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)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

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

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^#(f,x)	(10)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter,f),xs)	(6)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter,f),xs)	(6)

Using the Max-polynomial interpretation

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

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(10)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter,f),xs)	(6)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter,f),xs)	(6)

could be deleted.

The dependency pairs are split into 0 components.