Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/jambox1)

The rewrite relation of the following TRS is considered.

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

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the arctic semiring over the naturals

[c(x₁)]

0	0
-2	-2

· x₁ +

-∞

[b(x₁)]

0	0
0	0

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[c^#(x₁)]

6	8
6	8

· x₁ +

-∞

together with the usable rules

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

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

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

→

c^#(c(b(c(x1))))

(8)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1.1.1.1 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	4	4
0	0	4
0	0	4
-4	0	0

· x₁ +

-∞

[b(x₁)]

0	0	0	0
-4	-4	-4	0
-4	-4	-4	0
-4	-4	-4	-4

· x₁ +

-∞

[a(x₁)]

0	0	4	4
0	0	4	4
-4	0	0	0
-4	-4	0	0

· x₁ +

-∞

[c^#(x₁)]

4	6	7	7
4	6	7	7
4	6	7	7
4	6	7	7

· x₁ +

-∞

together with the usable rules

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

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

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

could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.