Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-150)

The rewrite relation of the following TRS is considered.

c(b(b(x1)))	→	a(c(a(x1)))	(1)
a(c(c(x1)))	→	a(a(b(x1)))	(2)
a(a(c(x1)))	→	b(b(a(x1)))	(3)
b(c(b(x1)))	→	a(b(b(x1)))	(4)
b(c(c(x1)))	→	c(b(b(x1)))	(5)
a(b(b(x1)))	→	a(c(b(x1)))	(6)
c(c(a(x1)))	→	a(b(c(x1)))	(7)

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

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

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(c(b(x1)))	→	c^#(x1)	(15)
c^#(c(b(x1)))	→	b^#(c(x1))	(16)
c^#(c(b(x1)))	→	b^#(b(c(x1)))	(17)
c^#(c(a(x1)))	→	b^#(a(a(x1)))	(18)
c^#(c(a(x1)))	→	a^#(a(x1))	(19)
c^#(a(a(x1)))	→	b^#(x1)	(20)
c^#(a(a(x1)))	→	b^#(b(x1))	(21)
c^#(a(a(x1)))	→	a^#(b(b(x1)))	(22)
b^#(c(b(x1)))	→	b^#(b(a(x1)))	(23)
b^#(c(b(x1)))	→	b^#(a(x1))	(24)
b^#(c(b(x1)))	→	a^#(x1)	(25)
b^#(b(c(x1)))	→	c^#(a(x1))	(26)
b^#(b(c(x1)))	→	a^#(x1)	(27)
b^#(b(c(x1)))	→	a^#(c(a(x1)))	(28)
b^#(b(a(x1)))	→	c^#(a(x1))	(29)
b^#(b(a(x1)))	→	b^#(c(a(x1)))	(30)
a^#(c(c(x1)))	→	c^#(b(a(x1)))	(31)
a^#(c(c(x1)))	→	b^#(a(x1))	(32)
a^#(c(c(x1)))	→	a^#(x1)	(33)

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

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

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

c^#(c(b(x1)))	→	c^#(x1)	(15)
c^#(c(b(x1)))	→	b^#(c(x1))	(16)
c^#(c(a(x1)))	→	a^#(a(x1))	(19)
c^#(a(a(x1)))	→	b^#(x1)	(20)
c^#(a(a(x1)))	→	b^#(b(x1))	(21)
b^#(c(b(x1)))	→	b^#(a(x1))	(24)
b^#(c(b(x1)))	→	a^#(x1)	(25)
b^#(b(c(x1)))	→	c^#(a(x1))	(26)
b^#(b(c(x1)))	→	a^#(x1)	(27)
b^#(b(a(x1)))	→	c^#(a(x1))	(29)
a^#(c(c(x1)))	→	b^#(a(x1))	(32)
a^#(c(c(x1)))	→	a^#(x1)	(33)

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^#(c(b(x1)))	→	b^#(b(c(x1)))	(17)
b^#(c(b(x1)))	→	b^#(b(a(x1)))	(23)
b^#(b(c(x1)))	→	a^#(c(a(x1)))	(28)
a^#(c(c(x1)))	→	c^#(b(a(x1)))	(31)
c^#(c(a(x1)))	→	b^#(a(a(x1)))	(18)
b^#(b(a(x1)))	→	b^#(c(a(x1)))	(30)
c^#(a(a(x1)))	→	a^#(b(b(x1)))	(22)

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)]

0	0
0	0

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[c^#(x₁)]

2	2
2	2

· x₁ +

-∞

[b^#(x₁)]

1	2
1	2

· x₁ +

-∞

[a^#(x₁)]

1	1
1	1

· x₁ +

-∞

together with the usable rules

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

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

b^#(c(b(x1)))	→	b^#(b(a(x1)))	(23)
b^#(b(c(x1)))	→	a^#(c(a(x1)))	(28)
a^#(c(c(x1)))	→	c^#(b(a(x1)))	(31)
c^#(c(a(x1)))	→	b^#(a(a(x1)))	(18)
c^#(a(a(x1)))	→	a^#(b(b(x1)))	(22)

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 +

together with the usable rules

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

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

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

→

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

(17)

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

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

→

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

(30)

1.1.1.1.1.1.1.1 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)]

4	4	8	8
4	4	4	4
4	4	4	4
0	0	4	4

· x₁ +

-∞

[b(x₁)]

4	4	4	4
0	4	4	4
0	4	4	4
0	4	4	4

· x₁ +

-∞

[a(x₁)]

4	4	4	4
4	4	4	4
0	0	0	0
0	0	0	0

· x₁ +

-∞

[b^#(x₁)]

2	2	2	6
2	2	2	6
2	2	2	6
2	2	2	6

· x₁ +

-∞

together with the usable rules

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

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

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

→

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

(30)

could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.