Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-60)

The rewrite relation of the following TRS is considered.

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

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(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

c^#(b(b(x1)))	→	a^#(b(b(x1)))	(13)
b^#(c(b(x1)))	→	c^#(c(a(x1)))	(14)
b^#(c(b(x1)))	→	c^#(a(x1))	(15)
b^#(c(b(x1)))	→	a^#(x1)	(16)
b^#(b(a(x1)))	→	b^#(x1)	(17)
b^#(b(a(x1)))	→	b^#(a(b(x1)))	(18)
b^#(b(a(x1)))	→	a^#(b(x1))	(19)
b^#(a(c(x1)))	→	b^#(a(x1))	(20)
b^#(a(c(x1)))	→	a^#(x1)	(21)
b^#(a(c(x1)))	→	a^#(b(a(x1)))	(22)
a^#(c(c(x1)))	→	c^#(b(b(x1)))	(23)
a^#(c(c(x1)))	→	b^#(x1)	(24)
a^#(c(c(x1)))	→	b^#(c(c(x1)))	(25)
a^#(c(c(x1)))	→	b^#(b(x1))	(26)

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(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

b^#(c(b(x1)))	→	c^#(a(x1))	(15)
b^#(c(b(x1)))	→	a^#(x1)	(16)
b^#(b(a(x1)))	→	b^#(x1)	(17)
b^#(b(a(x1)))	→	a^#(b(x1))	(19)
b^#(a(c(x1)))	→	b^#(a(x1))	(20)
b^#(a(c(x1)))	→	a^#(x1)	(21)
a^#(c(c(x1)))	→	b^#(x1)	(24)
a^#(c(c(x1)))	→	b^#(b(x1))	(26)

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(b(x1)))	→	a^#(b(b(x1)))	(13)
a^#(c(c(x1)))	→	c^#(b(b(x1)))	(23)
a^#(c(c(x1)))	→	b^#(c(c(x1)))	(25)
b^#(c(b(x1)))	→	c^#(c(a(x1)))	(14)
b^#(b(a(x1)))	→	b^#(a(b(x1)))	(18)
b^#(a(c(x1)))	→	a^#(b(a(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)]

4	6
4	4

· x₁ +

-∞

[b(x₁)]

4	4
4	4

· x₁ +

-∞

[a(x₁)]

4	4
4	4

· x₁ +

-∞

[c^#(x₁)]

3	5
3	5

· x₁ +

-∞

[b^#(x₁)]

3	5
3	5

· x₁ +

-∞

[a^#(x₁)]

5	5
5	5

· x₁ +

-∞

together with the usable rules

c(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

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

→

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

(23)

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

c^#(b(b(x1)))	→	a^#(b(b(x1)))	(13)
a^#(c(c(x1)))	→	b^#(c(c(x1)))	(25)
b^#(c(b(x1)))	→	c^#(c(a(x1)))	(14)
b^#(b(a(x1)))	→	b^#(a(b(x1)))	(18)
b^#(a(c(x1)))	→	a^#(b(a(x1)))	(22)

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 arctic semiring over the naturals

[c(x₁)]

2	4
2	2

· x₁ +

-∞

[b(x₁)]

2	2
2	2

· x₁ +

-∞

[a(x₁)]

2	2
2	2

· x₁ +

-∞

[c^#(x₁)]

1	3
1	3

· x₁ +

-∞

[b^#(x₁)]

2	3
2	3

· x₁ +

-∞

[a^#(x₁)]

3	3
3	3

· x₁ +

-∞

together with the usable rules

c(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

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

→

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

(14)

could be deleted.

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 +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

c(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

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

→

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

(13)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(c(c(x1)))	→	b^#(c(c(x1)))	(25)
b^#(b(a(x1)))	→	b^#(a(b(x1)))	(18)
b^#(a(c(x1)))	→	a^#(b(a(x1)))	(22)

1.1.1.1.1.1.1.1.1.1 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)]

3	3	6
3	3	6
0	3	3

· x₁ +

-∞

[b(x₁)]

3	3	3
0	3	3
0	3	3

· x₁ +

-∞

[a(x₁)]

3	6	6
0	3	3
0	3	3

· x₁ +

-∞

[b^#(x₁)]

10	10	10
10	10	10
10	10	10

· x₁ +

-∞

[a^#(x₁)]

10	12	12
10	12	12
10	12	12

· x₁ +

-∞

together with the usable rules

c(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

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

→

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

(25)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1 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 +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

c(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

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

→

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

(22)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

→

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

(18)

1.1.1.1.1.1.1.1.1.1.1.1.1 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	-3	0
-3	-3	-3

· x₁ +

-∞

[b(x₁)]

0	0	0
0	0	0
-3	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[b^#(x₁)]

1	2	3
1	2	3
1	2	3

· x₁ +

-∞

together with the usable rules

c(b(b(x1)))	→	a(b(b(x1)))	(7)
a(c(c(x1)))	→	c(b(b(x1)))	(8)
b(a(c(x1)))	→	a(b(a(x1)))	(9)
b(b(a(x1)))	→	b(a(b(x1)))	(10)
b(c(b(x1)))	→	c(c(a(x1)))	(11)
a(c(c(x1)))	→	b(c(c(x1)))	(12)

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

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

→

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

(18)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.