YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[b^#(x₁)]

0	-∞
-∞	-∞

· x₁ +

2	-∞
-∞	-∞

[b(x₁)]

0	-∞
0	-∞

· x₁ +

-∞	-∞
0	-∞

[c^#(x₁)]

0	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

1	-∞
0	0

· x₁ +

2	-∞
0	-∞

[c(x₁)]

-∞	0
1	1

· x₁ +

0	-∞
2	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(c(x0))

→

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

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[b(x₁)]

0	-∞
0	-∞

· x₁ +

0	-∞
0	-∞

[c^#(x₁)]

0	2
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

1	-∞
0	0

· x₁ +

1	-∞
0	-∞

[c(x₁)]

-∞	0
1	1

· x₁ +

0	-∞
1	-∞

together with the usable rules

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

(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.