Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/3-matchbox)

The rewrite relation of the following TRS is considered.

c(b(a(a(x1))))	→	a(a(b(c(x1))))	(1)
b(a(a(a(x1))))	→	a(a(a(b(x1))))	(2)
a(b(c(x1)))	→	c(b(a(x1)))	(3)
c(c(b(b(x1))))	→	b(b(c(c(x1))))	(4)

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^#(a(a(a(x1))))	→	a^#(a(b(x1)))	(5)
a^#(b(c(x1)))	→	b^#(a(x1))	(6)
c^#(c(b(b(x1))))	→	b^#(c(c(x1)))	(7)
c^#(b(a(a(x1))))	→	a^#(b(c(x1)))	(8)
c^#(b(a(a(x1))))	→	c^#(x1)	(9)
c^#(c(b(b(x1))))	→	b^#(b(c(c(x1))))	(10)
c^#(c(b(b(x1))))	→	c^#(c(x1))	(11)
a^#(b(c(x1)))	→	a^#(x1)	(12)
c^#(b(a(a(x1))))	→	b^#(c(x1))	(13)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(14)
b^#(a(a(a(x1))))	→	a^#(b(x1))	(15)
b^#(a(a(a(x1))))	→	a^#(a(a(b(x1))))	(16)
c^#(b(a(a(x1))))	→	a^#(a(b(c(x1))))	(17)
c^#(c(b(b(x1))))	→	c^#(x1)	(18)
b^#(a(a(a(x1))))	→	b^#(x1)	(19)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(a(a(a(x1))))	→	b^#(x1)	(19)
c^#(c(b(b(x1))))	→	c^#(c(x1))	(11)
c^#(c(b(b(x1))))	→	c^#(x1)	(18)
c^#(c(b(b(x1))))	→	b^#(b(c(c(x1))))	(10)
c^#(b(a(a(x1))))	→	a^#(a(b(c(x1))))	(17)
b^#(a(a(a(x1))))	→	a^#(a(a(b(x1))))	(16)
b^#(a(a(a(x1))))	→	a^#(b(x1))	(15)
c^#(b(a(a(x1))))	→	c^#(x1)	(9)
c^#(b(a(a(x1))))	→	a^#(b(c(x1)))	(8)
c^#(c(b(b(x1))))	→	b^#(c(c(x1)))	(7)
a^#(b(c(x1)))	→	b^#(a(x1))	(6)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(14)
c^#(b(a(a(x1))))	→	b^#(c(x1))	(13)
a^#(b(c(x1)))	→	a^#(x1)	(12)
b^#(a(a(a(x1))))	→	a^#(a(b(x1)))	(5)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 5854
[b(x₁)]	=	x₁ + 4685
[c(x₁)]	=	x₁ + 30613
[c^#(x₁)]	=	x₁ + 25928
[a^#(x₁)]	=	x₁ + 1169
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

c(c(b(b(x1))))	→	b(b(c(c(x1))))	(4)
c(b(a(a(x1))))	→	a(a(b(c(x1))))	(1)
a(b(c(x1)))	→	c(b(a(x1)))	(3)
b(a(a(a(x1))))	→	a(a(a(b(x1))))	(2)

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

b^#(a(a(a(x1))))	→	b^#(x1)	(19)
c^#(c(b(b(x1))))	→	c^#(c(x1))	(11)
c^#(c(b(b(x1))))	→	c^#(x1)	(18)
b^#(a(a(a(x1))))	→	a^#(b(x1))	(15)
c^#(b(a(a(x1))))	→	c^#(x1)	(9)
c^#(b(a(a(x1))))	→	a^#(b(c(x1)))	(8)
c^#(c(b(b(x1))))	→	b^#(c(c(x1)))	(7)
a^#(b(c(x1)))	→	b^#(a(x1))	(6)
c^#(b(a(a(x1))))	→	b^#(c(x1))	(13)
a^#(b(c(x1)))	→	a^#(x1)	(12)
b^#(a(a(a(x1))))	→	a^#(a(b(x1)))	(5)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(c(b(b(x1))))	→	b^#(b(c(c(x1))))	(10)
c^#(b(a(a(x1))))	→	a^#(a(b(c(x1))))	(17)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(14)
b^#(a(a(a(x1))))	→	a^#(a(a(b(x1))))	(16)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

0	1
0	0

· x₁ +

[b(x₁)]

0	0
1	0

· x₁ +

[c(x₁)]

1	0
0	0

· x₁ +

31302

[c^#(x₁)]

1	0
0	0

· x₁ +

[a^#(x₁)]

0	1
0	0

· x₁ +

[b^#(x₁)]

together with the usable rules

c(c(b(b(x1))))	→	b(b(c(c(x1))))	(4)
c(b(a(a(x1))))	→	a(a(b(c(x1))))	(1)
a(b(c(x1)))	→	c(b(a(x1)))	(3)
b(a(a(a(x1))))	→	a(a(a(b(x1))))	(2)

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

c^#(c(b(b(x1))))	→	b^#(b(c(c(x1))))	(10)
c^#(b(a(a(x1))))	→	a^#(a(b(c(x1))))	(17)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(14)
b^#(a(a(a(x1))))	→	a^#(a(a(b(x1))))	(16)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.