Certification Problem

Input (TPDB TRS_Standard/Applicative_05/TreeLevels)

The rewrite relation of the following TRS is considered.

app(app(map,f),nil)	→	nil	(1)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(2)
app(app(append,xs),nil)	→	xs	(3)
app(app(append,nil),ys)	→	ys	(4)
app(app(append,app(app(cons,x),xs)),ys)	→	app(app(cons,x),app(app(append,xs),ys))	(5)
app(app(zip,nil),yss)	→	yss	(6)
app(app(zip,xss),nil)	→	xss	(7)
app(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app(app(cons,app(app(append,xs),ys)),app(app(zip,xss),yss))	(8)
app(app(combine,xs),nil)	→	xs	(9)
app(app(combine,xs),app(app(cons,ys),yss))	→	app(app(combine,app(app(zip,xs),ys)),yss)	(10)
app(levels,app(app(node,x),xs))	→	app(app(cons,app(app(cons,x),nil)),app(app(combine,nil),app(app(map,levels),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^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(12)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(13)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(14)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(15)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(append,xs)	(16)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(app(append,xs),ys)	(17)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(app(cons,x),app(app(append,xs),ys))	(18)
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app^#(zip,xss)	(19)
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app^#(app(zip,xss),yss)	(20)
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app^#(append,xs)	(21)
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app^#(app(append,xs),ys)	(22)
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app^#(cons,app(app(append,xs),ys))	(23)
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss))	→	app^#(app(cons,app(app(append,xs),ys)),app(app(zip,xss),yss))	(24)
app^#(app(combine,xs),app(app(cons,ys),yss))	→	app^#(zip,xs)	(25)
app^#(app(combine,xs),app(app(cons,ys),yss))	→	app^#(app(zip,xs),ys)	(26)
app^#(app(combine,xs),app(app(cons,ys),yss))	→	app^#(combine,app(app(zip,xs),ys))	(27)
app^#(app(combine,xs),app(app(cons,ys),yss))	→	app^#(app(combine,app(app(zip,xs),ys)),yss)	(28)
app^#(levels,app(app(node,x),xs))	→	app^#(map,levels)	(29)
app^#(levels,app(app(node,x),xs))	→	app^#(app(map,levels),xs)	(30)
app^#(levels,app(app(node,x),xs))	→	app^#(combine,nil)	(31)
app^#(levels,app(app(node,x),xs))	→	app^#(app(combine,nil),app(app(map,levels),xs))	(32)
app^#(levels,app(app(node,x),xs))	→	app^#(cons,x)	(33)
app^#(levels,app(app(node,x),xs))	→	app^#(app(cons,x),nil)	(34)
app^#(levels,app(app(node,x),xs))	→	app^#(cons,app(app(cons,x),nil))	(35)
app^#(levels,app(app(node,x),xs))	→	app^#(app(cons,app(app(cons,x),nil)),app(app(combine,nil),app(app(map,levels),xs)))	(36)

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

app^#(levels,app(app(node,x),xs))	→	app^#(app(map,levels),xs)	(30)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(13)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(12)

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^#(levels,app(app(node,x),xs))	→	app^#(app(map,levels),xs)	(30)

2	>	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(13)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(12)

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(combine,xs),app(app(cons,ys),yss)) → app^#(app(combine,app(app(zip,xs),ys)),yss) (28)

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(combine,xs),app(app(cons,ys),yss)) → app^#(app(combine,app(app(zip,xs),ys)),yss) (28)

2 > 2

As there is no critical graph in the transitive closure, there are no infinite chains.
The 3^rd component contains the pair
app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss)) → app^#(app(zip,xss),yss) (20)

1.1.3 Size-Change Termination

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

app^#(app(zip,app(app(cons,xs),xss)),app(app(cons,ys),yss)) → app^#(app(zip,xss),yss) (20)

2 > 2

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

The 4^th component contains the pair

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

→

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

(17)

1.1.4 Reduction Pair Processor with Usable Rules

Using the

prec(app^#)	=	0	stat(app^#)	=	lex
prec(append)	=	0	stat(append)	=	lex
prec(cons)	=	0	stat(cons)	=	lex
prec(app)	=	1	stat(app)	=	lex

π(app^#)	=	[1]
π(append)	=	[]
π(cons)	=	[]
π(app)	=	[2]

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