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

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1.1 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

[b^#(x₁)]

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

· x₁ +

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

[b(x₁)]

0	0	0	-∞
0	-∞	0	-∞
0	0	0	0
0	0	0	0

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c^#(x₁)]

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

· x₁ +

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

[a(x₁)]

0	0	0	-∞
0	0	0	0
0	0	0	-∞
0	1	0	0

· x₁ +

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

[c(x₁)]

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

· x₁ +

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

together with the usable rules

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

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

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

remain.

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	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[b(x₁)]

0	-∞	-∞
1	0	-∞
-∞	0	-∞

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c^#(x₁)]

0	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[a(x₁)]

0	-∞	1
-∞	0	0
1	0	0

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c(x₁)]

0	1	-∞
1	0	0
1	0	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

together with the usable rules

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

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

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

remain.

1.1.1.1.1.1 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

[b^#(x₁)]

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

· x₁ +

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

[b(x₁)]

0	0	0
-∞	-∞	-∞
0	0	-∞
0	0	0

· x₁ +

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

[c^#(x₁)]

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

· x₁ +

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

[a(x₁)]

0	0	0	-∞
0	0	0	0
0	-∞	0	0
0	1	0	0

· x₁ +

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

[c(x₁)]

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

· x₁ +

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

together with the usable rules

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

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

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

→

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

remains.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.