Certification Problem

Input (TPDB TRS_Standard/Mixed_TRS/jones2)

The rewrite relation of the following TRS is considered.

f(empty,l)	→	l	(1)
f(cons(x,k),l)	→	g(k,l,cons(x,k))	(2)
g(a,b,c)	→	f(a,cons(b,c))	(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^#(a,b,c)	→	f^#(a,cons(b,c))	(4)
f^#(cons(x,k),l)	→	g^#(k,l,cons(x,k))	(5)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(cons(x,k),l) → g^#(k,l,cons(x,k)) (5)

g^#(a,b,c) → f^#(a,cons(b,c)) (4)

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

[f(x₁, x₂)] = 0

[f^#(x₁, x₂)] = x₁ + 0

[g^#(x₁, x₂, x₃)] = x₁ + 1

[cons(x₁, x₂)] = x₂ + 2

[empty] = 0

[g(x₁, x₂, x₃)] = 0

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

f^#(cons(x,k),l) → g^#(k,l,cons(x,k)) (5)

g^#(a,b,c) → f^#(a,cons(b,c)) (4)

could be deleted.
1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.