Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/013)

The rewrite relation of the following TRS is considered.

app(app(append,nil),ys)	→	ys	(1)
app(app(append,app(app(cons,x),xs)),ys)	→	app(app(cons,x),app(app(append,xs),ys))	(2)
app(app(flatwith,f),app(leaf,x))	→	app(app(cons,app(f,x)),nil)	(3)
app(app(flatwith,f),app(node,xs))	→	app(app(flatwithsub,f),xs)	(4)
app(app(flatwithsub,f),nil)	→	nil	(5)
app(app(flatwithsub,f),app(app(cons,x),xs))	→	app(app(append,app(app(flatwith,f),x)),app(app(flatwithsub,f),xs))	(6)

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(append,xs)	(7)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(app(append,xs),ys)	(8)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(app(cons,x),app(app(append,xs),ys))	(9)
app^#(app(flatwith,f),app(leaf,x))	→	app^#(f,x)	(10)
app^#(app(flatwith,f),app(leaf,x))	→	app^#(cons,app(f,x))	(11)
app^#(app(flatwith,f),app(leaf,x))	→	app^#(app(cons,app(f,x)),nil)	(12)
app^#(app(flatwith,f),app(node,xs))	→	app^#(flatwithsub,f)	(13)
app^#(app(flatwith,f),app(node,xs))	→	app^#(app(flatwithsub,f),xs)	(14)
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(flatwithsub,f),xs)	(15)
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(flatwith,f)	(16)
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(flatwith,f),x)	(17)
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(append,app(app(flatwith,f),x))	(18)
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(append,app(app(flatwith,f),x)),app(app(flatwithsub,f),xs))	(19)

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(flatwithsub,f),xs)	(15)
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(flatwith,f),x)	(17)
app^#(app(flatwith,f),app(node,xs))	→	app^#(app(flatwithsub,f),xs)	(14)
app^#(app(flatwith,f),app(leaf,x))	→	app^#(f,x)	(10)

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

app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(flatwithsub,f),xs)	(15)

2	>	2
1	≥	1
app^#(app(flatwithsub,f),app(app(cons,x),xs))	→	app^#(app(flatwith,f),x)	(17)

2	>	2
app^#(app(flatwith,f),app(node,xs))	→	app^#(app(flatwithsub,f),xs)	(14)

2	>	2
app^#(app(flatwith,f),app(leaf,x))	→	app^#(f,x)	(10)

2	>	2
1	>	1

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

The 2^nd component contains the pair

app^#(app(append,app(app(cons,x),xs)),ys)

→

app^#(app(append,xs),ys)

(8)

Using the

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

app^#(app(append,app(app(cons,x),xs)),ys)

→

app^#(app(append,xs),ys)

(8)

could be deleted.

There are no pairs anymore.