Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x05)

The rewrite relation of the following TRS is considered.

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

a(a(x1))	→	b(b(x1))	(1)
b(c(c(x1)))	→	a(c(d(x1)))	(5)
a(x1)	→	c(c(d(x1)))	(6)
d(c(x1))	→	c(b(x1))	(7)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(c(x1))	→	b^#(x1)	(8)
b^#(c(c(x1)))	→	d^#(x1)	(9)
b^#(c(c(x1)))	→	a^#(c(d(x1)))	(10)
a^#(x1)	→	d^#(x1)	(11)

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[d(x₁)]

0	0	0
0	0	0
-3	-3	-3

· x₁ +

-∞

[c(x₁)]

0	0	3
0	0	0
-3	0	0

· x₁ +

-∞

[b(x₁)]

0	0	3
-3	-3	0
-3	-3	0

· x₁ +

-∞

[a(x₁)]

3	3	3
0	0	0
0	0	0

· x₁ +

-∞

[d^#(x₁)]

5	7	7
5	7	7
5	7	7

· x₁ +

-∞

[b^#(x₁)]

7	7	8
7	7	8
7	7	8

· x₁ +

-∞

[a^#(x₁)]

6	7	7
6	7	7
6	7	7

· x₁ +

-∞

together with the usable rules

a(a(x1))	→	b(b(x1))	(1)
b(c(c(x1)))	→	a(c(d(x1)))	(5)
a(x1)	→	c(c(d(x1)))	(6)
d(c(x1))	→	c(b(x1))	(7)

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

b^#(c(c(x1)))	→	d^#(x1)	(9)
b^#(c(c(x1)))	→	a^#(c(d(x1)))	(10)

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

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

a(a(x1))	→	b(b(x1))	(1)
b(c(c(x1)))	→	a(c(d(x1)))	(5)
a(x1)	→	c(c(d(x1)))	(6)
d(c(x1))	→	c(b(x1))	(7)

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

d^#(c(x1))	→	b^#(x1)	(8)
a^#(x1)	→	d^#(x1)	(11)

and no rules could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.