Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/#3.16)

The rewrite relation of the following TRS is considered.

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

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^#(app(times,x),app(s,y))	→	app^#(app(times,x),y)	(13)
app^#(app(times,x),app(s,y))	→	app^#(plus,app(app(times,x),y))	(14)
app^#(app(times,x),app(s,y))	→	app^#(app(plus,app(app(times,x),y)),x)	(15)
app^#(app(plus,x),app(s,y))	→	app^#(app(plus,x),y)	(16)
app^#(app(plus,x),app(s,y))	→	app^#(s,app(app(plus,x),y))	(17)
app^#(app(plus,app(s,x)),y)	→	app^#(plus,x)	(18)
app^#(app(plus,app(s,x)),y)	→	app^#(app(plus,x),y)	(19)
app^#(app(plus,app(s,x)),y)	→	app^#(s,app(app(plus,x),y))	(20)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(21)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(22)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(23)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(24)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(25)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(26)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(27)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(28)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(29)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(30)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(31)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(32)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(33)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(34)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(35)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

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)	(29)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(35)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(25)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(31)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(22)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(21)

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^#(app(app(app(filter2,app(f,x)),f),x),xs)	(29)

2	>	2
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(35)

2	≥	2
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(25)

2	>	2
1	>	1
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(31)

2	≥	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(22)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(21)

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(times,x),app(s,y)) → app^#(app(times,x),y) (13)

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^#(app(times,x),app(s,y)) → app^#(app(times,x),y) (13)

2 > 2

1 ≥ 1

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

The 3^rd component contains the pair

app^#(app(plus,app(s,x)),y)	→	app^#(app(plus,x),y)	(19)
app^#(app(plus,x),app(s,y))	→	app^#(app(plus,x),y)	(16)

1.1.3 Subterm Criterion Processor

We use the projection

π(app^#)

and remove the pairs:

app^#(app(plus,x),app(s,y))

→

app^#(app(plus,x),y)

(16)

1.1.3.1 Reduction Pair Processor with Usable Rules

Using the

prec(app^#)	=	0	stat(app^#)	=	lex
prec(plus)	=	0	stat(plus)	=	lex
prec(s)	=	0	stat(s)	=	lex
prec(app)	=	1	stat(app)	=	lex

π(app^#)	=	[1]
π(plus)	=	[]
π(s)	=	[]
π(app)	=	[2]

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

app^#(app(plus,app(s,x)),y)

→