Certification Problem

Input (TPDB SRS_Standard/Zantema_06/01)

The rewrite relation of the following TRS is considered.

a(b(x1))	→	b(r(x1))	(1)
r(a(x1))	→	d(r(x1))	(2)
r(x1)	→	d(x1)	(3)
d(a(x1))	→	a(a(d(x1)))	(4)
d(x1)	→	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.

r^#(a(x1))	→	r^#(x1)	(6)
d^#(a(x1))	→	a^#(a(d(x1)))	(7)
d^#(a(x1))	→	a^#(d(x1))	(8)
a^#(b(x1))	→	r^#(x1)	(9)
d^#(x1)	→	a^#(x1)	(10)
r^#(a(x1))	→	d^#(r(x1))	(11)
d^#(a(x1))	→	d^#(x1)	(12)
r^#(x1)	→	d^#(x1)	(13)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

r^#(x1)	→	d^#(x1)	(13)
d^#(a(x1))	→	d^#(x1)	(12)
r^#(a(x1))	→	d^#(r(x1))	(11)
d^#(x1)	→	a^#(x1)	(10)
d^#(a(x1))	→	a^#(a(d(x1)))	(7)
a^#(b(x1))	→	r^#(x1)	(9)
d^#(a(x1))	→	a^#(d(x1))	(8)
r^#(a(x1))	→	r^#(x1)	(6)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[d(x₁)]	=	x₁ + 0
[b(x₁)]	=	x₁ + 3
[r(x₁)]	=	x₁ + 0
[d^#(x₁)]	=	x₁ + 1
[r^#(x₁)]	=	x₁ + 2
[a^#(x₁)]	=	x₁ + 0

together with the usable rules

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

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

r^#(x1)	→	d^#(x1)	(13)
r^#(a(x1))	→	d^#(r(x1))	(11)
d^#(x1)	→	a^#(x1)	(10)
d^#(a(x1))	→	a^#(a(d(x1)))	(7)
a^#(b(x1))	→	r^#(x1)	(9)
d^#(a(x1))	→	a^#(d(x1))	(8)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair
d^#(a(x1)) → d^#(x1) (12)

1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation

[a(x₁)] = x₁ + 1

[d(x₁)] = 0

[b(x₁)] = 0

[r(x₁)] = 0

[d^#(x₁)] = x₁ + 0

[r^#(x₁)] = 0

[a^#(x₁)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
d^#(a(x1)) → d^#(x1) (12)
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.
The 2^nd component contains the pair
r^#(a(x1)) → r^#(x1) (6)

1.1.1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation

[a(x₁)] = x₁ + 8367

[d(x₁)] = 8366

[b(x₁)] = x₁ + 3

[r(x₁)] = 0

[d^#(x₁)] = 1

[r^#(x₁)] = x₁ + 2

[a^#(x₁)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
r^#(a(x1)) → r^#(x1) (6)
could be deleted.
1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.