Certification Problem

Input (TPDB TRS_Innermost/Applicative_AG01_innermost/#4.15)

The rewrite relation of the following TRS is considered.

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

The evaluation strategy is innermost.

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

app^#(app(app(app(f,0),1),app(app(g,x),y)),z)	→	app^#(app(app(app(f,app(app(g,x),y)),app(app(g,x),y)),app(app(g,x),y)),app(h,x))	(11)
app^#(app(app(app(f,0),1),app(app(g,x),y)),z)	→	app^#(app(app(f,app(app(g,x),y)),app(app(g,x),y)),app(app(g,x),y))	(12)
app^#(app(app(app(f,0),1),app(app(g,x),y)),z)	→	app^#(app(f,app(app(g,x),y)),app(app(g,x),y))	(13)
app^#(app(app(app(f,0),1),app(app(g,x),y)),z)	→	app^#(f,app(app(g,x),y))	(14)
app^#(app(app(app(f,0),1),app(app(g,x),y)),z)	→	app^#(h,x)	(15)
app^#(h,app(app(g,x),y))	→	app^#(h,x)	(16)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(cons,app(fun,x)),app(app(map,fun),xs))	(17)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(cons,app(fun,x))	(18)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(19)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(20)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(21)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(fun,x)),fun),x)	(22)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(filter2,app(fun,x)),fun)	(23)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(filter2,app(fun,x))	(24)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(25)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(cons,x),app(app(filter,fun),xs))	(26)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(cons,x)	(27)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(28)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(filter,fun)	(29)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(30)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(filter,fun)	(31)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(20)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(19)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(25)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(28)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(30)

1.1.1 Usable Rules Processor

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

There are no rules.

1.1.1.1 Size-Change Termination

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)	(25)

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

1	>	1
2	>	2
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(20)

1	≥	1
2	>	2
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(28)

2	≥	2
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(30)

2	≥	2

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

The 2^nd component contains the pair
app^#(h,app(app(g,x),y)) → app^#(h,x) (16)

1.1.2 Usable Rules Processor

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

There are no rules.

1.1.2.1 Size-Change Termination

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

app^#(h,app(app(g,x),y)) → app^#(h,x) (16)

1 ≥ 1

2 > 2

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