Certification Problem

Input (TPDB TRS_Standard/SK90/4.41)

The rewrite relation of the following TRS is considered.

f(a,y)	→	f(y,g(y))	(1)
g(a)	→	b	(2)
g(b)	→	b	(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,y)	→	g^#(y)	(4)
f^#(a,y)	→	f^#(y,g(y))	(5)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(a,y) → f^#(y,g(y)) (5)

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

[a] = 2

[b] = 1

[f(x₁, x₂)] = 0

[f^#(x₁, x₂)] = x₁ + x₂ + 0

[g^#(x₁)] = 0

[g(x₁)] = 1

together with the usable rules

g(b) → b (3)

g(a) → b (2)

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

The dependency pairs are split into 0 components.