YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

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

[b(x₁)]	=	0 · x₁ + -∞
[d(x₁)]	=	0 · x₁ + -∞
[a(x₁)]	=	0 · x₁ + -∞
[c(x₁)]	=	8 · x₁ + -∞

the rules

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

remain.

1.1 Rule Removal

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
-∞	0

· x₁ +

-∞	-∞
-∞	-∞

[d(x₁)]

0	0
-∞	-∞

· x₁ +

-∞	-∞
-∞	-∞

[a(x₁)]

0	0
-∞	0

· x₁ +

-∞	-∞
-∞	-∞

[c(x₁)]

0	1
0	1

· x₁ +

-∞	-∞
-∞	-∞

the rules

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

remain.

1.1.1 Rule Removal

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 +

-∞	-∞
-∞	-∞

[d(x₁)]

0	0
1	1

· x₁ +

-∞	-∞
-∞	-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞	-∞
-∞	-∞

[c(x₁)]

0	-∞
0	-∞

· x₁ +

-∞	-∞
-∞	-∞

the rules

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

remain.

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

[a^#(x₁)]

0	-∞
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

-∞	0
0	0

· x₁ +

0	-∞
1	-∞

[b^#(x₁)]

0	0
-∞	-∞

· x₁ +

1	-∞
-∞	-∞

[a(x₁)]

0	0
-∞	-∞

· x₁ +

1	-∞
0	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1.1.1 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

[a^#(x₁)]

0	1	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

1	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[b(x₁)]

-∞	0	1
-∞	-∞	0
-∞	0	1

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
0	-∞	-∞

[b^#(x₁)]

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

1	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

1	1	1
-∞	1	0
-∞	0	-∞

· x₁ +

1	-∞	-∞
0	-∞	-∞
-∞	-∞	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

1.1.1.1.1.1.1.1.1 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

[a^#(x₁)]

0	-∞
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[b(x₁)]

-∞	0
0	1

· x₁ +

0	-∞
3	-∞

[a(x₁)]

2	1
0	-∞

· x₁ +

3	-∞
0	-∞

together with the usable rules

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

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

1.1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.