Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x01)

The rewrite relation of the following TRS is considered.

a(x1)	→	b(b(x1))	(1)
c(b(x1))	→	d(x1)	(2)
e(b(x1))	→	c(c(x1))	(3)
d(b(x1))	→	b(f(x1))	(4)
f(x1)	→	a(e(x1))	(5)
c(x1)	→	x1	(6)
a(a(x1))	→	f(x1)	(7)

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.

f^#(x1)	→	e^#(x1)	(8)
f^#(x1)	→	a^#(e(x1))	(9)
e^#(b(x1))	→	c^#(x1)	(10)
e^#(b(x1))	→	c^#(c(x1))	(11)
d^#(b(x1))	→	f^#(x1)	(12)
c^#(b(x1))	→	d^#(x1)	(13)
a^#(a(x1))	→	f^#(x1)	(14)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(x1)	→	e^#(x1)	(8)
e^#(b(x1))	→	c^#(x1)	(10)
c^#(b(x1))	→	d^#(x1)	(13)
d^#(b(x1))	→	f^#(x1)	(12)
f^#(x1)	→	a^#(e(x1))	(9)
a^#(a(x1))	→	f^#(x1)	(14)
e^#(b(x1))	→	c^#(c(x1))	(11)

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

[f(x₁)]

3	3	3
0	0	0
0	0	0

· x₁ +

-∞

[e(x₁)]

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

· x₁ +

-∞

[d(x₁)]

0	3	3
0	3	3
-3	0	0

· x₁ +

-∞

[c(x₁)]

0	0	3
0	0	3
-3	-3	0

· x₁ +

-∞

[b(x₁)]

0	0	3
0	0	0
-3	0	0

· x₁ +

-∞

[a(x₁)]

3	3	3
0	0	3
0	0	0

· x₁ +

-∞

[f^#(x₁)]

3	4	4
3	4	4
3	4	4

· x₁ +

-∞

[e^#(x₁)]

3	3	3
3	3	3
3	3	3

· x₁ +

-∞

[d^#(x₁)]

3	3	6
3	3	6
3	3	6

· x₁ +

-∞

[c^#(x₁)]

3	3	6
3	3	6
3	3	6

· x₁ +

-∞

[a^#(x₁)]

3	3	6
3	3	6
3	3	6

· x₁ +

-∞

together with the usable rules

a(x1)	→	b(b(x1))	(1)
c(b(x1))	→	d(x1)	(2)
e(b(x1))	→	c(c(x1))	(3)
d(b(x1))	→	b(f(x1))	(4)
f(x1)	→	a(e(x1))	(5)
c(x1)	→	x1	(6)
a(a(x1))	→	f(x1)	(7)

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

a^#(a(x1))

→

f^#(x1)

(14)

could be deleted.

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

[f(x₁)]

x₁ +

[e(x₁)]

x₁ +

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[f^#(x₁)]

x₁ +

[e^#(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[c^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

a(x1)	→	b(b(x1))	(1)
c(b(x1))	→	d(x1)	(2)
e(b(x1))	→	c(c(x1))	(3)
d(b(x1))	→	b(f(x1))	(4)
f(x1)	→	a(e(x1))	(5)
c(x1)	→	x1	(6)
a(a(x1))	→	f(x1)	(7)

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

f^#(x1)

→

a^#(e(x1))

(9)

and no rules 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

f^#(x1)	→	e^#(x1)	(8)
e^#(b(x1))	→	c^#(x1)	(10)
c^#(b(x1))	→	d^#(x1)	(13)
d^#(b(x1))	→	f^#(x1)	(12)
e^#(b(x1))	→	c^#(c(x1))	(11)

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

[f(x₁)]

3	3	3
0	0	0
0	0	0

· x₁ +

-∞

[e(x₁)]

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

· x₁ +

-∞

[d(x₁)]

0	3	3
0	3	3
-3	0	0

· x₁ +

-∞

[c(x₁)]

0	0	3
0	0	3
-3	-3	0

· x₁ +

-∞

[b(x₁)]

0	0	3
0	0	0
-3	0	0

· x₁ +

-∞

[a(x₁)]

3	3	3
0	0	3
0	0	0

· x₁ +

-∞

[f^#(x₁)]

14	14	14
14	14	14
14	14	14

· x₁ +

-∞

[e^#(x₁)]

11	11	11
11	11	11
11	11	11

· x₁ +

-∞

[d^#(x₁)]

11	14	14
11	14	14
11	14	14

· x₁ +

-∞

[c^#(x₁)]

11	11	14
11	11	14
11	11	14

· x₁ +

-∞

together with the usable rules

a(x1)	→	b(b(x1))	(1)
c(b(x1))	→	d(x1)	(2)
e(b(x1))	→	c(c(x1))	(3)
d(b(x1))	→	b(f(x1))	(4)
f(x1)	→	a(e(x1))	(5)
c(x1)	→	x1	(6)
a(a(x1))	→	f(x1)	(7)

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

f^#(x1)

→

e^#(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

[f(x₁)]

x₁ +

[e(x₁)]

x₁ +

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[f^#(x₁)]

x₁ +

[e^#(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[c^#(x₁)]

x₁ +

together with the usable rules

a(x1)	→	b(b(x1))	(1)
c(b(x1))	→	d(x1)	(2)
e(b(x1))	→	c(c(x1))	(3)
d(b(x1))	→	b(f(x1))	(4)
f(x1)	→	a(e(x1))	(5)
c(x1)	→	x1	(6)
a(a(x1))	→	f(x1)	(7)

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

e^#(b(x1))	→	c^#(x1)	(10)
c^#(b(x1))	→	d^#(x1)	(13)
d^#(b(x1))	→	f^#(x1)	(12)
e^#(b(x1))	→	c^#(c(x1))	(11)

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.