Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup17)

The rewrite relation of the following TRS is considered.

a(a(a(a(x1))))	→	b(b(x1))	(1)
b(b(a(a(x1))))	→	a(a(b(b(x1))))	(2)
b(b(b(b(c(c(x1))))))	→	c(c(a(a(x1))))	(3)
b(b(b(b(x1))))	→	a(a(a(a(a(a(x1))))))	(4)
c(c(a(a(x1))))	→	b(b(a(a(c(c(x1))))))	(5)

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(a(a(x1))))	→	a^#(b(b(x1)))	(6)
c^#(c(a(a(x1))))	→	b^#(a(a(c(c(x1)))))	(7)
b^#(b(b(b(c(c(x1))))))	→	a^#(a(x1))	(8)
c^#(c(a(a(x1))))	→	c^#(c(x1))	(9)
b^#(b(b(b(c(c(x1))))))	→	c^#(c(a(a(x1))))	(10)
b^#(b(b(b(c(c(x1))))))	→	c^#(a(a(x1)))	(11)
b^#(b(b(b(c(c(x1))))))	→	a^#(x1)	(12)
b^#(b(b(b(x1))))	→	a^#(a(a(x1)))	(13)
c^#(c(a(a(x1))))	→	a^#(a(c(c(x1))))	(14)
c^#(c(a(a(x1))))	→	c^#(x1)	(15)
c^#(c(a(a(x1))))	→	b^#(b(a(a(c(c(x1))))))	(16)
b^#(b(b(b(x1))))	→	a^#(x1)	(17)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(a(x1))))))	(18)
b^#(b(b(b(x1))))	→	a^#(a(a(a(x1))))	(19)
b^#(b(b(b(x1))))	→	a^#(a(x1))	(20)
a^#(a(a(a(x1))))	→	b^#(x1)	(21)
b^#(b(a(a(x1))))	→	b^#(b(x1))	(22)
b^#(b(a(a(x1))))	→	a^#(a(b(b(x1))))	(23)
c^#(c(a(a(x1))))	→	a^#(c(c(x1)))	(24)
a^#(a(a(a(x1))))	→	b^#(b(x1))	(25)
b^#(b(a(a(x1))))	→	b^#(x1)	(26)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(27)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(27)
b^#(b(a(a(x1))))	→	b^#(x1)	(26)
b^#(b(b(b(c(c(x1))))))	→	a^#(x1)	(12)
a^#(a(a(a(x1))))	→	b^#(b(x1))	(25)
b^#(b(b(b(c(c(x1))))))	→	c^#(a(a(x1)))	(11)
c^#(c(a(a(x1))))	→	a^#(c(c(x1)))	(24)
b^#(b(a(a(x1))))	→	a^#(a(b(b(x1))))	(23)
b^#(b(a(a(x1))))	→	b^#(b(x1))	(22)
a^#(a(a(a(x1))))	→	b^#(x1)	(21)
b^#(b(b(b(x1))))	→	a^#(a(x1))	(20)
b^#(b(b(b(c(c(x1))))))	→	c^#(c(a(a(x1))))	(10)
c^#(c(a(a(x1))))	→	c^#(c(x1))	(9)
b^#(b(b(b(c(c(x1))))))	→	a^#(a(x1))	(8)
b^#(b(b(b(x1))))	→	a^#(a(a(a(x1))))	(19)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(a(x1))))))	(18)
c^#(c(a(a(x1))))	→	b^#(a(a(c(c(x1)))))	(7)
b^#(b(b(b(x1))))	→	a^#(x1)	(17)
c^#(c(a(a(x1))))	→	b^#(b(a(a(c(c(x1))))))	(16)
c^#(c(a(a(x1))))	→	c^#(x1)	(15)
c^#(c(a(a(x1))))	→	a^#(a(c(c(x1))))	(14)
b^#(b(a(a(x1))))	→	a^#(b(b(x1)))	(6)
b^#(b(b(b(x1))))	→	a^#(a(a(x1)))	(13)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[b(x₁)]	=	x₁ + 0
[c(x₁)]	=	x₁ + 8366
[c^#(x₁)]	=	x₁ + 8366
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

b(b(b(b(x1))))	→	a(a(a(a(a(a(x1))))))	(4)
a(a(a(a(x1))))	→	b(b(x1))	(1)
b(b(b(b(c(c(x1))))))	→	c(c(a(a(x1))))	(3)
c(c(a(a(x1))))	→	b(b(a(a(c(c(x1))))))	(5)
b(b(a(a(x1))))	→	a(a(b(b(x1))))	(2)

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

b^#(b(b(b(c(c(x1))))))	→	a^#(x1)	(12)
b^#(b(b(b(c(c(x1))))))	→	c^#(a(a(x1)))	(11)
b^#(b(b(b(c(c(x1))))))	→	a^#(a(x1))	(8)
c^#(c(a(a(x1))))	→	c^#(x1)	(15)

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

b^#(b(b(b(x1))))	→	a^#(x1)	(17)
b^#(b(b(b(x1))))	→	a^#(a(x1))	(20)
b^#(b(b(b(x1))))	→	a^#(a(a(x1)))	(13)
b^#(b(b(b(x1))))	→	a^#(a(a(a(x1))))	(19)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(27)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(a(x1))))))	(18)
a^#(a(a(a(x1))))	→	b^#(x1)	(21)
a^#(a(a(a(x1))))	→	b^#(b(x1))	(25)
b^#(b(b(b(c(c(x1))))))	→	c^#(c(a(a(x1))))	(10)
c^#(c(a(a(x1))))	→	c^#(c(x1))	(9)
c^#(c(a(a(x1))))	→	a^#(c(c(x1)))	(24)
c^#(c(a(a(x1))))	→	a^#(a(c(c(x1))))	(14)
c^#(c(a(a(x1))))	→	b^#(a(a(c(c(x1)))))	(7)
c^#(c(a(a(x1))))	→	b^#(b(a(a(c(c(x1))))))	(16)
b^#(b(a(a(x1))))	→	b^#(x1)	(26)
b^#(b(a(a(x1))))	→	b^#(b(x1))	(22)
b^#(b(a(a(x1))))	→	a^#(b(b(x1)))	(6)
b^#(b(a(a(x1))))	→	a^#(a(b(b(x1))))	(23)

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)]

x₁ +

[b(x₁)]

x₁ +

[c(x₁)]

1	1
1	0

· x₁ +

39405

[c^#(x₁)]

1	0
0	0

· x₁ +

39405

[a^#(x₁)]

0	1
0	0

· x₁ +

[b^#(x₁)]

0	1
0	0

· x₁ +

together with the usable rules

b(b(b(b(x1))))	→	a(a(a(a(a(a(x1))))))	(4)
a(a(a(a(x1))))	→	b(b(x1))	(1)
b(b(b(b(c(c(x1))))))	→	c(c(a(a(x1))))	(3)
c(c(a(a(x1))))	→	b(b(a(a(c(c(x1))))))	(5)
b(b(a(a(x1))))	→	a(a(b(b(x1))))	(2)

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

b^#(b(b(b(x1))))	→	a^#(x1)	(17)
b^#(b(b(b(x1))))	→	a^#(a(x1))	(20)
b^#(b(b(b(x1))))	→	a^#(a(a(x1)))	(13)
b^#(b(b(b(x1))))	→	a^#(a(a(a(x1))))	(19)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(x1)))))	(27)
b^#(b(b(b(x1))))	→	a^#(a(a(a(a(a(x1))))))	(18)
a^#(a(a(a(x1))))	→	b^#(x1)	(21)
a^#(a(a(a(x1))))	→	b^#(b(x1))	(25)
b^#(b(b(b(c(c(x1))))))	→	c^#(c(a(a(x1))))	(10)
c^#(c(a(a(x1))))	→	c^#(c(x1))	(9)
c^#(c(a(a(x1))))	→	a^#(c(c(x1)))	(24)
c^#(c(a(a(x1))))	→	a^#(a(c(c(x1))))	(14)
c^#(c(a(a(x1))))	→	b^#(a(a(c(c(x1)))))	(7)
c^#(c(a(a(x1))))	→	b^#(b(a(a(c(c(x1))))))	(16)
b^#(b(a(a(x1))))	→	b^#(x1)	(26)
b^#(b(a(a(x1))))	→	b^#(b(x1))	(22)
b^#(b(a(a(x1))))	→	a^#(b(b(x1)))	(6)
b^#(b(a(a(x1))))	→	a^#(a(b(b(x1))))	(23)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.