Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/30)

The rewrite relation of the following TRS is considered.

app(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app(app(:,app(app(:,x),z)),app(app(:,app(app(:,app(app(:,x),y)),z)),u))	(1)
app(app(map,f),nil)	→	nil	(2)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(3)
app(app(filter,f),nil)	→	nil	(4)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(5)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(6)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(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(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,x),y)	(8)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(:,app(app(:,x),y))	(9)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,x),y)),z)	(10)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(:,app(app(:,app(app(:,x),y)),z))	(11)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,app(app(:,x),y)),z)),u)	(12)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(:,x)	(13)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,x),z)	(14)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(:,app(app(:,x),z))	(15)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,x),z)),app(app(:,app(app(:,app(app(:,x),y)),z)),u))	(16)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(17)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(18)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(19)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(20)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(21)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(22)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(23)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(24)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(25)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(26)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(27)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(28)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(29)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(30)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(31)

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)	(21)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(31)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(27)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(18)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(17)

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)	(21)

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

2	≥	2
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(27)

2	≥	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(18)

2	>	2
1	>	1
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(17)

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(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,x),z)),app(app(:,app(app(:,app(app(:,x),y)),z)),u))	(16)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,x),z)	(14)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,app(app(:,x),y)),z)),u)	(12)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,x),y)),z)	(10)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,x),y)	(8)

1.1.2 Reduction Pair Processor with Usable Rules

Using the

prec(app^#)	=	stat(app^#)	=	lex
prec(app)	=	stat(app)	=	lex
prec(C)	=	stat(C)	=	lex
prec(:)	=	stat(:)	=	lex

π(app^#)	=	1
π(app)	=	[1,2]
π(C)	=	[]
π(:)	=	[]

together with the usable rule

app(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)

→

app(app(:,app(app(:,x),z)),app(app(:,app(app(:,app(app(:,x),y)),z)),u))

(1)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,x),z)),app(app(:,app(app(:,app(app(:,x),y)),z)),u))	(16)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,x),z)	(14)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,app(app(:,x),y)),z)),u)	(12)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,app(app(:,x),y)),z)	(10)
app^#(app(:,app(app(:,app(app(:,app(app(:,C),x)),y)),z)),u)	→	app^#(app(:,x),y)	(8)

could be deleted.

1.1.2.1 P is empty

There are no pairs anymore.