Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/08)

The rewrite relation of the following TRS is considered.

app(D,t)	→	1	(1)
app(D,constant)	→	0	(2)
app(D,app(app(+,x),y))	→	app(app(+,app(D,x)),app(D,y))	(3)
app(D,app(app(*,x),y))	→	app(app(+,app(app(,y),app(D,x))),app(app(,x),app(D,y)))	(4)
app(D,app(app(-,x),y))	→	app(app(-,app(D,x)),app(D,y))	(5)
app(app(map,f),nil)	→	nil	(6)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(7)
app(app(filter,f),nil)	→	nil	(8)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(9)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(10)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(11)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

app^#(D,app(app(+,x),y))	→	app^#(D,y)	(12)
app^#(D,app(app(+,x),y))	→	app^#(D,x)	(13)
app^#(D,app(app(+,x),y))	→	app^#(+,app(D,x))	(14)
app^#(D,app(app(+,x),y))	→	app^#(app(+,app(D,x)),app(D,y))	(15)
app^#(D,app(app(*,x),y))	→	app^#(D,y)	(16)
app^#(D,app(app(*,x),y))	→	app^#(app(*,x),app(D,y))	(17)
app^#(D,app(app(*,x),y))	→	app^#(D,x)	(18)
app^#(D,app(app(*,x),y))	→	app^#(*,y)	(19)
app^#(D,app(app(*,x),y))	→	app^#(app(*,y),app(D,x))	(20)
app^#(D,app(app(*,x),y))	→	app^#(+,app(app(*,y),app(D,x)))	(21)
app^#(D,app(app(*,x),y))	→	app^#(app(+,app(app(,y),app(D,x))),app(app(,x),app(D,y)))	(22)
app^#(D,app(app(-,x),y))	→	app^#(D,y)	(23)
app^#(D,app(app(-,x),y))	→	app^#(D,x)	(24)
app^#(D,app(app(-,x),y))	→	app^#(-,app(D,x))	(25)
app^#(D,app(app(-,x),y))	→	app^#(app(-,app(D,x)),app(D,y))	(26)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(27)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(28)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(29)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(30)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(31)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(32)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(33)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(34)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(35)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(36)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(37)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(38)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(39)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(40)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(41)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(31)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(41)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(37)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(28)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(27)

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,f),app(app(cons,x),xs))	→	app^#(f,x)	(31)

2	>	2
1	>	1
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(41)

2	≥	2
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(37)

2	≥	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(28)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(27)

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^#(D,app(app(-,x),y))	→	app^#(D,x)	(24)
app^#(D,app(app(-,x),y))	→	app^#(D,y)	(23)
app^#(D,app(app(*,x),y))	→	app^#(D,x)	(18)
app^#(D,app(app(*,x),y))	→	app^#(D,y)	(16)
app^#(D,app(app(+,x),y))	→	app^#(D,x)	(13)
app^#(D,app(app(+,x),y))	→	app^#(D,y)	(12)

1.1.2 Size-Change Termination

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

app^#(D,app(app(-,x),y))	→	app^#(D,x)	(24)

2	>	2
1	≥	1
app^#(D,app(app(-,x),y))	→	app^#(D,y)	(23)

2	>	2
1	≥	1
app^#(D,app(app(*,x),y))	→	app^#(D,x)	(18)

2	>	2
1	≥	1
app^#(D,app(app(*,x),y))	→	app^#(D,y)	(16)

2	>	2
1	≥	1
app^#(D,app(app(+,x),y))	→	app^#(D,x)	(13)

2	>	2
1	≥	1
app^#(D,app(app(+,x),y))	→	app^#(D,y)	(12)

2	>	2
1	≥	1

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