Certification Problem

Input (TPDB TRS_Standard/Applicative_05/TreeMap)

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(treemap,f),app(app(node,x),xs))	→	app(app(node,app(f,x)),app(app(map,app(treemap,f)),xs))	(3)

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)	(4)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(5)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(6)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(7)
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(map,app(treemap,f))	(8)
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(app(map,app(treemap,f)),xs)	(9)
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(f,x)	(10)
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(node,app(f,x))	(11)
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(app(node,app(f,x)),app(app(map,app(treemap,f)),xs))	(12)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(app(map,app(treemap,f)),xs)	(9)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(5)
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(f,x)	(10)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(4)

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

app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(app(map,app(treemap,f)),xs)	(9)

2	>	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(5)

2	>	2
1	>	1
app^#(app(treemap,f),app(app(node,x),xs))	→	app^#(f,x)	(10)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(4)

2	>	2
1	≥	1

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