Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-559)

The rewrite relation of the following TRS is considered.

a(b(x1))	→	x1	(1)
a(c(x1))	→	c(b(c(c(x1))))	(2)
b(c(x1))	→	a(b(x1))	(3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(c(x1))	→	b^#(c(c(x1)))	(4)
b^#(c(x1))	→	a^#(b(x1))	(5)
b^#(c(x1))	→	b^#(x1)	(6)

1.1 Reduction Pair Processor

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

[a^#(x₁)]

-∞

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[c(x₁)]

-∞

-∞	-1	-∞
-∞	1	-∞
-1	0	1

· x₁

[b^#(x₁)]

-∞

-∞	-1	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[b(x₁)]

-∞

-1	0	1
-∞	-1	-∞
-∞	-∞	-1

· x₁

[a(x₁)]

-∞

1	-1	-∞
-∞	1	-∞
-∞	0	1

· x₁

the pair

b^#(c(x1))

→

a^#(b(x1))

(5)

could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
b^#(c(x1)) → b^#(x1) (6)

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[c(x₁)] = 1 · x₁

[b^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

b^#(c(x1)) → b^#(x1) (6)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.

Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-559)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Reduction Pair Processor

1.1.1 Dependency Graph Processor

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1 Size-Change Termination