Certification Problem

Input (TPDB TRS_Standard/Applicative_05/Ex8Polymorphic)

The rewrite relation of the following TRS is considered.

app(app(twice,f),x)	→	app(f,app(f,x))	(1)
app(app(map,f),nil)	→	nil	(2)
app(app(map,f),app(app(cons,h),t))	→	app(app(cons,app(f,h)),app(app(map,f),t))	(3)
app(app(fmap,nil),x)	→	nil	(4)
app(app(fmap,app(app(cons,f),t_f)),x)	→	app(app(cons,app(f,x)),app(app(fmap,t_f),x))	(5)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(twice,f),x)	→	app^#(f,x)	(6)
app^#(app(twice,f),x)	→	app^#(f,app(f,x))	(7)
app^#(app(map,f),app(app(cons,h),t))	→	app^#(app(map,f),t)	(8)
app^#(app(map,f),app(app(cons,h),t))	→	app^#(f,h)	(9)
app^#(app(map,f),app(app(cons,h),t))	→	app^#(cons,app(f,h))	(10)
app^#(app(map,f),app(app(cons,h),t))	→	app^#(app(cons,app(f,h)),app(app(map,f),t))	(11)
app^#(app(fmap,app(app(cons,f),t_f)),x)	→	app^#(fmap,t_f)	(12)
app^#(app(fmap,app(app(cons,f),t_f)),x)	→	app^#(app(fmap,t_f),x)	(13)
app^#(app(fmap,app(app(cons,f),t_f)),x)	→	app^#(f,x)	(14)
app^#(app(fmap,app(app(cons,f),t_f)),x)	→	app^#(cons,app(f,x))	(15)
app^#(app(fmap,app(app(cons,f),t_f)),x)	→	app^#(app(cons,app(f,x)),app(app(fmap,t_f),x))	(16)

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.