Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z067)

The rewrite relation of the following TRS is considered.

P(x1)	→	Q(Q(p(x1)))	(1)
p(p(x1))	→	q(q(x1))	(2)
p(Q(Q(x1)))	→	Q(Q(p(x1)))	(3)
Q(p(q(x1)))	→	q(p(Q(x1)))	(4)
q(q(p(x1)))	→	p(q(q(x1)))	(5)
q(Q(x1))	→	x1	(6)
Q(q(x1))	→	x1	(7)
p(P(x1))	→	x1	(8)
P(p(x1))	→	x1	(9)

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.

p^#(p(x1))	→	q^#(x1)	(10)
p^#(Q(Q(x1)))	→	Q^#(Q(p(x1)))	(11)
Q^#(p(q(x1)))	→	p^#(Q(x1))	(12)
P^#(x1)	→	Q^#(Q(p(x1)))	(13)
P^#(x1)	→	p^#(x1)	(14)
Q^#(p(q(x1)))	→	q^#(p(Q(x1)))	(15)
Q^#(p(q(x1)))	→	Q^#(x1)	(16)
p^#(Q(Q(x1)))	→	Q^#(p(x1))	(17)
P^#(x1)	→	Q^#(p(x1))	(18)
q^#(q(p(x1)))	→	q^#(x1)	(19)
q^#(q(p(x1)))	→	p^#(q(q(x1)))	(20)
p^#(p(x1))	→	q^#(q(x1))	(21)
p^#(Q(Q(x1)))	→	p^#(x1)	(22)
q^#(q(p(x1)))	→	q^#(q(x1))	(23)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

q^#(q(p(x1)))	→	q^#(q(x1))	(23)
Q^#(p(q(x1)))	→	Q^#(x1)	(16)
Q^#(p(q(x1)))	→	q^#(p(Q(x1)))	(15)
p^#(Q(Q(x1)))	→	p^#(x1)	(22)
p^#(p(x1))	→	q^#(q(x1))	(21)
q^#(q(p(x1)))	→	p^#(q(q(x1)))	(20)
Q^#(p(q(x1)))	→	p^#(Q(x1))	(12)
p^#(Q(Q(x1)))	→	Q^#(Q(p(x1)))	(11)
q^#(q(p(x1)))	→	q^#(x1)	(19)
p^#(Q(Q(x1)))	→	Q^#(p(x1))	(17)
p^#(p(x1))	→	q^#(x1)	(10)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[P^#(x₁)]	=	0
[Q(x₁)]	=	x₁ + 3253
[q(x₁)]	=	x₁ + 10451
[Q^#(x₁)]	=	x₁ + 0
[p^#(x₁)]	=	x₁ + 7199
[p(x₁)]	=	x₁ + 10452
[P(x₁)]	=	x₁ + 36466
[q^#(x₁)]	=	x₁ + 7198

together with the usable rules

Q(p(q(x1)))	→	q(p(Q(x1)))	(4)
p(P(x1))	→	x1	(8)
p(Q(Q(x1)))	→	Q(Q(p(x1)))	(3)
q(q(p(x1)))	→	p(q(q(x1)))	(5)
Q(q(x1))	→	x1	(7)
q(Q(x1))	→	x1	(6)
p(p(x1))	→	q(q(x1))	(2)

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

q^#(q(p(x1)))	→	q^#(q(x1))	(23)
Q^#(p(q(x1)))	→	Q^#(x1)	(16)
p^#(Q(Q(x1)))	→	p^#(x1)	(22)
p^#(p(x1))	→	q^#(q(x1))	(21)
Q^#(p(q(x1)))	→	p^#(Q(x1))	(12)
q^#(q(p(x1)))	→	q^#(x1)	(19)
p^#(Q(Q(x1)))	→	Q^#(p(x1))	(17)
p^#(p(x1))	→	q^#(x1)	(10)

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

Q^#(p(q(x1)))	→	q^#(p(Q(x1)))	(15)
p^#(Q(Q(x1)))	→	Q^#(Q(p(x1)))	(11)
q^#(q(p(x1)))	→	p^#(q(q(x1)))	(20)

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

[P^#(x₁)]

[Q(x₁)]

0	1
1	0

· x₁ +

[q(x₁)]

0	1
1	0

· x₁ +

[Q^#(x₁)]

0	1
0	0

· x₁ +

[p^#(x₁)]

1	1
0	0

· x₁ +

17625

[p(x₁)]

1	1
1	1

· x₁ +

17623

17627

[P(x₁)]

1	1
1	1

· x₁ +

26763

18345

[q^#(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

Q(p(q(x1)))	→	q(p(Q(x1)))	(4)
p(P(x1))	→	x1	(8)
p(Q(Q(x1)))	→	Q(Q(p(x1)))	(3)
q(q(p(x1)))	→	p(q(q(x1)))	(5)
Q(q(x1))	→	x1	(7)
q(Q(x1))	→	x1	(6)
p(p(x1))	→	q(q(x1))	(2)

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

Q^#(p(q(x1)))	→	q^#(p(Q(x1)))	(15)
p^#(Q(Q(x1)))	→	Q^#(Q(p(x1)))	(11)
q^#(q(p(x1)))	→	p^#(q(q(x1)))	(20)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.