Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/02)

The rewrite relation of the following TRS is considered.

app(app(.,1),x)	→	x	(1)
app(app(.,x),1)	→	x	(2)
app(app(.,app(i,x)),x)	→	1	(3)
app(app(.,x),app(i,x))	→	1	(4)
app(app(.,app(i,y)),app(app(.,y),z))	→	z	(5)
app(app(.,y),app(app(.,app(i,y)),z))	→	z	(6)
app(i,1)	→	1	(7)
app(i,app(i,x))	→	x	(8)
app(app(map,f),nil)	→	nil	(9)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(10)
app(app(filter,f),nil)	→	nil	(11)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(12)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(13)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(14)

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)	(15)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(16)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(17)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(18)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(19)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(20)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(21)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(22)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(23)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(24)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(25)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(26)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(27)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(28)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(29)

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)	(23)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(29)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(19)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(25)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(16)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(15)

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)	(23)

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

2	≥	2
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(19)

2	>	2
1	>	1
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(25)

2	≥	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(16)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(15)

2	>	2
1	≥	1

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