Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/3-matchbox)

The rewrite relation of the following TRS is considered.

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

a(a(b(c(x1))))	→	c(b(a(a(x1))))	(5)
a(a(a(b(x1))))	→	b(a(a(a(x1))))	(6)
c(b(a(x1)))	→	a(b(c(x1)))	(7)
b(b(c(c(x1))))	→	c(c(b(b(x1))))	(8)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(b(a(x1)))	→	c^#(x1)	(9)
c^#(b(a(x1)))	→	b^#(c(x1))	(10)
c^#(b(a(x1)))	→	a^#(b(c(x1)))	(11)
b^#(b(c(c(x1))))	→	c^#(c(b(b(x1))))	(12)
b^#(b(c(c(x1))))	→	c^#(b(b(x1)))	(13)
b^#(b(c(c(x1))))	→	b^#(x1)	(14)
b^#(b(c(c(x1))))	→	b^#(b(x1))	(15)
a^#(a(b(c(x1))))	→	c^#(b(a(a(x1))))	(16)
a^#(a(b(c(x1))))	→	b^#(a(a(x1)))	(17)
a^#(a(b(c(x1))))	→	a^#(x1)	(18)
a^#(a(b(c(x1))))	→	a^#(a(x1))	(19)
a^#(a(a(b(x1))))	→	b^#(a(a(a(x1))))	(20)
a^#(a(a(b(x1))))	→	a^#(x1)	(21)
a^#(a(a(b(x1))))	→	a^#(a(x1))	(22)
a^#(a(a(b(x1))))	→	a^#(a(a(x1)))	(23)

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

a(a(b(c(x1))))	→	c(b(a(a(x1))))	(5)
a(a(a(b(x1))))	→	b(a(a(a(x1))))	(6)
c(b(a(x1)))	→	a(b(c(x1)))	(7)
b(b(c(c(x1))))	→	c(c(b(b(x1))))	(8)

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

c^#(b(a(x1)))	→	c^#(x1)	(9)
c^#(b(a(x1)))	→	b^#(c(x1))	(10)
b^#(b(c(c(x1))))	→	c^#(b(b(x1)))	(13)
b^#(b(c(c(x1))))	→	b^#(x1)	(14)
b^#(b(c(c(x1))))	→	b^#(b(x1))	(15)
a^#(a(b(c(x1))))	→	b^#(a(a(x1)))	(17)
a^#(a(b(c(x1))))	→	a^#(x1)	(18)
a^#(a(b(c(x1))))	→	a^#(a(x1))	(19)
a^#(a(a(b(x1))))	→	a^#(x1)	(21)
a^#(a(a(b(x1))))	→	a^#(a(x1))	(22)
a^#(a(a(b(x1))))	→	a^#(a(a(x1)))	(23)

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

c^#(b(a(x1)))	→	a^#(b(c(x1)))	(11)
a^#(a(b(c(x1))))	→	c^#(b(a(a(x1))))	(16)
a^#(a(a(b(x1))))	→	b^#(a(a(a(x1))))	(20)
b^#(b(c(c(x1))))	→	c^#(c(b(b(x1))))	(12)

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

[c(x₁)]

9	9	9
9	9	9
6	9	9

· x₁ +

-∞

[b(x₁)]

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

· x₁ +

-∞

[a(x₁)]

6	6	6
6	6	6
3	3	3

· x₁ +

-∞

[c^#(x₁)]

12	13	14
12	13	14
12	13	14

· x₁ +

-∞

[b^#(x₁)]

2	2	5
2	2	5
2	2	5

· x₁ +

-∞

[a^#(x₁)]

9	9	9
9	9	9
9	9	9

· x₁ +

-∞

together with the usable rules

a(a(b(c(x1))))	→	c(b(a(a(x1))))	(5)
a(a(a(b(x1))))	→	b(a(a(a(x1))))	(6)
c(b(a(x1)))	→	a(b(c(x1)))	(7)
b(b(c(c(x1))))	→	c(c(b(b(x1))))	(8)

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

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

→

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

(20)

could be deleted.

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

a(a(b(c(x1))))	→	c(b(a(a(x1))))	(5)
a(a(a(b(x1))))	→	b(a(a(a(x1))))	(6)
c(b(a(x1)))	→	a(b(c(x1)))	(7)
b(b(c(c(x1))))	→	c(c(b(b(x1))))	(8)

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

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

→

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

(12)

and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(b(a(x1)))	→	a^#(b(c(x1)))	(11)
a^#(a(b(c(x1))))	→	c^#(b(a(a(x1))))	(16)

1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[c(x₁)]

0	0	0
-3	0	0
-3	0	0

· x₁ +

-∞

[b(x₁)]

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

· x₁ +

-∞

[a(x₁)]

6	6	6
6	6	6
6	6	6

· x₁ +

-∞

[c^#(x₁)]

1	1	1
1	1	1
1	1	1

· x₁ +

-∞

[a^#(x₁)]

7	7	8
7	7	8
7	7	8

· x₁ +

-∞

together with the usable rules

a(a(b(c(x1))))	→	c(b(a(a(x1))))	(5)
a(a(a(b(x1))))	→	b(a(a(a(x1))))	(6)
c(b(a(x1)))	→	a(b(c(x1)))	(7)
b(b(c(c(x1))))	→	c(c(b(b(x1))))	(8)

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

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

→

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

(16)

could be deleted.

1.1.1.1.1.1.1.1.1 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 +

[a^#(x₁)]

x₁ +

together with the usable rules

a(a(b(c(x1))))	→	c(b(a(a(x1))))	(5)
a(a(a(b(x1))))	→	b(a(a(a(x1))))	(6)
c(b(a(x1)))	→	a(b(c(x1)))	(7)
b(b(c(c(x1))))	→	c(c(b(b(x1))))	(8)

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

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

→

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

(11)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.