Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/022)

The rewrite relation of the following TRS is considered.

app(app(mapt,f),app(leaf,x))	→	app(leaf,app(f,x))	(1)
app(app(mapt,f),app(node,xs))	→	app(node,app(app(maptlist,f),xs))	(2)
app(app(maptlist,f),nil)	→	nil	(3)
app(app(maptlist,f),app(app(cons,x),xs))	→	app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))	(4)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(mapt,f),app(leaf,x))	→	app^#(f,x)	(5)
app^#(app(mapt,f),app(leaf,x))	→	app^#(leaf,app(f,x))	(6)
app^#(app(mapt,f),app(node,xs))	→	app^#(maptlist,f)	(7)
app^#(app(mapt,f),app(node,xs))	→	app^#(app(maptlist,f),xs)	(8)
app^#(app(mapt,f),app(node,xs))	→	app^#(node,app(app(maptlist,f),xs))	(9)
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(app(maptlist,f),xs)	(10)
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(mapt,f)	(11)
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(app(mapt,f),x)	(12)
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(cons,app(app(mapt,f),x))	(13)
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))	(14)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(app(maptlist,f),xs)	(10)
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(app(mapt,f),x)	(12)
app^#(app(mapt,f),app(node,xs))	→	app^#(app(maptlist,f),xs)	(8)
app^#(app(mapt,f),app(leaf,x))	→	app^#(f,x)	(5)

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

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

2	>	2
1	≥	1
app^#(app(maptlist,f),app(app(cons,x),xs))	→	app^#(app(mapt,f),x)	(12)

2	>	2
app^#(app(mapt,f),app(node,xs))	→	app^#(app(maptlist,f),xs)	(8)

2	>	2
app^#(app(mapt,f),app(leaf,x))	→	app^#(f,x)	(5)

2	>	2
1	>	1

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