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

Using the Max-polynomial interpretation

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

app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(17)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(app(takeWhile,p),xs)	(10)
app^#(app(takeWhile,p),app(app(cons,x),xs))	→	app^#(p,x)	(9)
app^#(app(dropWhile,p),app(app(cons,x),xs))	→	app^#(app(dropWhile,p),xs)	(13)

could be deleted.

The dependency pairs are split into 0 components.