Certification Problem

Input (TPDB TRS_Standard/Applicative_05/TreeFlatten)

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(flatten,app(app(node,x),xs))	→	app(app(cons,x),app(concat,app(app(map,flatten),xs)))	(3)
app(concat,nil)	→	nil	(4)
app(concat,app(app(cons,x),xs))	→	app(app(append,x),app(concat,xs))	(5)
app(app(append,nil),xs)	→	xs	(6)
app(app(append,app(app(cons,x),xs)),ys)	→	app(app(cons,x),app(app(append,xs),ys))	(7)

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)	(8)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(9)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(10)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(11)
app^#(flatten,app(app(node,x),xs))	→	app^#(map,flatten)	(12)
app^#(flatten,app(app(node,x),xs))	→	app^#(app(map,flatten),xs)	(13)
app^#(flatten,app(app(node,x),xs))	→	app^#(concat,app(app(map,flatten),xs))	(14)
app^#(flatten,app(app(node,x),xs))	→	app^#(cons,x)	(15)
app^#(flatten,app(app(node,x),xs))	→	app^#(app(cons,x),app(concat,app(app(map,flatten),xs)))	(16)
app^#(concat,app(app(cons,x),xs))	→	app^#(concat,xs)	(17)
app^#(concat,app(app(cons,x),xs))	→	app^#(append,x)	(18)
app^#(concat,app(app(cons,x),xs))	→	app^#(app(append,x),app(concat,xs))	(19)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(append,xs)	(20)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(app(append,xs),ys)	(21)
app^#(app(append,app(app(cons,x),xs)),ys)	→	app^#(app(cons,x),app(app(append,xs),ys))	(22)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

app^#(flatten,app(app(node,x),xs))	→	app^#(app(map,flatten),xs)	(13)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(9)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(8)

1.1.1 Subterm Criterion Processor

We use the projection

π(app^#)

and remove the pairs:

app^#(flatten,app(app(node,x),xs))	→	app^#(app(map,flatten),xs)	(13)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(9)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(8)

1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair
app^#(concat,app(app(cons,x),xs)) → app^#(concat,xs) (17)

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^#(concat,app(app(cons,x),xs)) → app^#(concat,xs) (17)

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

→

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

(21)

1.1.3 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