Certification Problem

Input (TPDB TRS_Standard/SK90/2.53)

The rewrite relation of the following TRS is considered.

f(x,y)	→	g(x,y)	(1)
g(h(x),y)	→	h(f(x,y))	(2)
g(h(x),y)	→	h(g(x,y))	(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.

g^#(h(x),y)	→	g^#(x,y)	(4)
f^#(x,y)	→	g^#(x,y)	(5)
g^#(h(x),y)	→	f^#(x,y)	(6)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

g^#(h(x),y) → f^#(x,y) (6)

f^#(x,y) → g^#(x,y) (5)

g^#(h(x),y) → g^#(x,y) (4)

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

[h(x₁)] = x₁ + 2

[f(x₁, x₂)] = 0

[f^#(x₁, x₂)] = x₁ + 1

[g^#(x₁, x₂)] = x₁ + 0

[g(x₁, x₂)] = 0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

g^#(h(x),y) → f^#(x,y) (6)

f^#(x,y) → g^#(x,y) (5)

g^#(h(x),y) → g^#(x,y) (4)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.