*%i)-7.963039367902653023965322255b-1
[x^4+a*x^2+b*x+c,
x^4-1.1137806681726386454165880796b0*x^2
+2.45929315488960353868620060765b-1*x+5.0b-1]
dist: 3.54929751662912295409283425277b-19
1 multi-roots:
9.23416931585132422243653899598b-1-6.27472747421841058511382108183b-18*%i
6.2747274742183885879114162131b-18*%i+9.23416931585132489749815540504b-1
[x^4+a*x^2+b*x+c,
x^4-1.9717229852393250817227538327b0*x^2
+4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 6.86627313685598614587092899877b-17
1 multi-roots:
(-1.99636848170413194768633600558b-15*%i)-7.49558352273655235457557488862b-1
1.99636848170413258293302923552b-15*%i-7.49558352273654292649918892678b-1
[x^4+a*x^2+b*x+c,
x^4-7.95576487395121227355033572426b-1*x^2
+4.91858630977920707737240121529b-1*x+5.0b-1]
dist: 4.10254003056179335802065997111b-15
1 multi-roots:
9.65994523816328465899290503781b-1-2.66934379792368063978571497834b-22*%i
2.66934379792368063978568047301b-22*%i+9.65994523816328465899323624718b-1
[x^4+a*x^2+b*x+c,
x^4-2.26361409430139643717899746966b0*x^2
+7.67624175513389459396781893818b-1*x+5.0b-1]
dist: 5.34895175649003922093343053329b-22
1 multi-roots:
(-4.29381502396318990315032909949b-19*%i)-6.9496377004645747104791949945b-1
4.29381502396139055973603979073b-19*%i-6.94963770046457469615178632778b-1
[x^4+a*x^2+b*x+c,
x^4-4.13672885217763849890412496004b-1*x^2
+7.67624175513389459396781893818b-1*x+5.0b-1]
dist: 1.67039524946424430892380307845b-18
1 multi-roots:
1.00617122801969079926553973356b0-1.60075239406241646868279412713b-19*%i
1.60075239406241646862513736191b-19*%i+1.00617122801969079936395710029b0
[x^4+a*x^2+b*x+c,
x^4-2.54325618867219212419273854599b0*x^2
+1.04338972004885821105632366611b0*x+5.0b-1]
dist: 3.34936273279166823298162816124b-19
1 multi-roots:
(-3.77185753862812953984534199095b-24*%i)-6.38943104246272475855369706517b-1
3.77185746499376146262928840184b-24*%i-6.38943104246272475855328903805b-1
[x^4+a*x^2+b*x+c,x^4+1.04338972004885821105632366611b0*x+5.0b-1]
dist: 4.14942038334135943047182928317b-23
1 multi-roots:
1.14953323588296799935017434052b0-1.25168566885221419821833160929b-18*%i
1.25168566885221419738370885239b-18*%i+1.14953323588296800153076203092b0
[x^4+a*x^2+b*x+c,
x^4-3.58590105561473348013717487389b0*x^2
+2.16812942838935673533957135587b0*x+5.0b-1]
dist: 3.31991426545575543023864496728b-18
1 multi-roots:
(-2.58295399767902797007565021885b-19*%i)-4.29760429000264573649957309613b-1
2.58295399767902797030577470367b-19*%i-4.29760429000264573499174648236b-1
[x^4+a*x^2+b*x+c,
x^4+2.15309806062091933677807113863b0*x^2
+2.16812942838935673533957135587b0*x+5.0b-1]
dist: 5.38146323165938018060210022577b-19
1 multi-roots:
1.2683963313168151249994797521b0-1.02128852191271037829509986173b-17*%i
1.02128852191271037157319651058b-17*%i+1.26839633131681514919485290902b0
[x^4+a*x^2+b*x+c,
x^4-4.51570275957450820467053128978b0*x^2
+3.29286913672985525962281904563b0*x+5.0b-1]
dist: 3.16643045115257576716717302982b-17
1 multi-roots:
8.98733613243950412511068357208b-21*%i-2.98842309414307218067753665713b-1
(-8.98733613243950412511912321165b-21*%i)-2.98842309414307218063144975543b-1
[x^4+a*x^2+b*x+c,
x^4+5.33076228416614959738464811916b0*x^2
+3.29286913672985525962281904563b0*x+5.0b-1]
dist: 1.85561005632311052538934339445b-20
1 multi-roots:
1.315498629312029794561387534b0-3.5935851244110502148689984252b-18*%i
3.5935851244110502062082304748b-18*%i+1.31549862931202980379594900061b0
[x^4+a*x^2+b*x+c,
x^4-4.90268221512206394838567282523b0*x^2
+3.79286913672985525962281904563b0*x+5.0b-1]
dist: 1.1701817878693290753327731022b-17
1 multi-roots:
2.54178527994425949722680256301b-23*%i-2.61198261984724332021510649724b-1
(-2.54178527994425949722681244282b-23*%i)-2.61198261984724332021482623075b-1
[x^4+a*x^2+b*x+c,
x^4+7.12406849810570318368614385351b0*x^2
+3.79286913672985525962281904563b0*x+5.0b-1]
dist: 5.80496512960753920779203206589b-23
[T,F]: [92,175]
Evaluation took 13.7400 seconds (13.7800 elapsed) using 1299.854 MB.
(%o2) [[c = root(c,1),b = root(126976*c^3-135*b^4,1),
root(8*a*c-9*b^2-2*a^3,1) <= a,true],
[root(c,1) < c,b < root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true],
[root(c,1) < c,b = root(b,1),
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,1)
<= a,true],
[root(c,1) < c,root(b,1) < b,
root(256*c^3-128*a^2*c^2+(144*a*b^2+16*a^4)*c-27*b^4-4*a^3*b^2,2)
<= a,true]]
CAD on maxima
SARAG https://github.com/andrejv/maxima/tree/master/share/contrib/sarag を待ちきれない方のために作ってみました．ただし，lifting のネックである根の分離は realroots に任せ，符号判定も近似的です．
使用例（wxmaxima 等で実行すると見易いと思います）
(vpeds ( [x,a,b],0,0,3,0 ),cnnm(cad ( '( a*x^2+b<0 and x>0 )) ) ); (vpeds ( [x,a,b],0,0,11,0 ),cnnm(cad ( '( a*x^2+b<0 and x>0 )) ) ); (vpeds ( [x,a,b],0,0,11,0 ),cnnm(cad ( '( a*x^5+b*x+1<0 and x>0 )) ) ); (vpeds ( [x,a,b],0,0,3,0 ),cnnm(cad ( '( sqrt(x)<a*x+b ) )) );
(%i1) (vpeds ( [x,a],0,0,11,0 ),cnnm(cad ( '( x*(x-2)*(x-3)*(x-5)*(x-7)=a ) )) );
connecting cells.
connecting cells.
matrix([a < root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,1),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)],
[a = root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,1),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)],
[root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,1) < a
and a < root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,
2),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,3)],
[a = root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,2),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,3)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,4)],
[root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,2) < a
and a < root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,
3),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,3)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,4)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,5)],
[a = root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,3),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,3)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,4)],
[root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,3) < a
and a < root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,
4),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,3)],
[a = root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,4),
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)
or x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,2)],
[root(3125*a^4+20592*a^3-16391308*a^2+244684800*a+2540160000,4) < a,
x = root(x^5-17*x^4+101*x^3-247*x^2+210*x-a,1)])
Evaluation took 2.3600 seconds (2.3600 elapsed) using 540.776 MB.
なお，例によって，コードの大半は結果の簡約に費やされています．
/* 20170324 22:04:46 */
showtime:on$
display2d:false$
prton2():=(prt:print,prt2:print)$
prton():=(prt:print)$
prtoff():=kill(prt,prt2)$
vpeds(v,p,e,d,s):=block([],
%vv:v,
algexact:true,
bsolve:false,
dtype:0,
if p=1 then (prtoff(),prton())
elseif p=2 then prton2()
else (ratprint:off,prtoff()),
if e=0 then delv0 (L,vv):=delv0s (L,vv)
else delv0 (L,vv):=delv0m (L,vv),
if d=0 then getsample (a,b,c):=getsample0 (a,b,c)
elseif d=1 then getsample (a,b,c):=getsample1 (a,b,c)
elseif d=11 then (dtype:11,getsample (a,b,c):=getsample11 (a,b,c))
elseif d=2 then getsample (a,b,c):=getsample2 (a,b,c)
else (getsample (a,b,c):=getsample1 (a,b,c), bsolve:true),
if s=1 then dispsample:true else dispsample:false,
usersimp:simp,
simp:false
)$
matchdeclare([aa,bb,cc],true)$
kill("implies")$
infix("implies",60,40)$
(x implies y):=(not(x) or y)$
%P:[]$
getlcP(f):=block([ff:expand(f),d,lc],
d:hipow(ff,%v),lc:coeff(ff,%v,d),/*print(%v),*/
if member(%v,listofvars(ff)) then %P:append(%P,[ff])
elseif listofvars(ff)#[] then %C:append(%C,[ff]),
if not(constantp(lc)) then
(%C:append(%C,[lc]),getlcP(ff-lc*(%v)^d))
)$
gcd2(f,g,v):=block([p1:f,p2:g,p3:1,k],
for k:0 unless p3=0 do
(p3:remainder (p1,p2,v),p1:p2,p2:p3),factor(content(p1,v)[2]))$
sqfree(f,v):=block([],
if member(v,listofvars (f)) then
content( divide (f,gcd2 (f,diff(f,v),v),v) [1] ,v) [2]
else f
)$
sqfreem(L,v):=block([],
if is(L=[]) then [] else
listify(setify( map(lambda([f],sqfree (f,v)),L) ))
)$
resultant1(f,v):=block ([g:sqfree(f,v)],
resultant (g,diff (g,v),v ) )$
resultant2(f1,f2,v):=block (
[g1: quotient (f1,gcd2 (f1,f2,v),v),
g2: quotient (f2,gcd2 (f1,f2,v),v)],
resultant (g1,g2,v ) )$
delv0m(L,v):=block([P:[],PP,i,lp],
delv0 (LL,vv):=delv0s (LL,vv),
%C:[],%v:v,
map(lambda([f],if member(%v,listofvars(f)) then P:append(P,[f])
else %C:append(%C,[f])),L),
PP:map("[",P),prt(PP),
/*
PP:map(lambda([f],(%P:[],getlcP(f),%P)),P),prt(PP),
*/
map(lambda([f],%C:append(%C,[resultant1(f,%v)])),flatten(PP)),
lp:length(PP),/*prt([%C,P]),*/
%C:append(%C,flatten(makelist(makelist(makelist(makelist(resultant2(PP[i][ii],PP[j][jj],%v),ii,1,length(PP[i])),jj,1,length(PP[j])),j,i+1,lp),i,1,lp-1))),
listify(setify(delete(false,map(lambda([f],if constantp(f) then false else f),%C))))
)$
delv0s(L,v):=block([P:[],PP,i,lp],
%C:[],%v:v,
map(lambda([f],if member(%v,listofvars(f)) then P:append(P,[f])
else %C:append(%C,[f])),L),
/*
PP:map("[",P),print(PP),
*/
PP:map(lambda([f],(%P:[],getlcP(f),%P)),P),prt(PP),
map(lambda([f],%C:append(%C,[resultant1(f,%v)])),flatten(PP)),
lp:length(PP),/*prt([%C,P]),*/
%C:append(%C,flatten(makelist(makelist(makelist(makelist(resultant2(PP[i][ii],PP[j][jj],%v),ii,1,length(PP[i])),jj,1,length(PP[j])),j,i+1,lp),i,1,lp-1))),
listify(setify(delete(false,map(lambda([f],if constantp(f) then false else f),%C))))
)$
load ("grobner")$
delv(L,vl):=block([p:[sqfreem(L,vl[1])],k],
for k:1 thru length(vl)-1 do
p:append(p,[sqfreem( delv0(last(p),vl[k]) ,vl[k+1])]),
p:append(rest (p,-1),
[listify(setify(
poly_normalize_list (last (p), [last (vl) ] )))]),
reverse(p)
)$
factors(f):=block ([g:factor(f)],
if atom(g) then g
elseif op(g)="+" then g else
(if op(g)="/" then g:args(g)[1],
if atom(g) then g
elseif op(g)="-" then g:-g,
if atom(g) then g
elseif op(g)="*" then
delete(0,
map(lambda([e],
if constantp(e) then 0
elseif atom(e) then e
elseif op(e)="^" then args(e)[1] else e),
args(g)))
elseif op(g)="^" then args(g)[1]
else g
)
)$
factorsm(L):=listify(setify(flatten(map(factors,L))))$
delv(L,vl):=block([p:[factorsm(L)],k],
for k:1 thru length(vl)-1 do
p:append(p,[factorsm( delv0(last(p),vl[k]) )]),
p:append(rest (p,-1),[factorsm(last(p))]),
reverse(p)
)$
realonly:on$
getsample0(sub,p,v):=block([p1,p2,q,k],
p1:listify(setify(flatten(map(lambda([f],algsys([f],listofvars(f))),subst(sub[2],p))))),
if p1=[] then p2:[0]
else (p1:sort(map(rhs,p1),"<"),q:map("[",p1),
p2:sort(listify(setify(append(p1,(append([first(p1)-1],p1)+append(p1,[last(p1)+1]))/2))),"<")),
makelist([[0.5*k],[v=p2[k]]],k,1,length(p2))
)$
getsample1(sub,p,v):=block([p1,sf,rt,T:[],S:[],p2,q,k],
p1:apply(append,map(lambda([f],
(sf: subst(sub[2],f),
rt:if constantp (sf) then []
else flatten(solve( sf ,v)),
rt:if rt=[] then []
else flatten(map(lambda([e],if is(imagpart(rhs(e))=0) then e else [] ), rt )),
prt2([p,sf,v,rt]),
rt:if rt=[] then []
else (rt:sort (map (rhs,rt),"<" ), makelist([[f,k ],rt[k]],k,1,length (rt) ))
)),p)),prt2("p1:"),prt2(p1),
if p1=[] then p2:[0]
else
(map (lambda ( [e], if not(member (e [2],T ))
then (T:append(T,[e[2]]),S:append(S,[e])) ),p1 ),
prt2 ("S:"), prt2(S),
if S=[] then (p1:[],p2:[0]) else (
S:sort (S,lambda ( [u,t], u[2]<t[2] ) ),
[q,p1]:[map (first,S),map (last,S)],
prt2 ("q:"), prt2(q),prt2 ("p1:"), prt2(p1),
p2:sort(listify(setify(append(p1,
(append([first(p1)-1],p1)+append(p1,[last(p1)+1]))/2))),"<")
) ),
prt2 ("p2:"), prt2(p2),
makelist([[
if p1=[] then true
elseif is(k=1) then v<first(q)
elseif is(k=length (p2) ) then last(q)<v
elseif evenp(k) then v=q[k/2]
else (q[(k-1)/2]<v and v<q[(k+1)/2])
],[v=p2[k]]],k,1,length(p2))
)$
load ("grobner") $
getrt(sub,f,v):=block([eqv,subv:listofvars (sub),p1,sf,rt00:[],rt01:[],rt0,rt:[],sl,k],prt2(sub,f,v),
p1:makelist (
if atom(sub [1] [k]) then sub[2][k][1]
elseif is (op (sub [1] [k] )="=") then args(rhs(sub [1] [k] ))[1]
else sub[2][k][1],
k,1,length (sub [2] ) ),
p1:append (p1, [f] ),prt2 ("p1:",p1),
sf:poly_reduced_grobner (p1,append (subv,[v] ) ),prt2 ("sf:",sf),
map (lambda ([e],if listofvars (e)=[v] then rt00:append (rt00, [e] ) else rt01:append (rt01, [e] ) ),sf ),
if rt00=[] then rt:[]
else
(rt0:apply(realroots,rt00),
if rt0=[] then rt:[] else
(sub2:apply("+",map(lambda ( [e],
abs (lhs (e[2])-rhs (e[2]) )
),sub [2] )),
/* cf. apply("+",[]) = 0 */
for eqv in rt0 do
(sl:algsys(subst (eqv,rt01),subv),
ans:apply(min, map (lambda ( [e], subst (e,sub2) ),sl )),
if is(ans<10^(-5)) then rt:append (rt, [eqv] ),
prt2(eqv,sl,"min:",float(ans)) )
)
),
rt:if rt=[] then []
else (rt:sort (map (rhs,rt),"<" ),
makelist([root(f,k),[last(rt00),rt[k]]],k,1,length (rt) )),
prt2("rt:",rt),rt
)$
rootsepsilon:1.0e-10$
getsample11(sub,p,v):=block([p1,S:[],T:[],p2,p0,q,k],
p1:apply(append,map(lambda([f],if member (v,listofvars (f) ) then getrt(sub,f,v) else []),p)),
prt2("p1 in 11:",p1),
if p1=[] then p2:[0]
else
(map (lambda ( [e], if not(member (e[2][2],T ))
then (S:append(S,[e]),T:append(T,[e[2][2]])) ),p1 ),
prt2 ("S:",S),
if S=[] then (p1:[],p2:[0]) else (
S:sort (S,lambda ( [u,t], u[2][2]<t[2][2] ) ),
[q,p0]:[map(first,S),map(last,S)],
p0p:map(first,p0),p1:map(last,p0),
prt2 ("q:"), prt2(q),prt2 ("p1:"), prt2(p1),
p2:sort(listify(setify(append(p1,
(append([first(p1)-1],p1)+append(p1,[last(p1)+1]))/2))),"<")
) ),
prt2 ("p2:"), prt2(p2),
makelist([[
if p1=[] then true
elseif is(k=1) then v<first(q)
elseif is(k=length (p2) ) then last(q)<v
elseif evenp(k) then v=q[k/2]
else (q[(k-1)/2]<v and v<q[(k+1)/2])
],[
[(if oddp(k) then v-p2[k] else p0p[k/2]),
(if oddp(k) then v=p2[k] else v=p1[k/2])]
]],k,1,length(p2))
)$
getsample2(sub,p,v):=block([sub1,rt,q,p1,p2,k],
sub1:delete([],map(lambda ( [e], if atom (e) then [] elseif op(e)="=" then e else []), sub[1] ) ),
rt:subst(sub1,p),
rt:flatten(map (lambda ( [e], if constantp (e) then [] else e ), rt )),
if rt=[] then rt:[]
else rt:flatten(map (lambda ( [e],solve (e,v) ),rt ) ),
if rt=[] then rt:[]
else
(rt:map (lambda ( [e],[e,ratsimp(subst (sub [2],e )) ] ), map(rhs,rt) ),
rt:delete([],map(lambda([e],if imagpart(e [2])=0 then e else [] ), rt )),
rt:listify(setify(rt))
),
if rt=[] then (p1:[],p2:[0])
else (
rt:sort (rt,lambda ( [u,t], u[2]<t[2] )),/*print (rt),*/
[q,p1]:[map (first,rt),map (last,rt)],
p2:sort(listify(setify(append(p1,
(append([first(p1)-1],p1)+append(p1,[last(p1)+1]))/2))),"<")
),
/*prt2 ("p2:"), prt2(p2),*/
makelist([[
if p1=[] then true
elseif is(k=1) then v<first(q)
elseif is(k=length (p2) ) then last(q)<v
elseif evenp(k) then v=q[k/2]
else (q[(k-1)/2]<v and v<q[(k+1)/2])
],[v=ratsimp( p2[k] )]],k,1,length(p2))
)$
is2(f):=
if atom (f) then f
elseif member (op (f), ["or","and"] ) then map (is,f)
else is (f)$
kill("Lo","Lc","Go","Gc","Eq")$
infix("Lo",60,40)$
infix("Lc",60,40)$
infix("Go",60,40)$
infix("Gc",60,40)$
infix("Eq",60,40)$
(x Lo y):=if member(extd,listofvars([x,y])) then false
else float(10^(-5)+x<y)$
(x Lc y):=if member(extd,listofvars([x,y])) then false
else float(x<y+10^(-5))$
(x Go y):=if member(extd,listofvars([x,y])) then false
else float(x>y+10^(-5))$
(x Gc y):=if member(extd,listofvars([x,y])) then false
else float(10^(-5)+x>y)$
(x Eq y):=if member(extd,listofvars([x,y])) then false
else abs(float(x-y))<10^(-5)$
kill("implies")$
infix(implies,60,40)$
"implies"(x,y):=((not(x)) or (y))$
defrule(imp2m, (aa) implies (bb), mor ( mnot((aa)),(bb) ));
defrule(noteq2m, (aa) # (bb), mnot (equal((aa),(bb))));
defrule(symneq2m, notequal((aa),(bb)), mnot (equal((aa),(bb))));
cad_monic(F0):=block ( [F,mpp:[],p,sx,sxl,i,k],
F:addD(subst(["="=equal,"and"=mand,"or"=mor,"not"=mnot],F0)),
F:apply1(F,imp2m,noteq2m,symneq2m),
simp:on,
for vk in %vv do F:sepaF(F,vk),
for vk in %vv do F:sepaS(sepaS(F,vk),vk),
for vk in %vv do F:sepaF(F,vk),
F:ev(subst([mand="and",mor="or",mnot="not"],F),eval),prt(F),
F:scanmap(lambda([f],if atom(f) then f
elseif member(op(f),["<","<=",">",">=",equal,notequal]) then
(p:dnf(sepalcv(f,%vv[1])),prt(p),
if op(p)="or" then (mpp:append (mpp, [args(p)] ),
concat(mp,length (mpp) ))
else f
)
else f),F),
print([F,mpp]),
if mpp=[] then [F] else
(
i:0,
sxl:map(lambda([n],(i:i+1,product(concat(sx,i)-k,k,1,n))),map(length,mpp)),
sxl:solve(sxl),
map(lambda([sxl],subst(ev(subst(mppp=mpp,subst(sxl,makelist(concat(mp,i)=mppp[i][concat(sx,i)],i,1,length(mpp)))),eval),F)),sxl)
)
) $
/* epsilon sgn-det ver. */
cad (F0) :=block ([F,vk,P2:{},pp,vv:reverse(%vv),pn:[[[],[]]],pn0],
F:addD(subst(["="=equal,"and"=mand,"or"=mor,"not"=mnot],F0)),
F:apply1(F,imp2m,noteq2m,symneq2m),
simp:on,
for vk in %vv do F:sepaF(F,vk),
for vk in %vv do F:sepaS(sepaS(F,vk),vk),
for vk in %vv do F:sepaF(F,vk),
F1:subst(["<"="Lo",">"="Go","<="="Lc",">="="Gc",equal="Eq"],F),
F:ev(subst([mand="and",mor="or",mnot="not"],F),eval),
F1:ev(subst([mand="and",mor="or",mnot="not"],F1),eval),
prt2(F1),
prt(F),
/*
F:subst(["<"="Lo",">"="Go","<="="Lc",">="="Gc",equal="Eq"],F),
*/
scanmap (lambda ( [f], if atom (f) then f
elseif member (op (f), ["<","<=",">",">=",equal,notequal]) then P2:append (P2, {lhs (f)-rhs (f) } ) else f ),F ),
simp:on,P2:listify (P2),
/*P2:map(lambda ([f], sqfree (f,(%vv)[1])),P2), */
pp:map(factor,delv(P2,%vv)), prt(pp),%pp:pp,
for n:1 thru length (vv) do
pn:apply(append,map(lambda([e],map(lambda([t],map(append,e,t)),getsample(e,pp[n],vv[n]))),pn)),prt(length (pn)),pn00:pn,
pn:
if is(dtype=11) then
delete(false,map(lambda([l],if is2( (subst( map(last,l[2]), F1)) ) then (if is(dispsample=true) or is(bsolve=true) then l else l[1])),pn))
else
delete(false,map(lambda([l],if is2( (subst(l[2], F1)) ) then (if is(dispsample=true) or is(bsolve=true) then l else l[1])),pn)),
/*print(pn),*/
simp:usersimp,
if bsolve=true and dispsample=true then cad2s (pn)
elseif bsolve=true and dispsample=false then cad2 (pn)
else pn
)$
cadm(f):=block([usr_display2d],
thisout:cad(f),
usr_display2d:display2d,
set_display('xml),
print(apply(matrix,thisout)),
display2d:usr_display2d,return(done)
)$
addD0(f):=block([DN],
if atom(f) then return(f)
elseif member(op(f),["<",">","<=",">=",equal]) then
(DN:{},
scanmap(lambda([t],if atom(t) then t
else (DN:ev(append(DN,{notequal(ratsimp(denom(t)),0)}),simp),t)),
f),prt2([f,DN]),
DN:ev(substpart("and",DN,0),eval),prt2([f,DN]),
if is(DN=false) then return(false)
else return( mand((f) , (apply1(DN,symneq2m))) ))
else return(f)
)$
addD(F):=block([addD0,S],
prt2("addD:"),
S:scanmap(addD0,F,bottomup),prt2(S),
return(S)
)$
sepaF0(f,v):=block([p,DN,NU],
if atom(f) then return(f)
elseif
member(op(f),["<",">","<=",">=",equal]) then
(p:factor(lhs(f)-rhs(f)),
NU:num(p),DN:denom(p),prt2([p,NU,DN]),
if member(v,listofvars(DN)) and member(op(f),["<",">"])
then return(apply(op(f),[NU*DN,0]))
elseif member(v,listofvars(DN)) and member(op(f),["<=",">="])
then return( mand( (apply(op(f),[NU*DN,0])) , (mnot(equal(DN,0))) ) )
elseif member(v,listofvars(DN)) and op(f)=equal
then return( mand( (equal(NU,0)) , (mnot(equal(DN,0))) ) )
else return(f))
else return(f))$
sepaF(F,v):=block([S],
prt2("sepaF for",v,":"),
S:scanmap(lambda([s],sepaF0(s,v)),F),
prt2(S),return(S)
)$
sepaSlocal(ff,v):=block([sepaSlocal0,getS,getSqrt,p,Sqrt,A,B,C],
sepaSlocal0(f):=block([],
if atom(f) or is(getS=true) then return(f)
elseif op(f)=sqrt and member(v,listofvars(f))
then (getS:true,getSqrt:f,return(Sqrt))
else return(f)),
if atom(ff) then return(ff)
elseif member(op(ff),["<",">","<=",">=",equal]) then
(
getS:false,p:scanmap(sepaSlocal0,lhs(ff)-rhs(ff)),
(if is(getS=true) then
(p:subst(getSqrt=Sqrt,p),
[A,B]:[coeff(expand(p),Sqrt,1),getSqrt^2],C:expand(A*Sqrt-p),
return(mand( [A,B,C,op(ff)] , (getSqrt^2 >= 0) )))
else return(ff))
)
else return(ff))$
defrule(sepaS1,[aa,bb,cc,equal],
mand(mor(mand(aa >= 0,cc >= 0),mand(aa < 0,cc <= 0)),/*bb >= 0,*/
equal(cc^2-aa^2*bb,0))
)$
defrule(sepaS2,[aa,bb,cc,"<="],
mand(mor(mand(aa < 0,aa^2*bb-cc^2 >= 0),
mand(cc >= 0,aa^2*bb-cc^2 <= 0))/*,bb >= 0*/)
)$
defrule(sepaS3,[aa,bb,cc,">="],
mand(mor(mand(aa > 0,cc^2-aa^2*bb <= 0),
mand(cc <= 0,cc^2-aa^2*bb >= 0))/*,bb >= 0*/)
)$
defrule(sepaS4,[aa,bb,cc,"<"],
mand(mor(mand(aa < 0,cc > 0),mand(aa < 0,aa^2*bb-cc^2 > 0),
mand(cc > 0,aa^2*bb-cc^2 < 0))/*,bb >= 0*/)
)$
defrule(sepaS5,[aa,bb,cc,">"],
mand(mor(mand(aa > 0,cc < 0),mand(aa > 0,cc^2-aa^2*bb < 0),
mand(cc < 0,cc^2-aa^2*bb > 0))/*,bb >= 0*/)
)$
sepaS(F,v):=block([S],
prt2("sepaS:"),
S:scanmap(lambda([s],sepaSlocal(s,v)),expand(F)),
prt2(S),
S:expand(apply1(S,sepaS1,sepaS2,sepaS3,sepaS4,sepaS5)),
prt2(S),return(S)
)$
sepalcv(f,v):=block([pp,dg,lc],
pp:expand(lhs(f)-rhs(f)),dg:hipow(pp,v),lc:ratcoef(pp,v,dg),
if atom(f) then return(f)
elseif member(op(f),["<",">","<=",">=",equal,notequal]) then
(if dg=0 then return(f)
else return(
(notequal(lc,0) and f) or
(equal(lc,0) and sepalcv(apply(op(f),[expand(pp-lc*v^dg),0]),v))
))
else return(f))$
/* cnf dnf */
kill("Or");
kill("And");
infix(Or,60,40)$
infix(And,60,40)$
matchdeclare([aa,bb,cc],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)$
cad2(F):=map(lambda([L], (U:[], map(lambda([f],
(if atom (f) then f
elseif op(f)="=" then
(sol:solve(subst(U,[first(rhs(f))]),lhs(f)),prt2(sol),
sol:lhs(f)=(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), map(rhs,sol) )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(rhs(f))][1],
U:append (U, [sol] ),sol)
elseif op(f)="<" and listp(rhs (f)) then
(sol:solve(subst(U,[first(rhs(f))]),lhs(f)),prt2(sol),
sol:lhs(f)<(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), map(rhs,sol) )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(rhs(f))][1])
elseif op(f)="<" and listp(lhs (f)) then
(sol:solve(subst(U,[first(lhs(f))]),rhs(f)),prt2(sol),
sol:(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), map(rhs,sol) )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(lhs(f))][1]<rhs(f))
elseif op(f)="and" then
(sol1:solve(subst(U,[first(lhs((args(f))[1]))]),rhs((args(f))[1])), sol1:map(rhs,sol1),
sol2:solve(subst(U,[first(rhs((args(f))[2]))]),lhs((args(f))[2])), sol2:map(rhs,sol2),
prt2([sol1,sol2]),
(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), sol1 )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(lhs((args(f))[1]))][1] < rhs((args(f))[1]) and
lhs((args(f))[2]) < (sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), sol2 )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(rhs((args(f))[2]))][1]
)
else f)
),L[1])) ),F)$
cad2s(F):=map(lambda([L], [(U:[], map(lambda([f],
(if atom (f) then f
elseif op(f)="=" then
(sol:solve(subst(U,[first(rhs(f))]),lhs(f)),prt2(sol),
sol:lhs(f)=(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), map(rhs,sol) )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(rhs(f))][1],
U:append (U, [sol] ),sol)
elseif op(f)="<" and listp(rhs (f)) then
(sol:solve(subst(U,[first(rhs(f))]),lhs(f)),prt2(sol),
sol:lhs(f)<(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), map(rhs,sol) )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(rhs(f))][1])
elseif op(f)="<" and listp(lhs (f)) then
(sol:solve(subst(U,[first(lhs(f))]),rhs(f)),prt2(sol),
sol:(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), map(rhs,sol) )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(lhs(f))][1]<rhs(f))
elseif op(f)="and" then
(sol1:solve(subst(U,[first(lhs((args(f))[1]))]),rhs((args(f))[1])), sol1:map(rhs,sol1),
sol2:solve(subst(U,[first(rhs((args(f))[2]))]),lhs((args(f))[2])), sol2:map(rhs,sol2),
prt2([sol1,sol2]),
(sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), sol1 )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(lhs((args(f))[1]))][1] < rhs((args(f))[1]) and
lhs((args(f))[2]) < (sort( delete([],map(lambda([e], if imagpart(subst(L[2],e))=0 then [e,subst(L[2],e)] else []), sol2 )) , lambda ( [ss,tt], ss[2]<tt[2] ) ))[last(rhs((args(f))[2]))][1]
)
else f)
),L[1])),L [2] ] ),F)$
cnn(LL0):=block([n,k,LL:subst("="=equal,LL0),LA,LB,LB0,s,t,L0,M],
print("connecting cells."),
if is(dispsample=true)
then
(n:length(LL[1][1]),
for k:n step -1 thru 1 do
(
LL:append(LL,[[append(makelist(0,n),[1]),makelist([0,0],n)]]),
L0:LL[1][1],t:delete(L0[k],L0),s:false,
LL:delete(0,map(lambda([L],
(LA:L[1],
if is(t=delete(LA[k],LA)) then (s:(s or LA[k]),LB:L[2],0)
else (M:subst(L0[k]=s,L0),L0:LA,t:delete(LA[k],LA),s:LA[k],LB0:LB,LB:L[2],append([M],[LB0]))
)
),LL))
),
thisout0:LL,
thisout:map(lambda([L],[map(lambda([e],if atom(e) then e elseif op(e)="or" then cnn0(e) else e),L[1]),L[2]] ),LL)
)
else
(n:length(LL[1]),
for k:n step -1 thru 1 do
(
LL:append(LL,[append(makelist(0,n),[1])]),
L0:LL[1],t:delete(L0[k],L0),s:false,
LL:delete(0,map(lambda([L],
if is(t=delete(L[k],L)) then (s:(s or L[k]),0)
else (M:subst(L0[k]=s,L0),L0:L,t:delete(L[k],L),s:L[k],M)
),LL))
),
thisout0:LL,
thisout:map(lambda([L],map(lambda([e],if atom(e) then e elseif op(e)="or" then cnn0(e) else e),L) ),LL)
),
thisout:subst([equal="="],thisout)
)$
distb(f):=block([],
if atom(f) then f
elseif member(op(f),["or","and"]) then map(distb,f)
elseif member(op(f),[equal,"<",">","<=",">="]) then apply(op(f),[{lhs(f)},{rhs(f)}])
else f
)$
cnn0(f):=block( [f1,S:[],v,Sb,T,f01],
if atom(f) then f
else
(
map(lambda([e],
(if not(member(lhs(e),S)) then S:append(S,[lhs(e)]),
if not(member(rhs(e),S)) then S:append(S,[rhs(e)]))),
args(subst("and"="or",f))),
v:if is("and"=op((args(f))[1])) then S[2] else S[1],
Sb:map("{",delete(v,S)),T:makelist({10^5*k},k,length(Sb)),
prt2(S,Sb,T,map("=",Sb,T),f,subst(map("=",Sb,T),distb(f))),
f01:ev(subst("{"="(",subst(map("=",Sb,T),distb(f))),eval),prt2(f01),
f01:ora(subst(["="=equal],cineqs2(f01))),
if atom (f01) then f01 else subst(map("=",append([{v}],T),append([v],delete(v,S))),distb(f01))
)
)$
cnnm(LL):=block([k,L0:LL,L1:cnn(LL),L2,usr_display2d],
for k:0 unless is(L0=L1) do (L2:cnn(L1),L0:L1,L1:L2),
usr_display2d:display2d,
set_display('xml),
print(apply(matrix,L1)),
display2d:usr_display2d,return(done)
)$
infix ("==",50,50)$
nary ("&&",50,50)$
nary ("||",50,50)$
sqrtdispflag:off$
mma(F):=subst (["="="==","and"="&&","or"="||"],ora(subst (["="="==",%i=I],F)))$
sqrtdispflag:on$
ora(f):=block([],
if not(listp(f)) then return(f),
if atom(f) then return(f)
else apply("or",map(lambda([e],apply("and",e)),subst("="=equal,f)))
)$
cineqs2(F):=block([lv,v,imagunit,L,M,N,k,K,J],
lv:listofvars(F),if length(lv)#1 then return([[is(F)]]) else v:first(lv),
L:sort(listify(setify(flatten(scanmap(
lambda([e],
if atom(e) then e
elseif member(op(e),["and","or"]) then substpart("[",e,0)
elseif member(op(e),["<",">","<=",">=","=","#",equal])
then map(lambda([t],if member(imagunit,listofvars(t))
then [] else rhs(t)),subst(%i=imagunit,solve(substpart("=",e,0),v)))
else e),
F)))),"<"),
if L=[] then return([[is2(subst(v=0,F))]]),
L:sort(append(L,(append([first(L)-1],L)+append(L,[last(L)+1]))/2),"<"),
M:map(lambda([e],if member(op(F),["and","or"]) then map(is2,rat(subst(v=e,F)))
else is2(rat(subst(v=e,F)))),L),
if length(setify(M))=1 then return([[is2(subst(v=0,F))]]),
L:append([0,0],L,[0,0]),M:append([true,true],M,[true,true]),
K:[],for k:3 thru length(M)-2 do (
if oddp(k) then
if part(M,k) then
(if not(part(M,k-1)) then K:append(K,[part(L,k-1)<v])
elseif not(part(M,k-2)) then K:append(K,[part(L,k-1)<=v]) else K:K,
if not(part(M,k+1)) then K:append(K,[v<part(L,k+1)])
elseif not(part(M,k+2)) then K:append(K,[v<=part(L,k+1)]) else K:K)
else K:K
else
if not(part(M,k-1)) and part(M,k) and not(part(M,k+1)) then
K:append(K,[v=part(L,k)]) else K:K),
N:[],J:[],for k in K do
if rhs(k)=v then J:[k]
else (N:append(N,[append(J,[k])]),J:[]),
delete([],append(N,[J]))
)$
