如何用Z3中的八元数进行计算

在上一篇文章中，给出了使用Z3计算四元数的代码。在这篇文章中，给出了一个用Z3中的八元数计算的代码

使用“Octonion代码”解决了以下示例：

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

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Proved that x(xy) = (xx)y
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Proved that (yx)x = y(xx)

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Proved that z(x(zy)) = ((zx)z)y
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Proved that x(z(yz)) = ((xz)y)z
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Proved that (zx)(yz) = (z(xy))z
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Proved that (zx)(yz) = z((xy)z)

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:    

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:
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Proved : ADA = DAD:
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Proved : BDB = DBD:
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Proved : CDC = DCD:
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Proved : AEA = EAE:
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Proved : BEB = EBE:
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Proved : CEC = ECE:
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Proved : DED = EDE:
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Proved : AFA = FAF:
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Proved : BFB = FBF:
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Proved : CFC = FCF:
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Proved : DFD = FDF:
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Proved : EFE = FEF:
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Proved : AGA = GAG:
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Proved : BGB = GBG:
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Proved : CGC = GCG:
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Proved : DGD = GDG:
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Proved : EGE = GEG:
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Proved : FGF = GFG:

使用Maple验证了该结果

其他例子：

代码：

x = Octonion("x")
y = Octonion("y")
s = Tactic('qfnra-nlsat').solver()
s.add(x*y + 30  == 0, x - y - y == 10)
print(s.check())
m = s.model()
print m

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

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Proved that x(xy) = (xx)y
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Proved that (yx)x = y(xx)

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Proved that z(x(zy)) = ((zx)z)y
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Proved that x(z(yz)) = ((xz)y)z
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Proved that (zx)(yz) = (z(xy))z
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Proved that (zx)(yz) = z((xy)z)

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:    

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:

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Proved : ABA = BAB:
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Proved : ACA = CAC:
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Proved : BCB = CBC:
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Proved : ADA = DAD:
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Proved : BDB = DBD:
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Proved : CDC = DCD:
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Proved : AEA = EAE:
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Proved : BEB = EBE:
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Proved : CEC = ECE:
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0
Proved : DED = EDE:
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Proved : AFA = FAF:
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Proved : BFB = FBF:
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Proved : CFC = FCF:
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Proved : DFD = FDF:
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Proved : EFE = FEF:
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Proved : AGA = GAG:
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Proved : BGB = GBG:
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Proved : CGC = GCG:
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Proved : DGD = GDG:
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Proved : EGE = GEG:
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Proved : FGF = GFG:

其他例子：

证明

x * y != y * x

(x * y) * z != x * (y * z)

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

其他例子：证明

x * y != y * x

(x * y) * z != x * (y * z)

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

另一个例子：证明八元数形成一个替代代数，它是说：

x * (x * y) = (x * x) * y
(y * x) * x = y * (x * x)

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

其他示例：证明八元数满足Moufang恒等式：

z * (x * (z * y)) = ((z * x) * z) * y
x * (z * (y * z)) = ((x * z) * y) * z
(z * x) * (y * z) = (z * (x * y)) * z
(z * x) * (y * z) = z * ((x * y) * z)

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

另一个例子：证明八元数

A = (1 + e1)/sqrt(2)
B = (1 + e2)/sqrt(2) 
C = (1 + e3)/sqrt(2) 

A = (1 + e4)/sqrt(2)
B = (1 + e5)/sqrt(2) 
C = (1 + e6)/sqrt(2) 

生成辫子群的表示，也就是说，我们有

ABA = BAB 
ACA = CAC 
BCB = CBC.

ABA = BAB 
ACA = CAC 
BCB = CBC

ABA = BAB 
ACA = CAC 
BCB = CBC

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

另一个例子：证明八元数

A = (1 + e1)/sqrt(2)
B = (1 + e2)/sqrt(2) 
C = (1 + e3)/sqrt(2) 

A = (1 + e4)/sqrt(2)
B = (1 + e5)/sqrt(2) 
C = (1 + e6)/sqrt(2) 

生成辫子群的表示，也就是说，我们有

ABA = BAB 
ACA = CAC 
BCB = CBC.

ABA = BAB 
ACA = CAC 
BCB = CBC

ABA = BAB 
ACA = CAC 
BCB = CBC

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

另一个例子：证明八元数

A = (1 + e5)/sqrt(2)
B = (1 + e6)/sqrt(2) 
C = (1 + e7)/sqrt(2) 

A = (1 + e1)/sqrt(2)         E = (1 + e5)/sqrt(2)
B = (1 + e2)/sqrt(2)         F = (1 + e6)/sqrt(2)
C = (1 + e3)/sqrt(2)         G = (1 + e7)/sqrt(2)
D = (1 + e4)/sqrt(2) 

生成辫子群的表示，也就是说，我们有

ABA = BAB     ACA = CAC     BCB = CBC    ADA = DAD     BDB = DBD
CDC = DCD     AEA = EAE     BEB = EBE    CEC = ECE     DED = EDE
AFA = FAF     BFB = FBF     CFC = FCF    DFD = FDF     EFE = FEF
AGA = GAG     BGB = GBG     CGC = GCG    DGD = GDg     EGE = GEG     FGF = GFG

x / x  = 1 

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

另一个例子：证明八元数

A = (1 + e5)/sqrt(2)
B = (1 + e6)/sqrt(2) 
C = (1 + e7)/sqrt(2) 

A = (1 + e1)/sqrt(2)         E = (1 + e5)/sqrt(2)
B = (1 + e2)/sqrt(2)         F = (1 + e6)/sqrt(2)
C = (1 + e3)/sqrt(2)         G = (1 + e7)/sqrt(2)
D = (1 + e4)/sqrt(2) 

生成辫子群的表示，也就是说，我们有

ABA = BAB     ACA = CAC     BCB = CBC    ADA = DAD     BDB = DBD
CDC = DCD     AEA = EAE     BEB = EBE    CEC = ECE     DED = EDE
AFA = FAF     BFB = FBF     CFC = FCF    DFD = FDF     EFE = FEF
AGA = GAG     BGB = GBG     CGC = GCG    DGD = GDg     EGE = GEG     FGF = GFG

x / x  = 1 

代码：

输出：

sat
[x.i3 = 1/2,
x.i7 = -1/2,
x.i2 = -1/2,
x.i5 = 1,
x.i1 = 0,
x.r = 0,
x.i6 = -1/2,
x.i4 = 0]

sat
[y.i3 = 1/2,
y.i7 = -1/2,
y.i2 = -1/2,
y.i5 = 1,
y.i1 = -1,
y.r = -5/2,
y.i6 = 1/2,
y.i4 = -2.3979157616?,
x.i7 = -1,
x.i6 = 1,
x.i5 = 2,
x.i4 = -4.7958315233?,
x.i3 = 1,
x.i2 = -1,
x.i1 = -2,
x.r = 5]

 0
-2·c1·e2 + -2·f1·g2 + 2·f2·g1 + 2·c2·e1 + 2·d2·h1 + -2·d1·h2
 2·d2·f1 + -2·g1·h2 + 2·g2·h1 + -2·b2·e1 + -2·d1·f2 + 2·b1·e2
 2·c1·f2 + 2·b1·h2 + -2·b2·h1 + 2·e2·g1 + -2·e1·g2 + -2·c2·f1
-2·d2·g1 + 2·b2·c1 + 2·d1·g2 + -2·b1·c2 + -2·f1·h2 + 2·f2·h1
-2·b2·g1 + 2·e1·h2 + -2·c1·d2 + 2·b1·g2 + -2·e2·h1 + 2·c2·d1
 2·d2·e1 + 2·b2·f1 + -2·d1·e2 + -2·c2·h1 + -2·b1·f2 + 2·c1·h2
-2·b1·d2 + -2·e1·f2 + -2·c1·g2 + 2·c2·g1 + 2·e2·f1 + 2·b2·d1

0
end r

-2·d2·e3·f1 + 2·e3·g1·h2 + -2·e3·g2·h1 + 2·d1·e3·f2 +-2·e1·g3·h2 + 2·c1·d2·g3 +
2·e2·g3·h1 + -2·c2·d1·g3 + 2·d2·e1·f3 + -2·d1·e2·f3 +-2·c2·f3·h1 + 2·c1·f3·h2 +
-2·c3·d2·g1 + 2·c3·d1·g2 + -2·c3·f1·h2 + 2·c3·f2·h1 +-2·d3·e1·f2 + -2·c1·d3·g2 +
2·c2·d3·g1 + 2·d3·e2·f1 + -2·c1·f2·h3 + -2·e2·g1·h3 + 2·e1·g2·h3 + 2·c2·f1·h3
end i1

-2·b2·d3·g1 + 2·d3·e1·h2 + 2·b1·d3·g2 + -2·d3·e2·h1 + -2·d2·e1·h3 + -2·b2·f1·h3 +
2·d1·e2·h3 + 2·b1·f2·h3 +-2·b1·d2·g3 + -2·e1·f2·g3 + 2·e2·f1·g3 + 2·b2·d1·g3 +
2·b3·d2·g1 + -2·b3·d1·g2 + 2·b3·f1·h2 + -2·b3·f2·h1 + -2·b1·f3·h2 + 2·b2·f3·h1 +
-2·e2·f3·g1 + 2·e1·f3·g2 + -2·e3·f1·g2 + 2·e3·f2·g1 + 2·d2·e3·h1 +-2·d1·e3·h2
end i2

-2·f3·g1·h2 + 2·f3·g2·h1 + -2·b2·e1·f3 + 2·b1·e2·f3 + -2·c1·e2·h3 +-2·f1·g2·h3 +
2·f2·g1·h3 + 2·c2·e1·h3 + 2·b3·e1·f2 + 2·b3·c1·g2 +-2·b3·c2·g1 + -2·b3·e2·f1 +
2·b2·e3·f1 +-2·c2·e3·h1 +  -2·b1·e3·f2 + 2·c1·e3·h2 + -2·b2·c1·g3 + 2·b1·c2·g3 +
2·f1·g3·h2 + -2·f2·g3·h1 + 2·b2·c3·g1 + -2·c3·e1·h2 + -2·b1·c3·g2 +  2·c3·e2·h1
end i3

-2·b2·d3·f1 + 2·c2·d3·h1 +2·b1·d3·f2 + -2·c1·d3·h2 + 2·b3·d2·f1 +-2·b3·g1·h2 +
2·b3·g2·h1 +-2·b3·d1·f2 +2·c1·f2·g3 +2·b1·g3·h2 +-2·b2·g3·h1 + -2·c2·f1·g3 +
2·c3·f1·g2 + -2·c3·f2·g1 +-2·c3·d2·h1 +2·c3·d1·h2 +2·b2·g1·h3 +2·c1·d2·h3 +
-2·b1·g2·h3 +-2·c2·d1·h3 +-2·b1·d2·f3 +-2·c1·f3·g2 +2·c2·f3·g1 +2·b2·d1·f3
end i4

-2·b3·d2·e1 + 2·b3·d1·e2 + 2·b3·c2·h1 + -2·b3·c1·h2 + -2·d2·g1·h3 + 2·b2·c1·h3 +
 2·d1·g2·h3 +-2·b1·c2·h3 +2·d3·g1·h2 +-2·d3·g2·h1 +2·b2·d3·e1 +-2·b1·d3·e2 +
-2·c1·e2·g3 + 2·c2·e1·g3 +2·d2·g3·h1 +-2·d1·g3·h2 +2·b1·d2·e3 + 2·c1·e3·g2 +
-2·c2·e3·g1 +-2·b2·d1·e3 +2·b1·c3·h2 +-2·b2·c3·h1 +2·c3·e2·g1 +-2·c3·e1·g2
 end i5

 2·b2·c1·d3 +-2·b1·c2·d3 +-2·d3·f1·h2 +2·d3·f2·h1 +2·b3·e1·h2 +-2·b3·c1·d2 +
 -2·b3·e2·h1 +2·b3·c2·d1 +-2·c1·e3·f2 +-2·b1·e3·h2 +2·b2·e3·h1 +2·c2·e3·f1 +
 2·b1·c3·d2 +2·c3·e1·f2 +-2·c3·e2·f1 +-2·b2·c3·d1 +2·c1·e2·f3 +-2·c2·e1·f3 +
 -2·d2·f3·h1 +2·d1·f3·h2 +2·d2·f1·h3 +-2·b2·e1·h3 +-2·d1·f2·h3 + 2·b1·e2·h3
 end i6

 2·c1·d3·e2 +2·d3·f1·g2 +-2·d3·f2·g1 +-2·c2·d3·e1 +2·d2·f3·g1 +-2·b2·c1·f3 +
 -2·d1·f3·g2 +2·b1·c2·f3 +-2·d2·f1·g3 +2·b2·e1·g3 +2·d1·f2·g3 +-2·b1·e2·g3 +
 2·c3·d2·e1 +2·b2·c3·f1 +-2·c3·d1·e2 +-2·b1·c3·f2 +-2·b2·e3·g1 +-2·c1·d2·e3 +
 2·b1·e3·g2 +2·c2·d1·e3 +2·b3·c1·f2 +2·b3·e2·g1 +-2·b3·e1·g2 +-2·b3·c2·f1
 end i7

0
0
0
0
0
0
0
0
Proved that x(xy) = (xx)y
0
0
0
0
0
0
0
0
Proved that (yx)x = y(xx)

0
0
0
0
0
0
0
0
Proved that z(x(zy)) = ((zx)z)y
0
0
0
0
0
0
0
0
Proved that x(z(yz)) = ((xz)y)z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = (z(xy))z
0
0
0
0
0
0
0
0
Proved that (zx)(yz) = z((xy)z)

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:    

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:

0
0
0
0
0
0
0
0
Proved : ABA = BAB:
0
0
0
0
0
0
0
0
Proved : ACA = CAC:
0
0
0
0
0
0
0
0
Proved : BCB = CBC:
0
0
0
0
0
0
0
0
Proved : ADA = DAD:
0
0
0
0
0
0
0
0
Proved : BDB = DBD:
0
0
0
0
0
0
0
0
Proved : CDC = DCD:
0
0
0
0
0
0
0
0
Proved : AEA = EAE:
0
0
0
0
0
0
0
0
Proved : BEB = EBE:
0
0
0
0
0
0
0
0
Proved : CEC = ECE:
0
0
0
0
0
0
0
0
Proved : DED = EDE:
0
0
0
0
0
0
0
0
Proved : AFA = FAF:
0
0
0
0
0
0
0
0
Proved : BFB = FBF:
0
0
0
0
0
0
0
0
Proved : CFC = FCF:
0
0
0
0
0
0
0
0
Proved : DFD = FDF:
0
0
0
0
0
0
0
0
Proved : EFE = FEF:
0
0
0
0
0
0
0
0
Proved : AGA = GAG:
0
0
0
0
0
0
0
0
Proved : BGB = GBG:
0
0
0
0
0
0
0
0
Proved : CGC = GCG:
0
0
0
0
0
0
0
0
Proved : DGD = GDG:
0
0
0
0
0
0
0
0
Proved : EGE = GEG:
0
0
0
0
0
0
0
0
Proved : FGF = GFG:

在线运行此示例

另一个例子：证明对于形式的可逆八元数

x = a + a1*e1 + a2*e2 + a3*e3 + a4*e4 + a5*e5 + a6*e6

我们有

ABA = BAB     ACA = CAC     BCB = CBC    ADA = DAD     BDB = DBD
CDC = DCD     AEA = EAE     BEB = EBE    CEC = ECE     DED = EDE
AFA = FAF     BFB = FBF     CFC = FCF    DFD = FDF     EFE = FEF
AGA = GAG     BGB = GBG     CGC = GCG    DGD = GDg     EGE = GEG     FGF = GFG

x / x  = 1 

在线运行此示例

请让我知道你对Octonion代码的看法，以及如何改进Octonion代码。非常感谢

请让我知道你对Octonion代码的看法，以及如何改进Octonion代码。非常感谢

我认为这个问题的一个恰当答案是它非常好