YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(b(x0))	→	c(d(x0))
d(d(x0))	→	b(e(x0))
b(x0)	→	d(c(x0))
d(x0)	→	x0
e(c(x0))	→	d(a(x0))
a(x0)	→	e(d(x0))

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

[e(x₁)]

-∞	0
-∞	0

· x₁ +

0	-∞
1	-∞

[a(x₁)]

0	0
0	0

· x₁ +

0	-∞
1	-∞

[c(x₁)]

0	0
1	1

· x₁ +

-∞	-∞
2	-∞

[a^#(x₁)]

3	2
-∞	-∞

· x₁ +

3	-∞
-∞	-∞

[b(x₁)]

2	2
1	1

· x₁ +

3	-∞
2	-∞

[e^#(x₁)]

0	2
-∞	-∞

· x₁ +

3	-∞
-∞	-∞

[d^#(x₁)]

2	1
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[d(x₁)]

1	1
0	0

· x₁ +

2	-∞
-∞	-∞

together with the usable rules

a(b(x0))	→	c(d(x0))
d(d(x0))	→	b(e(x0))
b(x0)	→	d(c(x0))
d(x0)	→	x0
e(c(x0))	→	d(a(x0))
a(x0)	→	e(d(x0))

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

e^#(c(x0))	→	a^#(x0)
a^#(x0)	→	e^#(d(x0))

remain.

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

[e(x₁)]

0	-∞
0	-∞

· x₁ +

0	-∞
1	-∞

[a(x₁)]

0	0
0	0

· x₁ +

0	-∞
1	-∞

[c(x₁)]

1	1
1	1

· x₁ +

2	-∞
-∞	-∞

[a^#(x₁)]

3	3
-∞	-∞

· x₁ +

3	-∞
-∞	-∞

[b(x₁)]

1	1
2	2

· x₁ +

2	-∞
3	-∞

[e^#(x₁)]

2	0
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[d(x₁)]

0	0
1	1

· x₁ +

0	-∞
2	-∞

together with the usable rules

a(b(x0))	→	c(d(x0))
d(d(x0))	→	b(e(x0))
b(x0)	→	d(c(x0))
d(x0)	→	x0
e(c(x0))	→	d(a(x0))
a(x0)	→	e(d(x0))

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

e^#(c(x0))

→

a^#(x0)

remains.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.