Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/015)

The rewrite relation of the following TRS is considered.

app(app(app(if,true),x),y)	→	x	(1)
app(app(app(if,false),x),y)	→	y	(2)
app(app(filter,f),nil)	→	nil	(3)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(if,app(f,x)),app(app(cons,x),app(app(filter,f),xs))),app(app(filter,f),xs))	(4)

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.

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter,f),xs)	(5)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(cons,x),app(app(filter,f),xs))	(6)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(7)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(if,app(f,x))	(8)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(if,app(f,x)),app(app(cons,x),app(app(filter,f),xs)))	(9)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(if,app(f,x)),app(app(cons,x),app(app(filter,f),xs))),app(app(filter,f),xs))	(10)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(filter,f),app(app(cons,x),xs)) → app^#(app(filter,f),xs) (5)

app^#(app(filter,f),app(app(cons,x),xs)) → app^#(f,x) (7)

1.1.1 Size-Change Termination

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

app^#(app(filter,f),app(app(cons,x),xs)) → app^#(app(filter,f),xs) (5)

2 > 2

1 ≥ 1

app^#(app(filter,f),app(app(cons,x),xs)) → app^#(f,x) (7)

2 > 2

1 > 1

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