Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/#3.45)

The rewrite relation of the following TRS is considered.

app(f,app(s,x))	→	app(f,x)	(1)
app(g,app(app(cons,0),y))	→	app(g,y)	(2)
app(g,app(app(cons,app(s,x)),y))	→	app(s,x)	(3)
app(h,app(app(cons,x),y))	→	app(h,app(g,app(app(cons,x),y)))	(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(s,x))	→	app^#(f,x)	(11)
app^#(g,app(app(cons,0),y))	→	app^#(g,y)	(12)
app^#(h,app(app(cons,x),y))	→	app^#(g,app(app(cons,x),y))	(13)
app^#(h,app(app(cons,x),y))	→	app^#(h,app(g,app(app(cons,x),y)))	(14)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(15)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(16)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(cons,app(fun,x))	(17)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(cons,app(fun,x)),app(app(map,fun),xs))	(18)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(19)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(filter2,app(fun,x))	(20)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(filter2,app(fun,x)),fun)	(21)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(fun,x)),fun),x)	(22)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(23)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(filter,fun)	(24)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(25)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(cons,x)	(26)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(cons,x),app(app(filter,fun),xs))	(27)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(filter,fun)	(28)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(29)

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(19)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(29)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(25)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(16)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(15)

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^#(fun,x)	(19)

2	>	2
1	>	1
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(29)

2	≥	2
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(25)

2	≥	2
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(16)

2	>	2
1	>	1
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(15)

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^#(f,app(s,x)) → app^#(f,x) (11)

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^#(f,app(s,x)) → app^#(f,x) (11)

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^#(h,app(app(cons,x),y))

→

app^#(h,app(g,app(app(cons,x),y)))

(14)

1.1.3 Reduction Pair Processor with Usable Rules

Using the

prec(app^#)	=	0	stat(app^#)	=	lex
prec(h)	=	0	stat(h)	=	lex
prec(0)	=	0	stat(0)	=	lex
prec(cons)	=	3	stat(cons)	=	lex
prec(g)	=	2	stat(g)	=	lex
prec(app)	=	0	stat(app)	=	lex
prec(s)	=	0	stat(s)	=	lex

π(app^#)	=	2
π(h)	=	[]
π(0)	=	[]
π(cons)	=	[]
π(g)	=	[]
π(app)	=	1
π(s)	=	[]

together with the usable rules

app(g,app(app(cons,0),y))	→	app(g,y)	(2)
app(g,app(app(cons,app(s,x)),y))	→	app(s,x)	(3)

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