Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x10)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

b^#(x1)	→	a^#(x1)	(5)
b^#(x1)	→	a^#(a(x1))	(6)
b^#(c(x1))	→	a^#(x1)	(7)
a^#(a(b(c(x1))))	→	b^#(b(a(a(x1))))	(8)
a^#(a(b(c(x1))))	→	b^#(a(a(x1)))	(9)
a^#(a(b(c(x1))))	→	a^#(x1)	(10)
a^#(a(b(c(x1))))	→	a^#(a(x1))	(11)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(x1)	→	a^#(x1)	(5)
a^#(a(b(c(x1))))	→	b^#(b(a(a(x1))))	(8)
b^#(x1)	→	a^#(a(x1))	(6)
a^#(a(b(c(x1))))	→	b^#(a(a(x1)))	(9)
b^#(c(x1))	→	a^#(x1)	(7)
a^#(a(b(c(x1))))	→	a^#(x1)	(10)
a^#(a(b(c(x1))))	→	a^#(a(x1))	(11)

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
-2	0

· x₁ +

-∞

[b(x₁)]

0	2
0	0

· x₁ +

-∞

[a(x₁)]

0	2
-2	0

· x₁ +

-∞

[b^#(x₁)]

7	9
7	9

· x₁ +

-∞

[a^#(x₁)]

7	9
7	9

· x₁ +

-∞

together with the usable rules

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

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

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

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

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

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
-2	0

· x₁ +

-∞

[b(x₁)]

0	2
0	0

· x₁ +

-∞

[a(x₁)]

0	2
-2	0

· x₁ +

-∞

[b^#(x₁)]

9	9
9	9

· x₁ +

-∞

[a^#(x₁)]

7	7
7	7

· x₁ +

-∞

together with the usable rules

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

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

b^#(x1)	→	a^#(x1)	(5)
b^#(c(x1))	→	a^#(x1)	(7)

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

a^#(a(b(c(x1))))	→	b^#(b(a(a(x1))))	(8)
b^#(x1)	→	a^#(a(x1))	(6)

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)]

0	2
0	0

· x₁ +

-∞

[b(x₁)]

2	4
0	2

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[b^#(x₁)]

2	2
2	2

· x₁ +

-∞

[a^#(x₁)]

1	1
1	1

· x₁ +

-∞

together with the usable rules

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

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

a^#(a(b(c(x1))))	→	b^#(b(a(a(x1))))	(8)
b^#(x1)	→	a^#(a(x1))	(6)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.