Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex15_Luc06_FR)

The rewrite relation of the following TRS is considered.

f(n__f(n__a))	→	f(n__g(n__f(n__a)))	(1)
f(X)	→	n__f(X)	(2)
a	→	n__a	(3)
g(X)	→	n__g(X)	(4)
activate(n__f(X))	→	f(X)	(5)
activate(n__a)	→	a	(6)
activate(n__g(X))	→	g(activate(X))	(7)
activate(X)	→	X	(8)

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.

activate^#(n__g(X))	→	g^#(activate(X))	(9)
f^#(n__f(n__a))	→	f^#(n__g(n__f(n__a)))	(10)
activate^#(n__g(X))	→	activate^#(X)	(11)
activate^#(n__a)	→	a^#	(12)
activate^#(n__f(X))	→	f^#(X)	(13)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

activate^#(n__g(X))

→

activate^#(X)

(11)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a]	=	0
[n__a]	=	0
[activate(x₁)]	=	0
[activate^#(x₁)]	=	x₁ + 0
[f(x₁)]	=	0
[n__f(x₁)]	=	0
[f^#(x₁)]	=	0
[g^#(x₁)]	=	0
[n__g(x₁)]	=	x₁ + 1
[a^#]	=	0
[g(x₁)]	=	0

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

activate^#(n__g(X))

→

activate^#(X)

(11)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.