Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup05)

The rewrite relation of the following TRS is considered.

a(a(s(s(x1))))	→	s(s(a(a(x1))))	(1)
b(b(a(a(b(b(s(s(x1))))))))	→	a(a(b(b(s(s(a(a(x1))))))))	(2)
b(b(a(a(b(b(b(b(x1))))))))	→	a(a(b(b(a(a(b(b(x1))))))))	(3)
a(a(b(b(a(a(a(a(x1))))))))	→	b(b(a(a(b(b(a(a(x1))))))))	(4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

s(s(a(a(x1))))	→	a(a(s(s(x1))))	(5)
s(s(b(b(a(a(b(b(x1))))))))	→	a(a(s(s(b(b(a(a(x1))))))))	(6)
b(b(b(b(a(a(b(b(x1))))))))	→	b(b(a(a(b(b(a(a(x1))))))))	(7)
a(a(a(a(b(b(a(a(x1))))))))	→	a(a(b(b(a(a(b(b(x1))))))))	(8)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

s^#(s(a(a(x1))))	→	s^#(x1)	(9)
s^#(s(a(a(x1))))	→	s^#(s(x1))	(10)
s^#(s(a(a(x1))))	→	a^#(s(s(x1)))	(11)
s^#(s(a(a(x1))))	→	a^#(a(s(s(x1))))	(12)
s^#(s(b(b(a(a(b(b(x1))))))))	→	a^#(x1)	(13)
s^#(s(b(b(a(a(b(b(x1))))))))	→	a^#(a(x1))	(14)
s^#(s(b(b(a(a(b(b(x1))))))))	→	b^#(a(a(x1)))	(15)
s^#(s(b(b(a(a(b(b(x1))))))))	→	b^#(b(a(a(x1))))	(16)
s^#(s(b(b(a(a(b(b(x1))))))))	→	s^#(b(b(a(a(x1)))))	(17)
s^#(s(b(b(a(a(b(b(x1))))))))	→	s^#(s(b(b(a(a(x1))))))	(18)
s^#(s(b(b(a(a(b(b(x1))))))))	→	a^#(s(s(b(b(a(a(x1)))))))	(19)
s^#(s(b(b(a(a(b(b(x1))))))))	→	a^#(a(s(s(b(b(a(a(x1))))))))	(20)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(x1)	(21)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(a(x1))	(22)
b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(a(a(x1)))	(23)
b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(b(a(a(x1))))	(24)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(b(b(a(a(x1)))))	(25)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(a(b(b(a(a(x1))))))	(26)
b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(a(a(b(b(a(a(x1)))))))	(27)
b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(b(a(a(b(b(a(a(x1))))))))	(28)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(x1)	(29)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(b(x1))	(30)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(b(b(x1)))	(31)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(a(b(b(x1))))	(32)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(a(a(b(b(x1)))))	(33)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(b(a(a(b(b(x1))))))	(34)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(b(b(a(a(b(b(x1)))))))	(35)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(a(b(b(a(a(b(b(x1))))))))	(36)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

s^#(s(a(a(x1))))	→	s^#(s(x1))	(10)
s^#(s(a(a(x1))))	→	s^#(x1)	(9)

1.1.1.1 Subterm Criterion Processor

We use the projection to multisets

π(s^#)	=	{ 1, 1 }
π(b)	=	{ 1 }
π(a)	=	{ 1 }
π(s)	=	{ 1, 1 }

to remove the pairs:

s^#(s(a(a(x1))))

→

s^#(x1)

(9)

1.1.1.1.1 Subterm Criterion Processor

We use the projection to multisets

π(s^#)	=	{ 1, 1 }
π(b)	=	{ 1, 1 }
π(a)	=	{ 1, 1 }
π(s)	=	{ 1, 1 }

to remove the pairs:

s^#(s(a(a(x1))))

→

s^#(s(x1))

(10)

1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

s^#(s(b(b(a(a(b(b(x1))))))))

→

s^#(s(b(b(a(a(x1))))))

(18)

1.1.1.2 Subterm Criterion Processor

We use the projection to multisets

π(s^#)	=	{ 1 }
π(b)	=	{ 1, 1 }
π(a)	=	{ 1, 1 }
π(s)	=	{ 1 }

to remove the pairs:

s^#(s(b(b(a(a(b(b(x1))))))))

→

s^#(s(b(b(a(a(x1))))))

(18)

1.1.1.2.1 P is empty

There are no pairs anymore.

The 3^rd component contains the pair

b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(b(a(a(b(b(a(a(x1))))))))	(28)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(a(b(b(a(a(x1))))))	(26)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(a(b(b(a(a(b(b(x1))))))))	(36)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(b(a(a(b(b(x1))))))	(34)
b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(b(a(a(x1))))	(24)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(a(x1))	(22)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(a(b(b(x1))))	(32)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(b(x1))	(30)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(x1)	(21)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(x1)	(29)

1.1.1.3 Subterm Criterion Processor

We use the projection to multisets

π(b^#)	=	{ 1, 1 }
π(a^#)	=	{ 1 }
π(b)	=	{ 1, 1 }
π(a)	=	{ 1, 1 }

to remove the pairs:

b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(a(b(b(a(a(x1))))))	(26)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(b(a(a(b(b(x1))))))	(34)
b^#(b(b(b(a(a(b(b(x1))))))))	→	b^#(b(a(a(x1))))	(24)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(a(x1))	(22)
a^#(a(a(a(b(b(a(a(x1))))))))	→	a^#(a(b(b(x1))))	(32)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(b(x1))	(30)
b^#(b(b(b(a(a(b(b(x1))))))))	→	a^#(x1)	(21)
a^#(a(a(a(b(b(a(a(x1))))))))	→	b^#(x1)	(29)

1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

a^#(a(a(a(b(b(a(a(x1))))))))

→

a^#(a(b(b(a(a(b(b(x1))))))))

(36)

1.1.1.3.1.1 Reduction Pair Processor with Usable Rules

Using the

prec(a^#)	=	stat(a^#)	=	lex
prec(b)	=	stat(b)	=	lex
prec(a)	=	stat(a)	=	lex

π(a^#)	=	1
π(b)	=	[]
π(a)	=	[1]

together with the usable rules

b(b(b(b(a(a(b(b(x1))))))))	→	b(b(a(a(b(b(a(a(x1))))))))	(7)
a(a(a(a(b(b(a(a(x1))))))))	→	a(a(b(b(a(a(b(b(x1))))))))	(8)

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

a^#(a(a(a(b(b(a(a(x1))))))))

→

a^#(a(b(b(a(a(b(b(x1))))))))

(36)

could be deleted.

1.1.1.3.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

b^#(b(b(b(a(a(b(b(x1))))))))

→

b^#(b(a(a(b(b(a(a(x1))))))))

(28)

1.1.1.3.1.2 Reduction Pair Processor with Usable Rules

Using the

prec(b^#)	=	stat(b^#)	=	lex
prec(b)	=	stat(b)	=	lex
prec(a)	=	stat(a)	=	lex

π(b^#)	=	1
π(b)	=	[1]
π(a)	=	[]

together with the usable rules

a(a(a(a(b(b(a(a(x1))))))))	→	a(a(b(b(a(a(b(b(x1))))))))	(8)
b(b(b(b(a(a(b(b(x1))))))))	→	b(b(a(a(b(b(a(a(x1))))))))	(7)

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

b^#(b(b(b(a(a(b(b(x1))))))))

→

b^#(b(a(a(b(b(a(a(x1))))))))

(28)

could be deleted.

1.1.1.3.1.2.1 P is empty

There are no pairs anymore.