YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

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	1
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[b(x₁)]

0	0
-∞	0

· x₁ +

-∞	-∞
-∞	-∞

[a^#(x₁)]

1	0
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[c^#(x₁)]

0	1
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[a(x₁)]

0	0
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[c(x₁)]

0	0
0	0

· x₁ +

-∞	-∞
-∞	-∞

together with the usable rules

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

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

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

[b^#(x₁)]

-∞	0	1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[b(x₁)]

0	0	-∞
0	0	-∞
1	0	0

· x₁ +

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

[a^#(x₁)]

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c^#(x₁)]

0	1	1
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

-∞	-∞	-∞
-∞	-∞	-∞
1	0	1

· x₁ +

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

[c(x₁)]

-∞	0	-∞
0	1	0
0	0	-∞

· x₁ +

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

together with the usable rules

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

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

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

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

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[b(x₁)]	=	2 · x₁ + 2
[a^#(x₁)]	=	0 · x₁ + 2
[a(x₁)]	=	4 · x₁ + 4
[c(x₁)]	=	-∞ · x₁ + 0

together with the usable rules

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

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

a^#(b(x0))

→

a^#(x0)

remains.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.