Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/002)

The rewrite relation of the following TRS is considered.

app(app(filter,f),nil)	→	nil	(1)
app(app(filter,f),app(app(cons,y),ys))	→	app(app(app(filtersub,app(f,y)),f),app(app(cons,y),ys))	(2)
app(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app(app(cons,y),app(app(filter,f),ys))	(3)
app(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app(app(filter,f),ys)	(4)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(filter,f),app(app(cons,y),ys))	→	app^#(f,y)	(5)
app^#(app(filter,f),app(app(cons,y),ys))	→	app^#(filtersub,app(f,y))	(6)
app^#(app(filter,f),app(app(cons,y),ys))	→	app^#(app(filtersub,app(f,y)),f)	(7)
app^#(app(filter,f),app(app(cons,y),ys))	→	app^#(app(app(filtersub,app(f,y)),f),app(app(cons,y),ys))	(8)
app^#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app^#(filter,f)	(9)
app^#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app^#(app(filter,f),ys)	(10)
app^#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app^#(app(cons,y),app(app(filter,f),ys))	(11)
app^#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app^#(filter,f)	(12)
app^#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app^#(app(filter,f),ys)	(13)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app^#(app(filter,f),ys)	(13)
app^#(app(filter,f),app(app(cons,y),ys))	→	app^#(f,y)	(5)
app^#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app^#(app(filter,f),ys)	(10)

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

app^#(app(app(filtersub,false),f),app(app(cons,y),ys))	→	app^#(app(filter,f),ys)	(13)

2	>	2
app^#(app(filter,f),app(app(cons,y),ys))	→	app^#(f,y)	(5)

2	>	2
1	>	1
app^#(app(app(filtersub,true),f),app(app(cons,y),ys))	→	app^#(app(filter,f),ys)	(10)

2	>	2

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