Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-304)

The rewrite relation of the following TRS is considered.

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

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^#(b(b(x1)))	→	a^#(x1)	(4)
a^#(b(b(x1)))	→	a^#(a(x1))	(5)
a^#(b(b(x1)))	→	c^#(a(a(x1)))	(6)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[b(x₁)]

0	0
1	0

· x₁ +

0	-∞
2	-∞

[a^#(x₁)]

-∞	1
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

0	-∞
0	1

· x₁ +

0	-∞
0	-∞

[c(x₁)]

0	-∞
0	-∞

· x₁ +

0	-∞
0	-∞

together with the usable rules

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

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

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

→

a^#(x1)

(4)

could be deleted.

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[b(x₁)]

0	1
0	0

· x₁ +

1	-∞
0	-∞

[a^#(x₁)]

-∞	1
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

1	2
-∞	0

· x₁ +

0	-∞
0	-∞

[c(x₁)]

-∞	0
-∞	0

· x₁ +

0	-∞
0	-∞

together with the usable rules

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

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

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

→

a^#(a(x1))

(5)

could be deleted.

1.1.1.1.1 P is empty

There are no pairs anymore.