Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z083)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals

[b(x₁)]	=	1 · x₁ + 1
[d(x₁)]	=	1 · x₁
[c(x₁)]	=	1 · x₁ + 1
[a(x₁)]	=	1 · x₁ + 1

all of the following rules can be deleted.

b(a(c(x1)))

→

b(c(x1))

(2)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(d(x1))	→	d^#(c(x1))	(8)
b^#(b(b(x1)))	→	a^#(b(c(x1)))	(9)
b^#(b(b(x1)))	→	b^#(c(x1))	(10)
d^#(c(x1))	→	b^#(d(x1))	(11)
d^#(c(x1))	→	d^#(x1)	(12)
d^#(c(x1))	→	d^#(b(d(x1)))	(13)
d^#(a(c(x1)))	→	b^#(b(x1))	(14)
d^#(a(c(x1)))	→	b^#(x1)	(15)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(c(x1))	→	d^#(b(d(x1)))	(13)
d^#(c(x1))	→	d^#(x1)	(12)

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[d(x₁)]	=	1 · x₁
[c(x₁)]	=	1 · x₁
[b(x₁)]	=	1 · x₁
[a(x₁)]	=	1 · x₁
[d^#(x₁)]	=	1 · x₁

together with the usable rules

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

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[d^#(x₁)]	=	1 · x₁
[c(x₁)]	=	1 + 1 · x₁
[b(x₁)]	=	1 + 1 · x₁
[d(x₁)]	=	1 · x₁
[a(x₁)]	=	1 + 1 · x₁

the pair

d^#(c(x1))

→

d^#(x1)

(12)

could be deleted.

1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[d^#(x₁)]

-∞

-∞	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

0	0	0
-∞	-∞	-∞
1	0	1

· x₁

[b(x₁)]

-∞

0	1	0
-∞	0	-∞
-∞	0	-∞

· x₁

[d(x₁)]

-∞	0	0
0	-∞	0
-∞	0	0

· x₁

[a(x₁)]

-∞

-∞	0	0
-∞	0	1
-∞	0	0

· x₁

the pair

d^#(c(x1))

→

d^#(b(d(x1)))

(13)

could be deleted.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.