Certification Problem

Input (TPDB TRS_Standard/Applicative_05/TakeDropWhile)

The rewrite relation of the following TRS is considered.

app(app(app(if,true),x),y)	→	x	(1)
app(app(app(if,true),x),y)	→	y	(2)
app(app(takeWhile,p),nil)	→	nil	(3)
app(app(takeWhile,p),app(app(cons,x),xs))	→	app(app(app(if,app(p,x)),app(app(cons,x),app(app(takeWhile,p),xs))),nil)	(4)
app(app(dropWhile,p),nil)	→	nil	(5)
app(app(dropWhile,p),app(app(cons,x),xs))	→	app(app(app(if,app(p,x)),app(app(dropWhile,p),xs)),app(app(cons,x),xs))	(6)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(takeWhile,p),xs)	(7)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(cons,x),app(app(takeWhile,p),xs))	(8)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(9)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(if,app(p,x))	(10)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(if,app(p,x)),app(app(cons,x),app(app(takeWhile,p),xs)))	(11)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(app(if,app(p,x)),app(app(cons,x),app(app(takeWhile,p),xs))),nil)	(12)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(app(dropWhile,p),xs)	(13)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(14)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(if,app(p,x))	(15)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(app(if,app(p,x)),app(app(dropWhile,p),xs))	(16)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(app(app(if,app(p,x)),app(app(dropWhile,p),xs)),app(app(cons,x),xs))	(17)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(app(dropWhile,p),xs)	(13)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(14)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(9)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(takeWhile,p),xs)	(7)

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

app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(app(dropWhile,p),xs)	(13)

2	>	2
1	≥	1
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(14)

2	>	2
1	>	1
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(9)

2	>	2
1	>	1
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(takeWhile,p),xs)	(7)

2	>	2
1	≥	1

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