Certification Problem

Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/Ex2_Luc02a_GM)

The rewrite relation of the following TRS is considered.

a__terms(N)	→	cons(recip(a__sqr(mark(N))),terms(s(N)))	(1)
a__sqr(0)	→	0	(2)
a__sqr(s(X))	→	s(a__add(a__sqr(mark(X)),a__dbl(mark(X))))	(3)
a__dbl(0)	→	0	(4)
a__dbl(s(X))	→	s(s(a__dbl(mark(X))))	(5)
a__add(0,X)	→	mark(X)	(6)
a__add(s(X),Y)	→	s(a__add(mark(X),mark(Y)))	(7)
a__first(0,X)	→	nil	(8)
a__first(s(X),cons(Y,Z))	→	cons(mark(Y),first(X,Z))	(9)
mark(terms(X))	→	a__terms(mark(X))	(10)
mark(sqr(X))	→	a__sqr(mark(X))	(11)
mark(add(X1,X2))	→	a__add(mark(X1),mark(X2))	(12)
mark(dbl(X))	→	a__dbl(mark(X))	(13)
mark(first(X1,X2))	→	a__first(mark(X1),mark(X2))	(14)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(15)
mark(recip(X))	→	recip(mark(X))	(16)
mark(s(X))	→	s(mark(X))	(17)
mark(0)	→	0	(18)
mark(nil)	→	nil	(19)
a__terms(X)	→	terms(X)	(20)
a__sqr(X)	→	sqr(X)	(21)
a__add(X1,X2)	→	add(X1,X2)	(22)
a__dbl(X)	→	dbl(X)	(23)
a__first(X1,X2)	→	first(X1,X2)	(24)

The evaluation strategy is innermost.

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a__terms^#(N)	→	a__sqr^#(mark(N))	(25)
a__terms^#(N)	→	mark^#(N)	(26)
a__sqr^#(s(X))	→	a__add^#(a__sqr(mark(X)),a__dbl(mark(X)))	(27)
a__sqr^#(s(X))	→	a__sqr^#(mark(X))	(28)
a__sqr^#(s(X))	→	mark^#(X)	(29)
a__sqr^#(s(X))	→	a__dbl^#(mark(X))	(30)
a__dbl^#(s(X))	→	a__dbl^#(mark(X))	(31)
a__dbl^#(s(X))	→	mark^#(X)	(32)
a__add^#(0,X)	→	mark^#(X)	(33)
a__add^#(s(X),Y)	→	a__add^#(mark(X),mark(Y))	(34)
a__add^#(s(X),Y)	→	mark^#(X)	(35)
a__add^#(s(X),Y)	→	mark^#(Y)	(36)
a__first^#(s(X),cons(Y,Z))	→	mark^#(Y)	(37)
mark^#(terms(X))	→	a__terms^#(mark(X))	(38)
mark^#(terms(X))	→	mark^#(X)	(39)
mark^#(sqr(X))	→	a__sqr^#(mark(X))	(40)
mark^#(sqr(X))	→	mark^#(X)	(41)
mark^#(add(X1,X2))	→	a__add^#(mark(X1),mark(X2))	(42)
mark^#(add(X1,X2))	→	mark^#(X1)	(43)
mark^#(add(X1,X2))	→	mark^#(X2)	(44)
mark^#(dbl(X))	→	a__dbl^#(mark(X))	(45)
mark^#(dbl(X))	→	mark^#(X)	(46)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(47)
mark^#(first(X1,X2))	→	mark^#(X1)	(48)
mark^#(first(X1,X2))	→	mark^#(X2)	(49)
mark^#(cons(X1,X2))	→	mark^#(X1)	(50)
mark^#(recip(X))	→	mark^#(X)	(51)
mark^#(s(X))	→	mark^#(X)	(52)

1.1 Reduction Pair Processor

Using the

prec(a__terms^#)	=	4		stat(a__terms^#)	=	lex
prec(a__sqr^#)	=	4		stat(a__sqr^#)	=	lex
prec(mark^#)	=	0		stat(mark^#)	=	mul
prec(s)	=	0		stat(s)	=	mul
prec(a__add^#)	=	1		stat(a__add^#)	=	lex
prec(a__sqr)	=	4		stat(a__sqr)	=	lex
prec(a__dbl)	=	2		stat(a__dbl)	=	lex
prec(0)	=	2		stat(0)	=	mul
prec(a__first^#)	=	0		stat(a__first^#)	=	mul
prec(terms)	=	4		stat(terms)	=	lex
prec(sqr)	=	4		stat(sqr)	=	lex
prec(add)	=	3		stat(add)	=	mul
prec(dbl)	=	2		stat(dbl)	=	lex
prec(first)	=	2		stat(first)	=	mul
prec(a__terms)	=	4		stat(a__terms)	=	lex
prec(a__add)	=	3		stat(a__add)	=	mul
prec(a__first)	=	2		stat(a__first)	=	mul
prec(nil)	=	2		stat(nil)	=	mul

π(a__terms^#)	=	[1]
π(a__sqr^#)	=	[1]
π(mark)	=	1
π(mark^#)	=	[1]
π(s)	=	[1]
π(a__add^#)	=	[2,1]
π(a__sqr)	=	[1]
π(a__dbl)	=	[1]
π(a__dbl^#)	=	1
π(0)	=	[]
π(a__first^#)	=	[2]
π(cons)	=	1
π(terms)	=	[1]
π(sqr)	=	[1]
π(add)	=	[1,2]
π(dbl)	=	[1]
π(first)	=	[1,2]
π(recip)	=	1
π(a__terms)	=	[1]
π(a__add)	=	[1,2]
π(a__first)	=	[1,2]
π(nil)	=	[]

the pairs

a__terms^#(N)	→	mark^#(N)	(26)
a__sqr^#(s(X))	→	a__add^#(a__sqr(mark(X)),a__dbl(mark(X)))	(27)
a__sqr^#(s(X))	→	a__sqr^#(mark(X))	(28)
a__sqr^#(s(X))	→	mark^#(X)	(29)
a__sqr^#(s(X))	→	a__dbl^#(mark(X))	(30)
a__dbl^#(s(X))	→	a__dbl^#(mark(X))	(31)
a__add^#(0,X)	→	mark^#(X)	(33)
a__add^#(s(X),Y)	→	a__add^#(mark(X),mark(Y))	(34)
a__add^#(s(X),Y)	→	mark^#(X)	(35)
a__add^#(s(X),Y)	→	mark^#(Y)	(36)
mark^#(terms(X))	→	a__terms^#(mark(X))	(38)
mark^#(terms(X))	→	mark^#(X)	(39)
mark^#(sqr(X))	→	a__sqr^#(mark(X))	(40)
mark^#(sqr(X))	→	mark^#(X)	(41)
mark^#(add(X1,X2))	→	a__add^#(mark(X1),mark(X2))	(42)
mark^#(add(X1,X2))	→	mark^#(X1)	(43)
mark^#(add(X1,X2))	→	mark^#(X2)	(44)
mark^#(dbl(X))	→	a__dbl^#(mark(X))	(45)
mark^#(dbl(X))	→	mark^#(X)	(46)
mark^#(first(X1,X2))	→	a__first^#(mark(X1),mark(X2))	(47)
mark^#(first(X1,X2))	→	mark^#(X1)	(48)
mark^#(first(X1,X2))	→	mark^#(X2)	(49)
mark^#(s(X))	→	mark^#(X)	(52)

could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(recip(X)) → mark^#(X) (51)

mark^#(cons(X1,X2)) → mark^#(X1) (50)

1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

There are no lhss.

1.1.1.1.1.1 Size-Change Termination

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

mark^#(recip(X)) → mark^#(X) (51)

1 > 1

mark^#(cons(X1,X2)) → mark^#(X1) (50)

1 > 1

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