Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-120)

The rewrite relation of the following TRS is considered.

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

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(c(c(x1)))	→	b(c(c(x1)))	(6)
c(b(b(x1)))	→	b(a(a(x1)))	(7)
b(a(b(x1)))	→	b(c(a(x1)))	(8)
a(a(c(x1)))	→	b(a(b(x1)))	(9)
c(c(c(x1)))	→	c(a(c(x1)))	(5)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

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(c(c(x1)))	→	b(c(c(x1)))	(6)
c(b(b(x1)))	→	b(a(a(x1)))	(7)
b(a(b(x1)))	→	b(c(a(x1)))	(8)
a(a(c(x1)))	→	b(a(b(x1)))	(9)
c(c(c(x1)))	→	c(a(c(x1)))	(5)

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

c^#(c(c(x1)))	→	a^#(c(x1))	(11)
c^#(b(b(x1)))	→	b^#(a(a(x1)))	(12)
c^#(b(b(x1)))	→	a^#(x1)	(13)
c^#(b(b(x1)))	→	a^#(a(x1))	(14)
b^#(a(b(x1)))	→	c^#(a(x1))	(15)
b^#(a(b(x1)))	→	a^#(x1)	(17)
a^#(c(c(x1)))	→	b^#(c(c(x1)))	(18)
a^#(a(c(x1)))	→	b^#(x1)	(19)
a^#(a(c(x1)))	→	b^#(a(b(x1)))	(20)
a^#(a(c(x1)))	→	a^#(b(x1))	(21)

and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

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

→

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

(10)

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	0
0	0

· x₁ +

-∞

[b(x₁)]

0	0
-2	-2

· x₁ +

-∞

[a(x₁)]

0	0
-2	-2

· x₁ +

-∞

[c^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

a(c(c(x1)))	→	b(c(c(x1)))	(6)
c(b(b(x1)))	→	b(a(a(x1)))	(7)
b(a(b(x1)))	→	b(c(a(x1)))	(8)
a(a(c(x1)))	→	b(a(b(x1)))	(9)
c(c(c(x1)))	→	c(a(c(x1)))	(5)

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

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

→

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

(10)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

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

→

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

(16)

1.1.1.1.2 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	4

· x₁ +

-∞

[b(x₁)]

2	4
2	4

· x₁ +

-∞

[a(x₁)]

4	6
2	4

· x₁ +

-∞

[b^#(x₁)]

3	3
3	3

· x₁ +

-∞

together with the usable rules

a(c(c(x1)))	→	b(c(c(x1)))	(6)
c(b(b(x1)))	→	b(a(a(x1)))	(7)
b(a(b(x1)))	→	b(c(a(x1)))	(8)
a(a(c(x1)))	→	b(a(b(x1)))	(9)
c(c(c(x1)))	→	c(a(c(x1)))	(5)

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

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

→

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

(16)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.