Certification Problem

Input (TPDB TRS_Standard/HirokawaMiddeldorp_04/t011)

The rewrite relation of the following TRS is considered.

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

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^#(f(x),y)	→	h^#(x,y)	(3)
h^#(x,y)	→	g^#(x,f(y))	(4)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

h^#(x,y) → g^#(x,f(y)) (4)

g^#(f(x),y) → h^#(x,y) (3)

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

[h(x₁, x₂)] = 0

[f(x₁)] = x₁ + 2

[h^#(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

h^#(x,y) → g^#(x,f(y)) (4)

g^#(f(x),y) → h^#(x,y) (3)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.