Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z117)

The rewrite relation of the following TRS is considered.

d(a(x1))	→	b(d(x1))	(1)
b(x1)	→	a(a(a(x1)))	(2)
c(d(c(x1)))	→	a(d(x1))	(3)
b(d(d(x1)))	→	c(c(d(d(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(d(x1))	→	d(b(x1))	(5)
b(x1)	→	a(a(a(x1)))	(2)
c(d(c(x1)))	→	d(a(x1))	(6)
d(d(b(x1)))	→	c(d(d(c(c(x1)))))	(7)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

d^#(d(b(x1)))	→	d^#(d(c(c(x1))))	(8)
d^#(d(b(x1)))	→	d^#(c(c(x1)))	(9)
d^#(d(b(x1)))	→	c^#(x1)	(10)
d^#(d(b(x1)))	→	c^#(d(d(c(c(x1)))))	(11)
d^#(d(b(x1)))	→	c^#(c(x1))	(12)
c^#(d(c(x1)))	→	d^#(a(x1))	(13)
c^#(d(c(x1)))	→	a^#(x1)	(14)
b^#(x1)	→	a^#(x1)	(15)
b^#(x1)	→	a^#(a(x1))	(16)
b^#(x1)	→	a^#(a(a(x1)))	(17)
a^#(d(x1))	→	d^#(b(x1))	(18)
a^#(d(x1))	→	b^#(x1)	(19)

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 +

[c^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

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

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

d^#(d(b(x1)))	→	d^#(c(c(x1)))	(9)
d^#(d(b(x1)))	→	c^#(x1)	(10)
d^#(d(b(x1)))	→	c^#(c(x1))	(12)
c^#(d(c(x1)))	→	a^#(x1)	(14)
b^#(x1)	→	a^#(x1)	(15)
b^#(x1)	→	a^#(a(x1))	(16)
b^#(x1)	→	a^#(a(a(x1)))	(17)
a^#(d(x1))	→	d^#(b(x1))	(18)
a^#(d(x1))	→	b^#(x1)	(19)

and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

1.1.1.1.1 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)]

6	6	6
3	3	6
3	3	3

· x₁ +

-∞

[c(x₁)]

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

· x₁ +

-∞

[b(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[d^#(x₁)]

13	14	15
13	14	15
13	14	15

· x₁ +

-∞

[c^#(x₁)]

8	8	8
8	8	8
8	8	8

· x₁ +

-∞

together with the usable rules

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

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

d^#(d(b(x1)))

→

d^#(d(c(c(x1))))

(8)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(d(b(x1)))	→	c^#(d(d(c(c(x1)))))	(11)
c^#(d(c(x1)))	→	d^#(a(x1))	(13)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[d(x₁)]

1	0
0	1

· x₁ +

[c(x₁)]

1	0
1	0

· x₁ +

[b(x₁)]

1	0
1	0

· x₁ +

[a(x₁)]

1	0
1	0

· x₁ +

[d^#(x₁)]

2	0
0	0

· x₁ +

[c^#(x₁)]

0	2
0	0

· x₁ +

together with the usable rules

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

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

d^#(d(b(x1)))	→	c^#(d(d(c(c(x1)))))	(11)
c^#(d(c(x1)))	→	d^#(a(x1))	(13)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.