Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup13)

The rewrite relation of the following TRS is considered.

a(a(a(a(x1))))	→	a(a(b(b(b(b(b(b(x1))))))))	(1)
b(b(a(a(x1))))	→	b(b(b(b(c(c(x1))))))	(2)
a(a(b(b(b(b(c(c(x1))))))))	→	a(a(a(a(a(a(b(b(x1))))))))	(3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

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

a(a(a(a(x1))))	→	b(b(b(b(b(b(a(a(x1))))))))	(4)
a(a(b(b(x1))))	→	c(c(b(b(b(b(x1))))))	(5)
c(c(b(b(b(b(a(a(x1))))))))	→	b(b(a(a(a(a(a(a(x1))))))))	(6)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(a(b(b(x1))))	→	c^#(b(b(b(b(x1)))))	(7)
a^#(a(b(b(x1))))	→	c^#(c(b(b(b(b(x1))))))	(8)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(9)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(10)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(a(x1)))))	(11)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(a(a(x1))))))	(12)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(a(a(x1))))))	(12)
a^#(a(b(b(x1))))	→	c^#(c(b(b(b(b(x1))))))	(8)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(a(x1)))))	(11)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(10)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(9)

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 naturals

[c^#(x₁)]

=

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

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[b(x₁)]

=

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

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a^#(x₁)]

=

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

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a(x₁)]

=

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

· x₁ +

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

[c(x₁)]

=

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

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

together with the usable rules

a(a(a(a(x1))))	→	b(b(b(b(b(b(a(a(x1))))))))	(4)
a(a(b(b(x1))))	→	c(c(b(b(b(b(x1))))))	(5)
c(c(b(b(b(b(a(a(x1))))))))	→	b(b(a(a(a(a(a(a(x1))))))))	(6)

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

c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(a(a(x1))))))	(12)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(a(x1)))))	(11)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(a(x1))))	(10)
c^#(c(b(b(b(b(a(a(x1))))))))	→	a^#(a(a(x1)))	(9)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.