Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/matchbox2)

The rewrite relation of the following TRS is considered.

t(o(x1))	→	m(a(x1))	(1)
t(e(x1))	→	n(s(x1))	(2)
a(l(x1))	→	a(t(x1))	(3)
o(m(a(x1)))	→	t(e(n(x1)))	(4)
s(a(x1))	→	l(a(t(o(m(a(t(e(x1))))))))	(5)
n(s(x1))	→	a(l(a(t(x1))))	(6)

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

o(t(x1))	→	a(m(x1))	(7)
e(t(x1))	→	s(n(x1))	(8)
l(a(x1))	→	t(a(x1))	(9)
a(m(o(x1)))	→	n(e(t(x1)))	(10)
a(s(x1))	→	e(t(a(m(o(t(a(l(x1))))))))	(11)
s(n(x1))	→	t(a(l(a(x1))))	(12)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

o^#(t(x1))	→	a^#(m(x1))	(13)
e^#(t(x1))	→	s^#(n(x1))	(14)
a^#(m(o(x1)))	→	e^#(t(x1))	(15)
a^#(s(x1))	→	o^#(t(a(l(x1))))	(16)
a^#(s(x1))	→	l^#(x1)	(17)
a^#(s(x1))	→	e^#(t(a(m(o(t(a(l(x1))))))))	(18)
a^#(s(x1))	→	a^#(m(o(t(a(l(x1))))))	(19)
a^#(s(x1))	→	a^#(l(x1))	(20)
s^#(n(x1))	→	l^#(a(x1))	(21)
s^#(n(x1))	→	a^#(x1)	(22)
s^#(n(x1))	→	a^#(l(a(x1)))	(23)

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

[o(x₁)]

x₁ +

[n(x₁)]

x₁ +

[m(x₁)]

x₁ +

[l(x₁)]

x₁ +

[e(x₁)]

x₁ +

[a(x₁)]

x₁ +

[t(x₁)]

x₁ +

[s(x₁)]

x₁ +

[o^#(x₁)]

x₁ +

[l^#(x₁)]

x₁ +

[e^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

[s^#(x₁)]

x₁ +

together with the usable rules

o(t(x1))	→	a(m(x1))	(7)
e(t(x1))	→	s(n(x1))	(8)
l(a(x1))	→	t(a(x1))	(9)
a(m(o(x1)))	→	n(e(t(x1)))	(10)
a(s(x1))	→	e(t(a(m(o(t(a(l(x1))))))))	(11)
s(n(x1))	→	t(a(l(a(x1))))	(12)

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

a^#(s(x1))	→	l^#(x1)	(17)
s^#(n(x1))	→	l^#(a(x1))	(21)

and no rules 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

o^#(t(x1))	→	a^#(m(x1))	(13)
a^#(m(o(x1)))	→	e^#(t(x1))	(15)
e^#(t(x1))	→	s^#(n(x1))	(14)
s^#(n(x1))	→	a^#(x1)	(22)
a^#(s(x1))	→	o^#(t(a(l(x1))))	(16)
a^#(s(x1))	→	e^#(t(a(m(o(t(a(l(x1))))))))	(18)
a^#(s(x1))	→	a^#(m(o(t(a(l(x1))))))	(19)
a^#(s(x1))	→	a^#(l(x1))	(20)
s^#(n(x1))	→	a^#(l(a(x1)))	(23)

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

[o(x₁)]

0	0
0	0

· x₁ +

-∞

[n(x₁)]

0	0
-2	-2

· x₁ +

-∞

[m(x₁)]

0	0
-2	0

· x₁ +

-∞

[l(x₁)]

0	0
-2	-2

· x₁ +

-∞

[e(x₁)]

0	2
0	0

· x₁ +

-∞

[a(x₁)]

0	0
-2	0

· x₁ +

-∞

[t(x₁)]

0	0
-2	-2

· x₁ +

-∞

[s(x₁)]

0	0
0	0

· x₁ +

-∞

[o^#(x₁)]

2	3
2	3

· x₁ +

-∞

[e^#(x₁)]

2	3
2	3

· x₁ +

-∞

[a^#(x₁)]

1	2
1	2

· x₁ +

-∞

[s^#(x₁)]

2	3
2	3

· x₁ +

-∞

together with the usable rules

o(t(x1))	→	a(m(x1))	(7)
e(t(x1))	→	s(n(x1))	(8)
l(a(x1))	→	t(a(x1))	(9)
a(m(o(x1)))	→	n(e(t(x1)))	(10)
a(s(x1))	→	e(t(a(m(o(t(a(l(x1))))))))	(11)
s(n(x1))	→	t(a(l(a(x1))))	(12)

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

a^#(s(x1))	→	a^#(l(x1))	(20)
s^#(n(x1))	→	a^#(l(a(x1)))	(23)

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

o^#(t(x1))	→	a^#(m(x1))	(13)
a^#(m(o(x1)))	→	e^#(t(x1))	(15)
e^#(t(x1))	→	s^#(n(x1))	(14)
s^#(n(x1))	→	a^#(x1)	(22)
a^#(s(x1))	→	o^#(t(a(l(x1))))	(16)
a^#(s(x1))	→	e^#(t(a(m(o(t(a(l(x1))))))))	(18)
a^#(s(x1))	→	a^#(m(o(t(a(l(x1))))))	(19)

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

[o(x₁)]

2	2
0	0

· x₁ +

-∞

[n(x₁)]

0	0
-2	-2

· x₁ +

-∞

[m(x₁)]

0	0
-2	-2

· x₁ +

-∞

[l(x₁)]

0	0
-2	-2

· x₁ +

-∞

[e(x₁)]

2	2
2	2

· x₁ +

-∞

[a(x₁)]

0	2
0	2

· x₁ +

-∞

[t(x₁)]

0	0
-2	-2

· x₁ +

-∞

[s(x₁)]

2	2
2	2

· x₁ +

-∞

[o^#(x₁)]

15	15
15	15

· x₁ +

-∞

[e^#(x₁)]

14	16
14	16

· x₁ +

-∞

[a^#(x₁)]

14	14
14	14

· x₁ +

-∞

[s^#(x₁)]

14	14
14	14

· x₁ +

-∞

together with the usable rules

o(t(x1))	→	a(m(x1))	(7)
e(t(x1))	→	s(n(x1))	(8)
l(a(x1))	→	t(a(x1))	(9)
a(m(o(x1)))	→	n(e(t(x1)))	(10)
a(s(x1))	→	e(t(a(m(o(t(a(l(x1))))))))	(11)
s(n(x1))	→	t(a(l(a(x1))))	(12)

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

o^#(t(x1))	→	a^#(m(x1))	(13)
a^#(m(o(x1)))	→	e^#(t(x1))	(15)
a^#(s(x1))	→	o^#(t(a(l(x1))))	(16)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

e^#(t(x1))	→	s^#(n(x1))	(14)
s^#(n(x1))	→	a^#(x1)	(22)
a^#(s(x1))	→	e^#(t(a(m(o(t(a(l(x1))))))))	(18)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[o(x₁)]

0	3	3
0	0	0
-3	0	0

· x₁ +

-∞

[n(x₁)]

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

· x₁ +

-∞

[m(x₁)]

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

· x₁ +

-∞

[l(x₁)]

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

· x₁ +

-∞

[e(x₁)]

0	3	3
0	0	0
0	0	0

· x₁ +

-∞

[a(x₁)]

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

· x₁ +

-∞

[t(x₁)]

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

· x₁ +

-∞

[s(x₁)]

0	3	3
0	0	3
0	0	3

· x₁ +

-∞

[e^#(x₁)]

16	16	19
16	16	19
16	16	19

· x₁ +

-∞

[a^#(x₁)]

13	15	16
13	15	16
13	15	16

· x₁ +

-∞

[s^#(x₁)]

15	18	18
15	18	18
15	18	18

· x₁ +

-∞

together with the usable rules

o(t(x1))	→	a(m(x1))	(7)
e(t(x1))	→	s(n(x1))	(8)
l(a(x1))	→	t(a(x1))	(9)
a(m(o(x1)))	→	n(e(t(x1)))	(10)
a(s(x1))	→	e(t(a(m(o(t(a(l(x1))))))))	(11)
s(n(x1))	→	t(a(l(a(x1))))	(12)

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

e^#(t(x1))

→

s^#(n(x1))

(14)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[o(x₁)]

x₁ +

[n(x₁)]

x₁ +

[m(x₁)]

x₁ +

[l(x₁)]

x₁ +

[e(x₁)]

x₁ +

[a(x₁)]

x₁ +

[t(x₁)]

x₁ +

[s(x₁)]

x₁ +

[e^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

[s^#(x₁)]

x₁ +

together with the usable rules

o(t(x1))	→	a(m(x1))	(7)
e(t(x1))	→	s(n(x1))	(8)
l(a(x1))	→	t(a(x1))	(9)
a(m(o(x1)))	→	n(e(t(x1)))	(10)
a(s(x1))	→	e(t(a(m(o(t(a(l(x1))))))))	(11)
s(n(x1))	→	t(a(l(a(x1))))	(12)

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

s^#(n(x1))	→	a^#(x1)	(22)
a^#(s(x1))	→	e^#(t(a(m(o(t(a(l(x1))))))))	(18)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.