Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-435)

The rewrite relation of the following TRS is considered.

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

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.

c^#(a(b(x1)))	→	c^#(x1)	(4)
c^#(a(b(x1)))	→	a^#(c(x1))	(5)
c^#(a(b(x1)))	→	a^#(a(c(x1)))	(6)
a^#(x1)	→	c^#(x1)	(7)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(a(b(x1)))	→	c^#(x1)	(4)
c^#(a(b(x1)))	→	a^#(c(x1))	(5)
a^#(x1)	→	c^#(x1)	(7)
c^#(a(b(x1)))	→	a^#(a(c(x1)))	(6)

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
0	0

· x₁ +

-∞

[a(x₁)]

0	2
0	0

· x₁ +

-∞

[c^#(x₁)]

1	2
1	2

· x₁ +

-∞

[a^#(x₁)]

1	3
1	3

· x₁ +

-∞

together with the usable rules

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

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

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

→

c^#(x1)

(4)

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

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
0	0

· x₁ +

-∞

[a(x₁)]

0	2
0	0

· x₁ +

-∞

[c^#(x₁)]

8	9
8	9

· x₁ +

-∞

[a^#(x₁)]

8	9
8	9

· x₁ +

-∞

together with the usable rules

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

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

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

→

a^#(c(x1))

(5)

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^#(x1)	→	c^#(x1)	(7)
c^#(a(b(x1)))	→	a^#(a(c(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	0
-2	-2

· x₁ +

-∞

[b(x₁)]

0	2
0	0

· x₁ +

-∞

[a(x₁)]

2	4
0	2

· x₁ +

-∞

[c^#(x₁)]

1	2
1	2

· x₁ +

-∞

[a^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

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

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

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

→

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

(6)

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 +

[a^#(x₁)]

x₁ +

together with the usable rules

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

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

a^#(x1)

→

c^#(x1)

(7)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.