Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-546)

The rewrite relation of the following TRS is considered.

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

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

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[b(x₁)]	=	x₁ + 1
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 2

together with the usable rules

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

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

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

1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(a^#) = 1
in combination with the following Weighted Path Order with the following precedence and status

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

prec(b) = 1 status(b) = [] list-extension(b) = Lex

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

and the following Max-polynomial interpretation

[a(x₁)] = x₁ + 1

[b(x₁)] = x₁ + 1

[b^#(x₁)] = 1

together with the usable rules

b(b(b(b(x1)))) → a(a(b(b(x1)))) (1)

b(a(a(a(x1)))) → a(b(a(b(x1)))) (3)

a(a(b(a(x1)))) → a(a(a(b(x1)))) (2)

(w.r.t. the implicit argument filter of the reduction pair), the pair
a^#(a(b(a(x1)))) → a^#(a(a(b(x1)))) (11)
could be deleted.
1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.