Certification Problem

Input (TPDB TRS_Innermost/Applicative_AG01_innermost/#4.13)

The rewrite relation of the following TRS is considered.

app(app(app(f,0),1),x)	→	app(app(app(f,x),x),x)	(1)
app(app(app(f,x),y),z)	→	2	(2)
0	→	2	(3)
1	→	2	(4)
app(app(app(g,x),x),y)	→	y	(5)
app(app(app(g,x),y),y)	→	x	(6)
app(app(map,fun),nil)	→	nil	(7)
app(app(map,fun),app(app(cons,x),xs))	→	app(app(cons,app(fun,x)),app(app(map,fun),xs))	(8)
app(app(filter,fun),nil)	→	nil	(9)
app(app(filter,fun),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(fun,x)),fun),x),xs)	(10)
app(app(app(app(filter2,true),fun),x),xs)	→	app(app(cons,x),app(app(filter,fun),xs))	(11)
app(app(app(app(filter2,false),fun),x),xs)	→	app(app(filter,fun),xs)	(12)

The evaluation strategy is innermost.

Prove or disprove termination.

Yes.

The following set of initial dependency pairs has been identified.

app^#(app(app(f,0),1),x)	→	app^#(app(app(f,x),x),x)	(13)
app^#(app(app(f,0),1),x)	→	app^#(app(f,x),x)	(14)
app^#(app(app(f,0),1),x)	→	app^#(f,x)	(15)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(cons,app(fun,x)),app(app(map,fun),xs))	(16)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(cons,app(fun,x))	(17)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(18)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(19)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(20)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(fun,x)),fun),x)	(21)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(filter2,app(fun,x)),fun)	(22)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(filter2,app(fun,x))	(23)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(24)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(cons,x),app(app(filter,fun),xs))	(25)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(cons,x)	(26)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(27)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(filter,fun)	(28)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(29)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(filter,fun)	(30)

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(19)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(18)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(20)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(27)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(24)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(29)

We restrict the rewrite rules to the following usable rules of the DP problem.

app(app(app(f,x),y),z)	→	2	(2)
app(app(app(g,x),x),y)	→	y	(5)
app(app(app(g,x),y),y)	→	x	(6)
app(app(map,fun),nil)	→	nil	(7)
app(app(map,fun),app(app(cons,x),xs))	→	app(app(cons,app(fun,x)),app(app(map,fun),xs))	(8)
app(app(filter,fun),nil)	→	nil	(9)
app(app(filter,fun),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(fun,x)),fun),x),xs)	(10)
app(app(app(app(filter2,false),fun),x),xs)	→	app(app(filter,fun),xs)	(12)
app(app(app(app(filter2,true),fun),x),xs)	→	app(app(cons,x),app(app(filter,fun),xs))	(11)

We restrict the innermost strategy to the following left hand sides.

app(app(app(f,x0),x1),x2)

app(app(app(g,x0),x0),x1)

app(app(app(g,x0),x1),x1)

app(app(map,x0),nil)

app(app(map,x0),app(app(cons,x1),x2))

app(app(filter,x0),nil)

app(app(filter,x0),app(app(cons,x1),x2))

app(app(app(app(filter2,true),x0),x1),x2)

app(app(app(app(filter2,false),x0),x1),x2)

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

app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(24)

1	>	1
2	>	2
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(18)

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

1	≥	1
2	>	2
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(20)

2	>	2
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(27)

2	≥	2
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(29)

2	≥	2

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