Certification Problem

Input (TPDB SRS_Standard/Zantema_06/08)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(x1)	→	b^#(x1)	(4)
c^#(x1)	→	b^#(a(a(b(x1))))	(5)
b^#(a(a(b(a(x1)))))	→	c^#(x1)	(6)
b^#(a(a(b(a(x1)))))	→	c^#(c(x1))	(7)
b^#(a(a(b(a(x1)))))	→	c^#(c(c(x1)))	(8)
b^#(a(a(b(a(x1)))))	→	c^#(c(c(c(x1))))	(9)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(x1)	→	b^#(x1)	(4)
b^#(a(a(b(a(x1)))))	→	c^#(x1)	(6)
c^#(x1)	→	b^#(a(a(b(x1))))	(5)
b^#(a(a(b(a(x1)))))	→	c^#(c(x1))	(7)
b^#(a(a(b(a(x1)))))	→	c^#(c(c(x1)))	(8)
b^#(a(a(b(a(x1)))))	→	c^#(c(c(c(x1))))	(9)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 4 with strict dimension 1 over the arctic semiring over the naturals

[c(x₁)]

0	-∞	0	-∞
1	0	1	-∞
0	-∞	0	-∞
1	1	2	-∞

· x₁ +

[b(x₁)]

-∞	-∞	0	-∞
-∞	0	1	-∞
-∞	-∞	0	-∞
0	-∞	0	-∞

· x₁ +

-∞

[a(x₁)]

1	-∞	1	-∞
-∞	-∞	0	0
-∞	0	-∞	-∞
0	-∞	0	-∞

· x₁ +

-∞

[c^#(x₁)]

7	3	7	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

-∞

[b^#(x₁)]

0	3	7	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞
-∞	-∞	-∞	-∞

· x₁ +

-∞

together with the usable rules

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

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

b^#(a(a(b(a(x1)))))	→	c^#(x1)	(6)
b^#(a(a(b(a(x1)))))	→	c^#(c(x1))	(7)
b^#(a(a(b(a(x1)))))	→	c^#(c(c(x1)))	(8)
b^#(a(a(b(a(x1)))))	→	c^#(c(c(c(x1))))	(9)

could be deleted.

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[c^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

together with the usable rules

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

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

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

and no rules could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.