Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z081)

The rewrite relation of the following TRS is considered.

b(c(a(x1)))	→	a(b(x1))	(1)
b(b(b(x1)))	→	c(a(c(x1)))	(2)
c(d(x1))	→	d(c(x1))	(3)
c(d(b(x1)))	→	d(c(c(x1)))	(4)
d(c(x1))	→	b(b(b(x1)))	(5)
c(b(x1))	→	d(a(x1))	(6)
d(b(c(x1)))	→	a(a(x1))	(7)
d(a(x1))	→	b(x1)	(8)

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(x1)))	→	c^#(x1)	(9)
d^#(c(x1))	→	b^#(x1)	(10)
c^#(d(b(x1)))	→	c^#(x1)	(11)
b^#(c(a(x1)))	→	b^#(x1)	(12)
c^#(d(b(x1)))	→	d^#(c(c(x1)))	(13)
c^#(d(b(x1)))	→	c^#(c(x1))	(14)
c^#(d(x1))	→	d^#(c(x1))	(15)
d^#(a(x1))	→	b^#(x1)	(16)
d^#(c(x1))	→	b^#(b(x1))	(17)
c^#(b(x1))	→	d^#(a(x1))	(18)
b^#(b(b(x1)))	→	c^#(a(c(x1)))	(19)
c^#(d(x1))	→	c^#(x1)	(20)
d^#(c(x1))	→	b^#(b(b(x1)))	(21)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(c(x1))	→	b^#(b(b(x1)))	(21)
c^#(d(b(x1)))	→	c^#(c(x1))	(14)
c^#(d(x1))	→	c^#(x1)	(20)
c^#(b(x1))	→	d^#(a(x1))	(18)
c^#(d(b(x1)))	→	d^#(c(c(x1)))	(13)
b^#(c(a(x1)))	→	b^#(x1)	(12)
c^#(d(b(x1)))	→	c^#(x1)	(11)
d^#(c(x1))	→	b^#(x1)	(10)
d^#(c(x1))	→	b^#(b(x1))	(17)
d^#(a(x1))	→	b^#(x1)	(16)
c^#(d(x1))	→	d^#(c(x1))	(15)
b^#(b(b(x1)))	→	c^#(x1)	(9)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[d(x₁)]	=	x₁ + 61230
[b(x₁)]	=	x₁ + 30615
[c(x₁)]	=	x₁ + 30615
[d^#(x₁)]	=	x₁ + 30617
[c^#(x₁)]	=	x₁ + 4
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

c(d(b(x1)))	→	d(c(c(x1)))	(4)
d(a(x1))	→	b(x1)	(8)
b(c(a(x1)))	→	a(b(x1))	(1)
c(d(x1))	→	d(c(x1))	(3)
d(c(x1))	→	b(b(b(x1)))	(5)
d(b(c(x1)))	→	a(a(x1))	(7)
c(b(x1))	→	d(a(x1))	(6)
b(b(b(x1)))	→	c(a(c(x1)))	(2)

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

d^#(c(x1))	→	b^#(b(b(x1)))	(21)
c^#(d(b(x1)))	→	c^#(c(x1))	(14)
c^#(d(x1))	→	c^#(x1)	(20)
c^#(b(x1))	→	d^#(a(x1))	(18)
c^#(d(b(x1)))	→	d^#(c(c(x1)))	(13)
b^#(c(a(x1)))	→	b^#(x1)	(12)
c^#(d(b(x1)))	→	c^#(x1)	(11)
d^#(c(x1))	→	b^#(x1)	(10)
d^#(c(x1))	→	b^#(b(x1))	(17)
d^#(a(x1))	→	b^#(x1)	(16)
c^#(d(x1))	→	d^#(c(x1))	(15)
b^#(b(b(x1)))	→	c^#(x1)	(9)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.