Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-85)

The rewrite relation of the following TRS is considered.

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

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(x1)	→	x1	(1)
a(a(x1))	→	b(c(c(b(a(b(x1))))))	(4)
b(c(x1))	→	a(x1)	(5)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

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	3	3
0	3	3
0	3	3

· x₁ +

-∞

[b(x₁)]

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

· x₁ +

-∞

[a(x₁)]

0	3	3
0	3	3
-3	0	0

· x₁ +

-∞

[b^#(x₁)]

16	16	16
16	16	16
16	16	16

· x₁ +

-∞

[a^#(x₁)]

16	19	19
16	19	19
16	19	19

· x₁ +

-∞

together with the usable rules

a(x1)	→	x1	(1)
a(a(x1))	→	b(c(c(b(a(b(x1))))))	(4)
b(c(x1))	→	a(x1)	(5)

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

a^#(a(x1))	→	b^#(x1)	(7)
a^#(a(x1))	→	b^#(a(b(x1)))	(9)
a^#(a(x1))	→	a^#(b(x1))	(10)

could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

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

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
0	0	0

· x₁ +

-∞

[a(x₁)]

0	0	0
0	0	0
0	0	0

· x₁ +

-∞

[b^#(x₁)]

1	4	4
1	4	4
1	4	4

· x₁ +

-∞

[a^#(x₁)]

1	1	3
1	1	3
1	1	3

· x₁ +

-∞

together with the usable rules

a(x1)	→	x1	(1)
a(a(x1))	→	b(c(c(b(a(b(x1))))))	(4)
b(c(x1))	→	a(x1)	(5)

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

a^#(a(x1))

→

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

(8)

could be deleted.

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

a(x1)	→	x1	(1)
a(a(x1))	→	b(c(c(b(a(b(x1))))))	(4)
b(c(x1))	→	a(x1)	(5)

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

b^#(c(x1))

→

a^#(x1)

(6)

and no rules could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.