Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/2-matchbox)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(b(x1)))))))	(4)
b^#(b(b(a(x1))))	→	b^#(b(b(x1)))	(5)
c^#(c(x1))	→	b^#(b(x1))	(6)
b^#(b(c(c(x1))))	→	c^#(c(c(a(a(a(a(x1)))))))	(7)
b^#(b(c(c(x1))))	→	c^#(a(a(a(a(x1)))))	(8)
c^#(c(x1))	→	b^#(b(b(x1)))	(9)
c^#(c(x1))	→	b^#(x1)	(10)
b^#(b(b(a(x1))))	→	b^#(b(x1))	(11)
b^#(b(c(c(x1))))	→	c^#(c(a(a(a(a(x1))))))	(12)
b^#(b(b(a(x1))))	→	b^#(b(b(b(x1))))	(13)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(x1))))))	(14)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(b(b(x1))))))))	(15)
b^#(b(b(a(x1))))	→	b^#(x1)	(16)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(x1)))))	(17)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(b(a(x1))))	→	b^#(b(b(b(b(x1)))))	(17)
c^#(c(x1))	→	b^#(x1)	(10)
c^#(c(x1))	→	b^#(b(b(x1)))	(9)
b^#(b(b(a(x1))))	→	b^#(x1)	(16)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(b(b(x1))))))))	(15)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(x1))))))	(14)
b^#(b(c(c(x1))))	→	c^#(c(c(a(a(a(a(x1)))))))	(7)
c^#(c(x1))	→	b^#(b(x1))	(6)
b^#(b(b(a(x1))))	→	b^#(b(b(x1)))	(5)
b^#(b(b(a(x1))))	→	b^#(b(b(b(x1))))	(13)
b^#(b(c(c(x1))))	→	c^#(c(a(a(a(a(x1))))))	(12)
b^#(b(b(a(x1))))	→	b^#(b(x1))	(11)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(b(x1)))))))	(4)

1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals

[a(x₁)]

=

0	0
0	1

· x₁ +

1002

2961

[b(x₁)]

=

0	1
0	1

· x₁ +

23508

592

[c(x₁)]

=

0	1
1	0

· x₁ +

12256

182

[c^#(x₁)]

=

1	0
0	0

· x₁ +

0

0

[b^#(x₁)]

=

0	1
0	0

· x₁ +

11071

0

together with the usable rules

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

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

b^#(b(b(a(x1))))	→	b^#(b(b(b(b(x1)))))	(17)
c^#(c(x1))	→	b^#(x1)	(10)
c^#(c(x1))	→	b^#(b(b(x1)))	(9)
b^#(b(b(a(x1))))	→	b^#(x1)	(16)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(b(b(x1))))))))	(15)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(x1))))))	(14)
b^#(b(c(c(x1))))	→	c^#(c(c(a(a(a(a(x1)))))))	(7)
c^#(c(x1))	→	b^#(b(x1))	(6)
b^#(b(b(a(x1))))	→	b^#(b(b(x1)))	(5)
b^#(b(b(a(x1))))	→	b^#(b(b(b(x1))))	(13)
b^#(b(c(c(x1))))	→	c^#(c(a(a(a(a(x1))))))	(12)
b^#(b(b(a(x1))))	→	b^#(b(x1))	(11)
b^#(b(b(a(x1))))	→	b^#(b(b(b(b(b(b(x1)))))))	(4)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.