Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/torpa1)

The rewrite relation of the following TRS is considered.

a(d(x1))	→	d(b(c(b(d(x1)))))	(1)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	a(d(x1))	(3)
d(f(x1))	→	b(d(x1))	(4)

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

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

f^#(d(x1))	→	d^#(b(x1))	(8)
f^#(d(x1))	→	b^#(x1)	(9)
d^#(a(x1))	→	d^#(x1)	(10)
d^#(a(x1))	→	d^#(b(c(b(d(x1)))))	(11)
d^#(a(x1))	→	b^#(d(x1))	(12)
d^#(a(x1))	→	b^#(c(b(d(x1))))	(13)
b^#(d(b(x1)))	→	d^#(a(x1))	(14)
b^#(d(b(x1)))	→	a^#(x1)	(15)
a^#(x1)	→	f^#(b(b(x1)))	(16)
a^#(x1)	→	b^#(x1)	(17)
a^#(x1)	→	b^#(f(b(b(x1))))	(18)
a^#(x1)	→	b^#(b(x1))	(19)
a^#(x1)	→	b^#(b(f(b(b(x1)))))	(20)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(d(x1))	→	d^#(b(x1))	(8)
d^#(a(x1))	→	d^#(x1)	(10)
d^#(a(x1))	→	b^#(d(x1))	(12)
b^#(d(b(x1)))	→	d^#(a(x1))	(14)
b^#(d(b(x1)))	→	a^#(x1)	(15)
a^#(x1)	→	f^#(b(b(x1)))	(16)
f^#(d(x1))	→	b^#(x1)	(9)
a^#(x1)	→	b^#(x1)	(17)
a^#(x1)	→	b^#(f(b(b(x1))))	(18)
a^#(x1)	→	b^#(b(x1))	(19)
a^#(x1)	→	b^#(b(f(b(b(x1)))))	(20)

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

[f(x₁)]

0	2
-2	0

· x₁ +

-∞

[d(x₁)]

0	2
0	2

· x₁ +

-∞

[c(x₁)]

0	0
-2	-2

· x₁ +

-∞

[b(x₁)]

0	0
-2	0

· x₁ +

-∞

[a(x₁)]

0	2
-2	0

· x₁ +

-∞

[f^#(x₁)]

9	10
9	10

· x₁ +

-∞

[d^#(x₁)]

10	12
10	12

· x₁ +

-∞

[b^#(x₁)]

10	10
10	10

· x₁ +

-∞

[a^#(x₁)]

10	12
10	12

· x₁ +

-∞

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

a^#(x1)

→

f^#(b(b(x1)))

(16)

could be deleted.

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 +

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[f^#(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

f^#(d(x1))	→	d^#(b(x1))	(8)
f^#(d(x1))	→	b^#(x1)	(9)

and no rules 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

d^#(a(x1))	→	d^#(x1)	(10)
d^#(a(x1))	→	b^#(d(x1))	(12)
b^#(d(b(x1)))	→	d^#(a(x1))	(14)
b^#(d(b(x1)))	→	a^#(x1)	(15)
a^#(x1)	→	b^#(x1)	(17)
a^#(x1)	→	b^#(f(b(b(x1))))	(18)
a^#(x1)	→	b^#(b(x1))	(19)
a^#(x1)	→	b^#(b(f(b(b(x1)))))	(20)

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

[f(x₁)]

0	0
-2	0

· x₁ +

-∞

[d(x₁)]

0	2
0	2

· x₁ +

-∞

[c(x₁)]

0	0
-2	-2

· x₁ +

-∞

[b(x₁)]

0	0
-2	0

· x₁ +

-∞

[a(x₁)]

0	0
-2	0

· x₁ +

-∞

[d^#(x₁)]

6	8
6	8

· x₁ +

-∞

[b^#(x₁)]

4	6
4	6

· x₁ +

-∞

[a^#(x₁)]

5	6
5	6

· x₁ +

-∞

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

b^#(d(b(x1)))

→

a^#(x1)

(15)

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

[f(x₁)]

x₁ +

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

a^#(x1)	→	b^#(x1)	(17)
a^#(x1)	→	b^#(f(b(b(x1))))	(18)
a^#(x1)	→	b^#(b(x1))	(19)
a^#(x1)	→	b^#(b(f(b(b(x1)))))	(20)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(a(x1))	→	d^#(x1)	(10)
d^#(a(x1))	→	b^#(d(x1))	(12)
b^#(d(b(x1)))	→	d^#(a(x1))	(14)

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[f(x₁)]

1	0
0	0

· x₁ +

[d(x₁)]

0	1
0	0

· x₁ +

[c(x₁)]

0	0
0	0

· x₁ +

[b(x₁)]

1	1
0	1

· x₁ +

[a(x₁)]

1	2
0	1

· x₁ +

[d^#(x₁)]

0	1
0	0

· x₁ +

[b^#(x₁)]

1	0
0	0

· x₁ +

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

d^#(a(x1))

→

d^#(x1)

(10)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

d^#(a(x1))	→	b^#(d(x1))	(12)
b^#(d(b(x1)))	→	d^#(a(x1))	(14)

1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[f(x₁)]

1	0
0	0

· x₁ +

[d(x₁)]

0	1
0	0

· x₁ +

[c(x₁)]

0	0
0	0

· x₁ +

[b(x₁)]

0	0
0	1

· x₁ +

[a(x₁)]

0	1
0	0

· x₁ +

[d^#(x₁)]

1	0
0	0

· x₁ +

[b^#(x₁)]

1	1
0	0

· x₁ +

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

d^#(a(x1))

→

b^#(d(x1))

(12)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

[f(x₁)]

x₁ +

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

together with the usable rules

d(a(x1))	→	d(b(c(b(d(x1)))))	(5)
a(x1)	→	b(b(f(b(b(x1)))))	(2)
b(d(b(x1)))	→	d(a(x1))	(6)
f(d(x1))	→	d(b(x1))	(7)

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

b^#(d(b(x1)))

→

d^#(a(x1))

(14)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.