Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/01)

The rewrite relation of the following TRS is considered.

app(app(\,x),x)	→	e	(1)
app(app(\,e),x)	→	x	(2)
app(app(\,x),app(app(.,x),y))	→	y	(3)
app(app(\,app(app(/,x),y)),x)	→	y	(4)
app(app(/,x),x)	→	e	(5)
app(app(/,x),e)	→	x	(6)
app(app(/,app(app(.,y),x)),x)	→	y	(7)
app(app(/,x),app(app(\,y),x))	→	y	(8)
app(app(.,e),x)	→	x	(9)
app(app(.,x),e)	→	x	(10)
app(app(.,x),app(app(\,x),y))	→	y	(11)
app(app(.,app(app(/,y),x)),x)	→	y	(12)
app(app(map,f),nil)	→	nil	(13)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(14)
app(app(filter,f),nil)	→	nil	(15)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(16)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(17)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(18)

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)	(19)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(20)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(21)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(22)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(23)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(24)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(25)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(26)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(27)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(28)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(29)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(30)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(31)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(32)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(33)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(27)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(33)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(23)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(29)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(20)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(19)

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

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(27)

2	>	2
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(33)

2	≥	2
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(23)

2	>	2
1	>	1
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(29)

2	≥	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(20)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(19)

2	>	2
1	≥	1

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