Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-354)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(b(b(a(x1))))	→	a^#(b(b(x1)))	(4)
b^#(b(a(b(x1))))	→	b^#(b(a(x1)))	(5)
a^#(b(b(a(x1))))	→	b^#(a(b(a(x1))))	(6)
b^#(b(a(b(x1))))	→	b^#(a(x1))	(7)
a^#(b(b(a(x1))))	→	a^#(b(a(x1)))	(8)
b^#(b(a(b(x1))))	→	b^#(b(b(a(x1))))	(9)
b^#(b(b(a(x1))))	→	b^#(b(x1))	(10)
b^#(b(b(a(x1))))	→	a^#(a(b(b(x1))))	(11)
b^#(b(a(b(x1))))	→	a^#(x1)	(12)
b^#(b(b(a(x1))))	→	b^#(x1)	(13)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(b(a(x1))))	→	b^#(x1)	(13)
b^#(b(a(b(x1))))	→	a^#(x1)	(12)
b^#(b(b(a(x1))))	→	a^#(a(b(b(x1))))	(11)
b^#(b(b(a(x1))))	→	b^#(b(x1))	(10)
a^#(b(b(a(x1))))	→	b^#(a(b(a(x1))))	(6)
b^#(b(a(b(x1))))	→	b^#(b(a(x1)))	(5)
b^#(b(a(b(x1))))	→	b^#(b(b(a(x1))))	(9)
a^#(b(b(a(x1))))	→	a^#(b(a(x1)))	(8)
b^#(b(a(b(x1))))	→	b^#(a(x1))	(7)
b^#(b(b(a(x1))))	→	a^#(b(b(x1)))	(4)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[b(x₁)]	=	x₁ + 2440
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 1

together with the usable rules

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

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

b^#(b(b(a(x1))))	→	b^#(x1)	(13)
b^#(b(a(b(x1))))	→	a^#(x1)	(12)
b^#(b(b(a(x1))))	→	a^#(a(b(b(x1))))	(11)
b^#(b(b(a(x1))))	→	b^#(b(x1))	(10)
a^#(b(b(a(x1))))	→	b^#(a(b(a(x1))))	(6)
b^#(b(a(b(x1))))	→	b^#(b(a(x1)))	(5)
a^#(b(b(a(x1))))	→	a^#(b(a(x1)))	(8)
b^#(b(a(b(x1))))	→	b^#(a(x1))	(7)
b^#(b(b(a(x1))))	→	a^#(b(b(x1)))	(4)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

(9)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

in combination with the following Weighted Path Order with the following precedence and status

prec(a)	=	1		status(a)	=	[]		list-extension(a)	=	Lex
prec(b)	=	0		status(b)	=	[1]		list-extension(b)	=	Lex
prec(a^#)	=	0		status(a^#)	=	[]		list-extension(a^#)	=	Lex
prec(b^#)	=	0		status(b^#)	=	[1]		list-extension(b^#)	=	Lex

and the following Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[b(x₁)]	=	x₁ + 30614
[a^#(x₁)]	=	1
[b^#(x₁)]	=	x₁ + 1

together with the usable rules

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

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

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

→

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

(9)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.