Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/hydra)

The rewrite relation of the following TRS is considered.

app(f,app(app(cons,nil),y))	→	y	(1)
app(f,app(app(cons,app(f,app(app(cons,nil),y))),z))	→	app(app(app(copy,n),y),z)	(2)
app(app(app(copy,0),y),z)	→	app(f,z)	(3)
app(app(app(copy,app(s,x)),y),z)	→	app(app(app(copy,x),y),app(app(cons,app(f,y)),z))	(4)
app(app(map,fun),nil)	→	nil	(5)
app(app(map,fun),app(app(cons,x),xs))	→	app(app(cons,app(fun,x)),app(app(map,fun),xs))	(6)
app(app(filter,fun),nil)	→	nil	(7)
app(app(filter,fun),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(fun,x)),fun),x),xs)	(8)
app(app(app(app(filter2,true),fun),x),xs)	→	app(app(cons,x),app(app(filter,fun),xs))	(9)
app(app(app(app(filter2,false),fun),x),xs)	→	app(app(filter,fun),xs)	(10)

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^#(f,app(app(cons,app(f,app(app(cons,nil),y))),z))	→	app^#(copy,n)	(11)
app^#(f,app(app(cons,app(f,app(app(cons,nil),y))),z))	→	app^#(app(copy,n),y)	(12)
app^#(f,app(app(cons,app(f,app(app(cons,nil),y))),z))	→	app^#(app(app(copy,n),y),z)	(13)
app^#(app(app(copy,0),y),z)	→	app^#(f,z)	(14)
app^#(app(app(copy,app(s,x)),y),z)	→	app^#(f,y)	(15)
app^#(app(app(copy,app(s,x)),y),z)	→	app^#(cons,app(f,y))	(16)
app^#(app(app(copy,app(s,x)),y),z)	→	app^#(app(cons,app(f,y)),z)	(17)
app^#(app(app(copy,app(s,x)),y),z)	→	app^#(copy,x)	(18)
app^#(app(app(copy,app(s,x)),y),z)	→	app^#(app(copy,x),y)	(19)
app^#(app(app(copy,app(s,x)),y),z)	→	app^#(app(app(copy,x),y),app(app(cons,app(f,y)),z))	(20)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(21)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(22)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(cons,app(fun,x))	(23)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(cons,app(fun,x)),app(app(map,fun),xs))	(24)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(25)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(filter2,app(fun,x))	(26)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(filter2,app(fun,x)),fun)	(27)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(fun,x)),fun),x)	(28)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(29)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(filter,fun)	(30)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(31)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(cons,x)	(32)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(cons,x),app(app(filter,fun),xs))	(33)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(filter,fun)	(34)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(35)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

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

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

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

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

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

2	>	2
1	>	1
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),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(app(copy,app(s,x)),y),z)

→

app^#(app(app(copy,x),y),app(app(cons,app(f,y)),z))

(20)

1.1.2 Reduction Pair Processor with Usable Rules

Using the

prec(app^#)	=	0	stat(app^#)	=	lex
prec(s)	=	4	stat(s)	=	lex
prec(n)	=	0	stat(n)	=	lex
prec(copy)	=	0	stat(copy)	=	lex
prec(app)	=	0	stat(app)	=	lex
prec(nil)	=	0	stat(nil)	=	lex
prec(cons)	=	1	stat(cons)	=	lex
prec(f)	=	0	stat(f)	=	lex

π(app^#)	=	1
π(s)	=	[]
π(n)	=	[]
π(copy)	=	[]
π(app)	=	[1,2]
π(nil)	=	[]
π(cons)	=	[]
π(f)	=	[]

together with the usable rules

app(f,app(app(cons,nil),y))	→	y	(1)
app(f,app(app(cons,app(f,app(app(cons,nil),y))),z))	→	app(app(app(copy,n),y),z)	(2)

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

app^#(app(app(copy,app(s,x)),y),z)

→

app^#(app(app(copy,x),y),app(app(cons,app(f,y)),z))

(20)

could be deleted.

1.1.2.1 P is empty

There are no pairs anymore.