Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/jambox4)

The rewrite relation of the following TRS is considered.

c(c(x1))	→	a(b(x1))	(1)
b(x1)	→	a(a(x1))	(2)
b(b(b(x1)))	→	a(c(b(x1)))	(3)
a(c(a(x1)))	→	a(c(c(x1)))	(4)

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(c(x1))	→	b(a(x1))	(5)
b(x1)	→	a(a(x1))	(2)
b(b(b(x1)))	→	b(c(a(x1)))	(6)
a(c(a(x1)))	→	c(c(a(x1)))	(7)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(c(x1))	→	b^#(a(x1))	(8)
c^#(c(x1))	→	a^#(x1)	(9)
b^#(x1)	→	a^#(x1)	(10)
b^#(x1)	→	a^#(a(x1))	(11)
b^#(b(b(x1)))	→	c^#(a(x1))	(12)
b^#(b(b(x1)))	→	b^#(c(a(x1)))	(13)
b^#(b(b(x1)))	→	a^#(x1)	(14)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(15)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(c(x1))	→	b^#(a(x1))	(8)
b^#(x1)	→	a^#(x1)	(10)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(15)
c^#(c(x1))	→	a^#(x1)	(9)
b^#(x1)	→	a^#(a(x1))	(11)
b^#(b(b(x1)))	→	c^#(a(x1))	(12)
b^#(b(b(x1)))	→	b^#(c(a(x1)))	(13)
b^#(b(b(x1)))	→	a^#(x1)	(14)

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	2
0	0

· x₁ +

-∞

[b(x₁)]

0	2
0	2

· x₁ +

-∞

[a(x₁)]

0	2
-2	0

· x₁ +

-∞

[c^#(x₁)]

5	5
5	5

· x₁ +

-∞

[b^#(x₁)]

5	5
5	5

· x₁ +

-∞

[a^#(x₁)]

3	5
3	5

· x₁ +

-∞

together with the usable rules

c(c(x1))	→	b(a(x1))	(5)
b(x1)	→	a(a(x1))	(2)
b(b(b(x1)))	→	b(c(a(x1)))	(6)
a(c(a(x1)))	→	c(c(a(x1)))	(7)

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

c^#(c(x1))	→	a^#(x1)	(9)
b^#(b(b(x1)))	→	c^#(a(x1))	(12)
b^#(b(b(x1)))	→	b^#(c(a(x1)))	(13)
b^#(b(b(x1)))	→	a^#(x1)	(14)

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^#(c(x1))	→	b^#(a(x1))	(8)
b^#(x1)	→	a^#(x1)	(10)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(15)
b^#(x1)	→	a^#(a(x1))	(11)

1.1.1.1.1.1 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	2
0	0

· x₁ +

-∞

[b(x₁)]

2	2
0	0

· x₁ +

-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞

[c^#(x₁)]

1	2
1	2

· x₁ +

-∞

[b^#(x₁)]

2	2
2	2

· x₁ +

-∞

[a^#(x₁)]

1	1
1	1

· x₁ +

-∞

together with the usable rules

c(c(x1))	→	b(a(x1))	(5)
b(x1)	→	a(a(x1))	(2)
b(b(b(x1)))	→	b(c(a(x1)))	(6)
a(c(a(x1)))	→	c(c(a(x1)))	(7)

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

b^#(x1)	→	a^#(x1)	(10)
b^#(x1)	→	a^#(a(x1))	(11)

could be deleted.

1.1.1.1.1.1.1 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 +

[a^#(x₁)]

x₁ +

together with the usable rules

c(c(x1))	→	b(a(x1))	(5)
b(x1)	→	a(a(x1))	(2)
b(b(b(x1)))	→	b(c(a(x1)))	(6)
a(c(a(x1)))	→	c(c(a(x1)))	(7)

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

c^#(c(x1))	→	b^#(a(x1))	(8)
a^#(c(a(x1)))	→	c^#(c(a(x1)))	(15)

and no rules could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.