YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(a(b(b(b(b(a(a(x0))))))))	→	a(a(c(c(a(a(b(b(x0))))))))
a(a(c(c(x0))))	→	c(c(c(c(a(a(x0))))))
c(c(c(c(c(c(x0))))))	→	b(b(c(c(b(b(x0))))))

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(a(b(b(b(b(a(a(x0))))))))	→	b(b(a(a(c(c(a(a(x0))))))))
c(c(a(a(x0))))	→	a(a(c(c(c(c(x0))))))
c(c(c(c(c(c(x0))))))	→	b(b(c(c(b(b(x0))))))

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(a(b(b(b(b(a(a(x0))))))))	→	c^#(a(a(x0)))
a^#(a(b(b(b(b(a(a(x0))))))))	→	c^#(c(a(a(x0))))
a^#(a(b(b(b(b(a(a(x0))))))))	→	a^#(c(c(a(a(x0)))))
a^#(a(b(b(b(b(a(a(x0))))))))	→	a^#(a(c(c(a(a(x0))))))
c^#(c(a(a(x0))))	→	c^#(x0)
c^#(c(a(a(x0))))	→	c^#(c(x0))
c^#(c(a(a(x0))))	→	c^#(c(c(x0)))
c^#(c(a(a(x0))))	→	c^#(c(c(c(x0))))
c^#(c(a(a(x0))))	→	a^#(c(c(c(c(x0)))))
c^#(c(a(a(x0))))	→	a^#(a(c(c(c(c(x0))))))
c^#(c(c(c(c(c(x0))))))	→	c^#(b(b(x0)))
c^#(c(c(c(c(c(x0))))))	→	c^#(c(b(b(x0))))

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

c^#(c(a(a(x0))))	→	c^#(c(c(c(x0))))
c^#(c(a(a(x0))))	→	a^#(a(c(c(c(c(x0))))))
a^#(a(b(b(b(b(a(a(x0))))))))	→	a^#(a(c(c(a(a(x0))))))
a^#(a(b(b(b(b(a(a(x0))))))))	→	a^#(c(c(a(a(x0)))))
a^#(a(b(b(b(b(a(a(x0))))))))	→	c^#(c(a(a(x0))))
c^#(c(a(a(x0))))	→	a^#(c(c(c(c(x0)))))
c^#(c(a(a(x0))))	→	c^#(c(c(x0)))
c^#(c(a(a(x0))))	→	c^#(c(x0))
c^#(c(a(a(x0))))	→	c^#(x0)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight function

prec(c^#)	=	weight(c^#)	=	1
prec(a^#)	=	weight(a^#)	=	0
prec(c)	=	weight(c)	=	0
prec(b)	=	weight(b)	=	0
prec(a)	=	weight(a)	=	0

in combination with the following argument filter

π(c^#) = 1

π(a^#) = 1

π(c) = 1

π(b) = 1

π(a) = [1]

together with the usable rules

a(a(b(b(b(b(a(a(x0))))))))	→	b(b(a(a(c(c(a(a(x0))))))))
c(c(a(a(x0))))	→	a(a(c(c(c(c(x0))))))
c(c(c(c(c(c(x0))))))	→	b(b(c(c(b(b(x0))))))

(w.r.t. the implicit argument filter of the reduction pair), the pair

a^#(a(b(b(b(b(a(a(x0))))))))

→

a^#(a(c(c(a(a(x0))))))

remains.

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

[b(x₁)]

0	0	1	0
0	0	0	1
0	0	0	0
1	1	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a^#(x₁)]

0	0	0	1
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a(x₁)]

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	1

· x₁ +

0	0	0	0
0	0	0	0
1	0	0	0
0	0	0	0

[c(x₁)]

0	0	0
0	0	1
1	1	0
0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

all pairs could be removed.

1.1.1.1.1.1 P is empty

There are no pairs anymore.