Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_9_Luc02c_FR)

The rewrite relation of the following TRS is considered.

2nd(cons1(X,cons(Y,Z)))	→	Y	(1)
2nd(cons(X,X1))	→	2nd(cons1(X,activate(X1)))	(2)
from(X)	→	cons(X,n__from(n__s(X)))	(3)
from(X)	→	n__from(X)	(4)
s(X)	→	n__s(X)	(5)
activate(n__from(X))	→	from(activate(X))	(6)
activate(n__s(X))	→	s(activate(X))	(7)
activate(X)	→	X	(8)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

2nd^#(cons(X,X1))	→	activate^#(X1)	(9)
2nd^#(cons(X,X1))	→	2nd^#(cons1(X,activate(X1)))	(10)
activate^#(n__from(X))	→	activate^#(X)	(11)
activate^#(n__from(X))	→	from^#(activate(X))	(12)
activate^#(n__s(X))	→	activate^#(X)	(13)
activate^#(n__s(X))	→	s^#(activate(X))	(14)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

activate^#(n__from(X)) → activate^#(X) (11)

activate^#(n__s(X)) → activate^#(X) (13)

1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

activate^#(n__from(X)) → activate^#(X) (11)

1 > 1

activate^#(n__s(X)) → activate^#(X) (13)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.