Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-14)

The rewrite relation of the following TRS is considered.

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

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.

a^#(a(a(b(x1))))	→	a^#(x1)	(3)
a^#(a(a(b(x1))))	→	a^#(a(x1))	(4)
a^#(a(a(b(x1))))	→	a^#(a(a(x1)))	(5)
a^#(a(a(b(x1))))	→	b^#(a(a(a(x1))))	(6)
a^#(a(a(b(x1))))	→	a^#(b(a(a(a(x1)))))	(7)
a^#(a(a(b(x1))))	→	b^#(a(b(a(a(a(x1))))))	(8)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(a(a(b(x1))))	→	a^#(a(a(x1)))	(5)
a^#(a(a(b(x1))))	→	a^#(x1)	(3)
a^#(a(a(b(x1))))	→	a^#(a(x1))	(4)
a^#(a(a(b(x1))))	→	a^#(b(a(a(a(x1)))))	(7)

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers

[a(x₁)]

-∞	0	-∞
-∞	-∞	0
0	-∞	-∞

· x₁ +

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

[b(x₁)]

1	0	0
0	-∞	-∞
1	0	0

· x₁ +

1	-∞	-∞
0	-∞	-∞
1	-∞	-∞

[a^#(x₁)]

-∞	0	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

together with the usable rules

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

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

a^#(a(a(b(x1))))	→	a^#(a(a(x1)))	(5)
a^#(a(a(b(x1))))	→	a^#(b(a(a(a(x1)))))	(7)

could be deleted.

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[a(x₁)]	=	1 · x₁ + 5
[b(x₁)]	=	-1 · x₁ + 1
[a^#(x₁)]	=	0 · x₁ + -∞

together with the usable rules

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

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

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

→

a^#(x1)

(3)

could be deleted.

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the rationals with delta = 1/64

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

together with the usable rules

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

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

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

→

a^#(a(x1))

(4)

could be deleted.

1.1.1.1.1.1 P is empty

There are no pairs anymore.