Certification Problem

Input (TPDB SRS_Standard/Zantema_06/beans4)

The rewrite relation of the following TRS is considered.

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

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 8856
[b(x₁)]	=	x₁ + 2942
[c(x₁)]	=	x₁ + 8856
[c^#(x₁)]	=	x₁ + 5914
[a^#(x₁)]	=	x₁ + 5914
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

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

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

a^#(c(b(x1)))	→	b^#(a(x1))	(18)
b^#(c(a(x1)))	→	c^#(x1)	(17)
b^#(a(a(x1)))	→	c^#(x1)	(10)
c^#(b(x1))	→	a^#(x1)	(8)
a^#(c(b(x1)))	→	a^#(x1)	(14)
b^#(a(a(x1)))	→	b^#(c(x1))	(13)
b^#(c(a(x1)))	→	b^#(c(x1))	(12)
c^#(a(x1))	→	c^#(x1)	(6)

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^#(b(x1))	→	b^#(a(x1))	(11)
b^#(a(a(x1)))	→	a^#(b(c(x1)))	(9)
b^#(c(a(x1)))	→	a^#(b(c(x1)))	(7)
a^#(c(b(x1)))	→	c^#(b(a(x1)))	(15)
c^#(a(x1))	→	a^#(c(x1))	(16)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[b(x₁)]	=	1
[c(x₁)]	=	25909
[c^#(x₁)]	=	25909
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	2

together with the usable rules

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

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

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

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(c(b(x1)))	→	c^#(b(a(x1)))	(15)
c^#(a(x1))	→	a^#(c(x1))	(16)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

0	1
1	1

· x₁ +

41067

[b(x₁)]

0	1
1	1

· x₁ +

41069

[c(x₁)]

0	1
1	1

· x₁ +

41065

[c^#(x₁)]

1	1
1	1

· x₁ +

[a^#(x₁)]

1	1
1	1

· x₁ +

[b^#(x₁)]

together with the usable rules

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

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

a^#(c(b(x1)))	→	c^#(b(a(x1)))	(15)
c^#(a(x1))	→	a^#(c(x1))	(16)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.