Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z115)

The rewrite relation of the following TRS is considered.

a(a(x1))	→	d(c(x1))	(1)
a(b(x1))	→	c(c(c(x1)))	(2)
b(b(x1))	→	a(c(c(x1)))	(3)
c(c(x1))	→	b(x1)	(4)
c(d(x1))	→	a(a(x1))	(5)
d(d(x1))	→	b(a(c(x1)))	(6)

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.

a^#(b(x1))	→	c^#(c(x1))	(7)
d^#(d(x1))	→	c^#(x1)	(8)
a^#(a(x1))	→	c^#(x1)	(9)
b^#(b(x1))	→	a^#(c(c(x1)))	(10)
b^#(b(x1))	→	c^#(x1)	(11)
a^#(a(x1))	→	d^#(c(x1))	(12)
c^#(c(x1))	→	b^#(x1)	(13)
c^#(d(x1))	→	a^#(a(x1))	(14)
b^#(b(x1))	→	c^#(c(x1))	(15)
d^#(d(x1))	→	a^#(c(x1))	(16)
a^#(b(x1))	→	c^#(x1)	(17)
a^#(b(x1))	→	c^#(c(c(x1)))	(18)
c^#(d(x1))	→	a^#(x1)	(19)
d^#(d(x1))	→	b^#(a(c(x1)))	(20)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(d(x1))	→	b^#(a(c(x1)))	(20)
c^#(c(x1))	→	b^#(x1)	(13)
a^#(a(x1))	→	d^#(c(x1))	(12)
c^#(d(x1))	→	a^#(x1)	(19)
a^#(b(x1))	→	c^#(c(c(x1)))	(18)
a^#(b(x1))	→	c^#(x1)	(17)
b^#(b(x1))	→	c^#(x1)	(11)
b^#(b(x1))	→	a^#(c(c(x1)))	(10)
a^#(a(x1))	→	c^#(x1)	(9)
d^#(d(x1))	→	c^#(x1)	(8)
d^#(d(x1))	→	a^#(c(x1))	(16)
b^#(b(x1))	→	c^#(c(x1))	(15)
c^#(d(x1))	→	a^#(a(x1))	(14)
a^#(b(x1))	→	c^#(c(x1))	(7)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 19502
[d(x₁)]	=	x₁ + 26816
[b(x₁)]	=	x₁ + 21939
[c(x₁)]	=	x₁ + 12188
[d^#(x₁)]	=	x₁ + 9751
[c^#(x₁)]	=	x₁ + 0
[a^#(x₁)]	=	x₁ + 2438
[b^#(x₁)]	=	x₁ + 4876

together with the usable rules

c(c(x1))	→	b(x1)	(4)
a(a(x1))	→	d(c(x1))	(1)
b(b(x1))	→	a(c(c(x1)))	(3)
c(d(x1))	→	a(a(x1))	(5)
d(d(x1))	→	b(a(c(x1)))	(6)
a(b(x1))	→	c(c(c(x1)))	(2)

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

d^#(d(x1))	→	b^#(a(c(x1)))	(20)
c^#(c(x1))	→	b^#(x1)	(13)
a^#(a(x1))	→	d^#(c(x1))	(12)
c^#(d(x1))	→	a^#(x1)	(19)
a^#(b(x1))	→	c^#(c(c(x1)))	(18)
a^#(b(x1))	→	c^#(x1)	(17)
b^#(b(x1))	→	c^#(x1)	(11)
b^#(b(x1))	→	a^#(c(c(x1)))	(10)
a^#(a(x1))	→	c^#(x1)	(9)
d^#(d(x1))	→	c^#(x1)	(8)
d^#(d(x1))	→	a^#(c(x1))	(16)
b^#(b(x1))	→	c^#(c(x1))	(15)
c^#(d(x1))	→	a^#(a(x1))	(14)
a^#(b(x1))	→	c^#(c(x1))	(7)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.