Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/026)

The rewrite relation of the following TRS is considered.

app(app(map,f),nil)	→	nil	(1)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(2)
app(app(app(comp,f),g),x)	→	app(f,app(g,x))	(3)
app(twice,f)	→	app(app(comp,f),f)	(4)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(5)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(6)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(7)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(8)
app^#(app(app(comp,f),g),x)	→	app^#(g,x)	(9)
app^#(app(app(comp,f),g),x)	→	app^#(f,app(g,x))	(10)
app^#(twice,f)	→	app^#(comp,f)	(11)
app^#(twice,f)	→	app^#(app(comp,f),f)	(12)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

We use the projection

π(app^#)

and remove the pairs:

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

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