YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over (5 x 5)-matrices with strict dimension 1 over the naturals

[b^#(x₁)]

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

· x₁ +

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

[b(x₁)]

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

· x₁ +

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

[c^#(x₁)]

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

[a(x₁)]

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

· x₁ +

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

[c(x₁)]

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

· x₁ +

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

the pairs

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

remain.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(x0))

→

b^#(c(x0))

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers

[b^#(x₁)]

0	1	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[b(x₁)]

-∞	1	0
-∞	0	1
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
0	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

0	0	1
0	-∞	0
-∞	-∞	0

· x₁ +

0	-∞	-∞
0	-∞	-∞
-∞	-∞	-∞

[c(x₁)]

-∞	0	1
-∞	-∞	0
-∞	0	1

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
0	-∞	-∞

together with the usable rules

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

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

1.1.1.1.1.1 P is empty

There are no pairs anymore.