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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1 Reduction Pair Processor with Usable Rules

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

[c^#(x₁)]

1	0	0	0
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[c(x₁)]

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

· x₁ +

-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
0	-∞	-∞	-∞

[a^#(x₁)]

0	0	0	0
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

0	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

[a(x₁)]

0	-∞	-∞
0	0	0
1	0	0
-∞	-∞	0

· x₁ +

-∞	-∞	-∞	-∞
0	-∞	-∞	-∞
0	-∞	-∞	-∞
0	-∞	-∞	-∞

[b(x₁)]

0	0	-∞	0
0	0	0	0
0	0	-∞	0
0	0	0	0

· x₁ +

-∞	-∞	-∞	-∞
0	-∞	-∞	-∞
0	-∞	-∞	-∞
0	-∞	-∞	-∞

together with the usable rules

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

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

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

remain.

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

[c^#(x₁)]

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c(x₁)]

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a^#(x₁)]

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

0	0	-∞
1	0	-∞
0	0	0

· x₁ +

-∞	-∞	-∞
1	-∞	-∞
-∞	-∞	-∞

[b(x₁)]

1	-∞	1
0	-∞	0
0	1	0

· x₁ +

-∞	-∞	-∞
0	-∞	-∞
1	-∞	-∞

together with the usable rules

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

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

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

→

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

remains.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.