Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove3)

The rewrite relation of the following TRS is considered.

p(0(x1))	→	s(s(0(s(s(p(x1))))))	(1)
p(s(0(x1)))	→	0(x1)	(2)
p(s(s(x1)))	→	s(p(s(x1)))	(3)
f(s(x1))	→	g(q(i(x1)))	(4)
g(x1)	→	f(p(p(x1)))	(5)
q(i(x1))	→	q(s(x1))	(6)
q(s(x1))	→	s(s(x1))	(7)
i(x1)	→	s(x1)	(8)

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

0(p(x1))	→	p(s(s(0(s(s(x1))))))	(9)
0(s(p(x1)))	→	0(x1)	(10)
s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

0^#(s(p(x1)))	→	0^#(x1)	(16)
0^#(p(x1))	→	0^#(s(s(x1)))	(17)
0^#(p(x1))	→	s^#(x1)	(18)
0^#(p(x1))	→	s^#(0(s(s(x1))))	(19)
0^#(p(x1))	→	s^#(s(x1))	(20)
0^#(p(x1))	→	s^#(s(0(s(s(x1)))))	(21)
i^#(x1)	→	s^#(x1)	(22)
i^#(q(x1))	→	s^#(q(x1))	(23)
s^#(f(x1))	→	i^#(q(g(x1)))	(24)
s^#(f(x1))	→	g^#(x1)	(25)
s^#(s(p(x1)))	→	s^#(x1)	(26)
s^#(s(p(x1)))	→	s^#(p(s(x1)))	(27)
s^#(q(x1))	→	s^#(x1)	(28)
s^#(q(x1))	→	s^#(s(x1))	(29)

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

[0(x₁)]

x₁ +

[i(x₁)]

x₁ +

[g(x₁)]

x₁ +

[f(x₁)]

x₁ +

[s(x₁)]

x₁ +

[q(x₁)]

x₁ +

[p(x₁)]

x₁ +

[0^#(x₁)]

x₁ +

[i^#(x₁)]

x₁ +

[g^#(x₁)]

x₁ +

[s^#(x₁)]

x₁ +

together with the usable rules

0(p(x1))	→	p(s(s(0(s(s(x1))))))	(9)
0(s(p(x1)))	→	0(x1)	(10)
s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

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

0^#(p(x1))	→	s^#(x1)	(18)
0^#(p(x1))	→	s^#(0(s(s(x1))))	(19)
0^#(p(x1))	→	s^#(s(x1))	(20)
0^#(p(x1))	→	s^#(s(0(s(s(x1)))))	(21)
s^#(f(x1))	→	g^#(x1)	(25)

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

0^#(s(p(x1)))	→	0^#(x1)	(16)
0^#(p(x1))	→	0^#(s(s(x1)))	(17)

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

[i(x₁)]

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

· x₁ +

-∞

[g(x₁)]

3	3	6
3	3	6
3	3	6

· x₁ +

-∞

[f(x₁)]

3	3	6
0	0	3
0	0	3

· x₁ +

-∞

[s(x₁)]

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

· x₁ +

-∞

[q(x₁)]

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

· x₁ +

-∞

[p(x₁)]

0	0	3
0	0	0
-3	0	0

· x₁ +

-∞

[0^#(x₁)]

1	3	4
1	3	4
1	3	4

· x₁ +

-∞

together with the usable rules

s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

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

0^#(p(x1))

→

0^#(s(s(x1)))

(17)

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
0^#(s(p(x1))) → 0^#(x1) (16)

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

[s(x₁)] = x₁ +

1

[p(x₁)] = x₁ +

1

[0^#(x₁)] = x₁ +

0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
0^#(s(p(x1))) → 0^#(x1) (16)
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.

The 2^nd component contains the pair

i^#(x1)	→	s^#(x1)	(22)
s^#(f(x1))	→	i^#(q(g(x1)))	(24)
i^#(q(x1))	→	s^#(q(x1))	(23)
s^#(q(x1))	→	s^#(x1)	(28)
s^#(s(p(x1)))	→	s^#(x1)	(26)
s^#(q(x1))	→	s^#(s(x1))	(29)

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

[i(x₁)]

0	0
0	0

· x₁ +

-∞

[g(x₁)]

2	4
0	2

· x₁ +

-∞

[f(x₁)]

2	4
2	4

· x₁ +

-∞

[s(x₁)]

0	0
0	0

· x₁ +

-∞

[q(x₁)]

0	2
0	0

· x₁ +

-∞

[p(x₁)]

0	0
-2	-2

· x₁ +

-∞

[i^#(x₁)]

1	2
1	2

· x₁ +

-∞

[s^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

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

s^#(q(x1))

→

s^#(x1)

(28)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

i^#(x1)	→	s^#(x1)	(22)
s^#(f(x1))	→	i^#(q(g(x1)))	(24)
i^#(q(x1))	→	s^#(q(x1))	(23)
s^#(q(x1))	→	s^#(s(x1))	(29)
s^#(s(p(x1)))	→	s^#(x1)	(26)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

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

[i(x₁)]

0	3	3
0	3	3
-3	0	0

· x₁ +

-∞

[g(x₁)]

6	6	9
3	3	6
3	3	6

· x₁ +

-∞

[f(x₁)]

6	6	9
6	6	9
6	6	9

· x₁ +

-∞

[s(x₁)]

0	0	3
0	0	3
-3	0	0

· x₁ +

-∞

[q(x₁)]

0	3	3
0	3	3
-3	0	0

· x₁ +

-∞

[p(x₁)]

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

· x₁ +

-∞

[i^#(x₁)]

9	9	12
9	9	12
9	9	12

· x₁ +

-∞

[s^#(x₁)]

9	9	12
9	9	12
9	9	12

· x₁ +

-∞

together with the usable rules

s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

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

s^#(f(x1))

→

i^#(q(g(x1)))

(24)

could be deleted.

1.1.1.1.2.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

[i(x₁)]

x₁ +

[g(x₁)]

x₁ +

[f(x₁)]

x₁ +

[s(x₁)]

x₁ +

[q(x₁)]

x₁ +

[p(x₁)]

x₁ +

[i^#(x₁)]

x₁ +

[s^#(x₁)]

x₁ +

together with the usable rules

s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

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

i^#(x1)	→	s^#(x1)	(22)
i^#(q(x1))	→	s^#(q(x1))	(23)

and no rules could be deleted.

1.1.1.1.2.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

s^#(q(x1))	→	s^#(s(x1))	(29)
s^#(s(p(x1)))	→	s^#(x1)	(26)

1.1.1.1.2.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 2 with strict dimension 1 over the arctic semiring over the naturals

[i(x₁)]

0	2
0	2

· x₁ +

-∞

[g(x₁)]

2	4
0	2

· x₁ +

-∞

[f(x₁)]

2	4
2	4

· x₁ +

-∞

[s(x₁)]

0	2
0	2

· x₁ +

-∞

[q(x₁)]

2	4
0	2

· x₁ +

-∞

[p(x₁)]

0	0
-2	-2

· x₁ +

-∞

[s^#(x₁)]

8	9
8	9

· x₁ +

-∞

together with the usable rules

s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

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

s^#(q(x1))

→

s^#(s(x1))

(29)

could be deleted.

1.1.1.1.2.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
s^#(s(p(x1))) → s^#(x1) (26)

1.1.1.1.2.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

[s(x₁)] = x₁ +

1

[p(x₁)] = x₁ +

1

[s^#(x₁)] = x₁ +

0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
s^#(s(p(x1))) → s^#(x1) (26)
and no rules could be deleted.
1.1.1.1.2.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove3)

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

1.1 Dependency Pair Transformation

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.2 Reduction Pair Processor with Usable Rules

1.1.1.1.2.1 Dependency Graph Processor

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.2.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.2.1.1.1.1 Dependency Graph Processor

1.1.1.1.2.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.2.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.2.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.2.1.1.1.1.1.1.1.1 Dependency Graph Processor