Certification Problem

Input (TPDB TRS_Standard/Various_04/25)

The rewrite relation of the following TRS is considered.

g(a)	→	g(b)	(1)
b	→	f(a,a)	(2)
f(a,a)	→	g(d)	(3)

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.

f^#(a,a)	→	g^#(d)	(4)
g^#(a)	→	g^#(b)	(5)
b^#	→	f^#(a,a)	(6)
g^#(a)	→	b^#	(7)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
g^#(a) → g^#(b) (5)

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

[a] = 2

[d] = 1

[b] = 1

[f(x₁, x₂)] = 1

[f^#(x₁, x₂)] = 0

[g^#(x₁)] = x₁ + 0

[g(x₁)] = 0

[b^#] = 0

together with the usable rules

g(a) → g(b) (1)

f(a,a) → g(d) (3)

b → f(a,a) (2)

(w.r.t. the implicit argument filter of the reduction pair), the pair
g^#(a) → g^#(b) (5)
could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.