Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x04)

The rewrite relation of the following TRS is considered.

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
c(a(x1))	→	a(b(a(a(c(x1)))))	(2)
b(b(b(x1)))	→	a(b(x1))	(3)
c(b(x1))	→	a(a(c(x1)))	(4)
c(b(x1))	→	b(a(d(x1)))	(5)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	a(b(b(x1)))	(8)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

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

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

all of the following rules can be deleted.

c(b(x1))

→

b(a(d(x1)))

(5)

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

d^#(d(x1))	→	d^#(b(d(x1)))	(13)
d^#(d(x1))	→	d^#(b(d(b(d(x1)))))	(14)
d^#(d(x1))	→	b^#(d(x1))	(15)
d^#(d(x1))	→	b^#(d(b(d(x1))))	(16)
c^#(c(x1))	→	d^#(c(x1))	(17)
c^#(c(x1))	→	c^#(d(c(x1)))	(18)
b^#(c(x1))	→	c^#(a(a(x1)))	(19)
b^#(c(x1))	→	a^#(x1)	(20)
b^#(c(x1))	→	a^#(a(x1))	(21)
b^#(b(b(x1)))	→	b^#(a(x1))	(22)
b^#(b(b(x1)))	→	a^#(x1)	(23)
a^#(c(x1))	→	c^#(a(a(b(a(x1)))))	(24)
a^#(c(x1))	→	b^#(a(x1))	(25)
a^#(c(x1))	→	a^#(x1)	(26)
a^#(c(x1))	→	a^#(b(a(x1)))	(27)
a^#(c(x1))	→	a^#(a(b(a(x1))))	(28)
a^#(a(x1))	→	b^#(a(x1))	(29)
a^#(a(x1))	→	b^#(a(b(a(x1))))	(30)
a^#(a(x1))	→	a^#(b(a(x1)))	(31)
a^#(a(x1))	→	a^#(b(a(b(a(x1)))))	(32)
a^#(a(a(x1)))	→	b^#(b(a(x1)))	(33)
a^#(a(a(x1)))	→	b^#(a(x1))	(34)

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

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[d^#(x₁)]

x₁ +

[c^#(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

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

d^#(d(x1))	→	b^#(d(x1))	(15)
d^#(d(x1))	→	b^#(d(b(d(x1))))	(16)
c^#(c(x1))	→	d^#(c(x1))	(17)
b^#(c(x1))	→	c^#(a(a(x1)))	(19)
b^#(c(x1))	→	a^#(x1)	(20)
b^#(c(x1))	→	a^#(a(x1))	(21)
a^#(c(x1))	→	c^#(a(a(b(a(x1)))))	(24)
a^#(c(x1))	→	b^#(a(x1))	(25)
a^#(c(x1))	→	a^#(x1)	(26)
a^#(c(x1))	→	a^#(b(a(x1)))	(27)
a^#(c(x1))	→	a^#(a(b(a(x1))))	(28)

and no rules could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

d^#(d(x1))	→	d^#(b(d(x1)))	(13)
d^#(d(x1))	→	d^#(b(d(b(d(x1)))))	(14)

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

[d(x₁)]

0	0
0	0

· x₁ +

-∞

[c(x₁)]

6	6
4	4

· x₁ +

-∞

[b(x₁)]

0	0
-2	-2

· x₁ +

-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞

[d^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

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

d^#(d(x1))	→	d^#(b(d(x1)))	(13)
d^#(d(x1))	→	d^#(b(d(b(d(x1)))))	(14)

could be deleted.

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

c^#(c(x1))

→

c^#(d(c(x1)))

(18)

1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

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

[d(x₁)]

0	0
-2	-2

· x₁ +

-∞

[c(x₁)]

0	0
0	0

· x₁ +

-∞

[b(x₁)]

0	0
0	0

· x₁ +

-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞

[c^#(x₁)]

1	2
1	2

· x₁ +

-∞

together with the usable rules

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

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

c^#(c(x1))

→

c^#(d(c(x1)))

(18)

could be deleted.

1.1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

b^#(b(b(x1)))	→	b^#(a(x1))	(22)
b^#(b(b(x1)))	→	a^#(x1)	(23)
a^#(a(x1))	→	b^#(a(x1))	(29)
a^#(a(x1))	→	b^#(a(b(a(x1))))	(30)
a^#(a(x1))	→	a^#(b(a(x1)))	(31)
a^#(a(x1))	→	a^#(b(a(b(a(x1)))))	(32)
a^#(a(a(x1)))	→	b^#(b(a(x1)))	(33)
a^#(a(a(x1)))	→	b^#(a(x1))	(34)

1.1.1.1.1.3 Reduction Pair Processor with Usable Rules

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

[d(x₁)]

0	0
-2	0

· x₁ +

-∞

[c(x₁)]

0	0
-2	-2

· x₁ +

-∞

[b(x₁)]

0	0
-2	-2

· x₁ +

-∞

[a(x₁)]

0	0
0	0

· x₁ +

-∞

[b^#(x₁)]

11	11
11	11

· x₁ +

-∞

[a^#(x₁)]

9	11
9	11

· x₁ +

-∞

together with the usable rules

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

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

a^#(a(x1))	→	a^#(b(a(x1)))	(31)
a^#(a(x1))	→	a^#(b(a(b(a(x1)))))	(32)

could be deleted.

1.1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b^#(b(b(x1)))	→	b^#(a(x1))	(22)
b^#(b(b(x1)))	→	a^#(x1)	(23)
a^#(a(x1))	→	b^#(a(x1))	(29)
a^#(a(x1))	→	b^#(a(b(a(x1))))	(30)
a^#(a(a(x1)))	→	b^#(b(a(x1)))	(33)
a^#(a(a(x1)))	→	b^#(a(x1))	(34)

1.1.1.1.1.3.1.1 Reduction Pair Processor with Usable Rules

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

[d(x₁)]

-∞	-∞
-∞	-∞

· x₁ +

-∞

[c(x₁)]

-∞	-∞
-∞	-∞

· x₁ +

[b(x₁)]

8	0
0	-∞

· x₁ +

-∞

[a(x₁)]

-∞	0
3	8

· x₁ +

[b^#(x₁)]

0	-∞
-∞	-∞

· x₁ +

-∞

[a^#(x₁)]

8	-∞
-∞	-∞

· x₁ +

-∞

together with the usable rules

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

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

b^#(b(b(x1)))	→	b^#(a(x1))	(22)
a^#(a(x1))	→	b^#(a(x1))	(29)
a^#(a(x1))	→	b^#(a(b(a(x1))))	(30)
a^#(a(a(x1)))	→	b^#(b(a(x1)))	(33)
a^#(a(a(x1)))	→	b^#(a(x1))	(34)

could be deleted.

1.1.1.1.1.3.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

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

[d(x₁)]

x₁ +

[c(x₁)]

x₁ +

[b(x₁)]

x₁ +

[a(x₁)]

x₁ +

[b^#(x₁)]

x₁ +

[a^#(x₁)]

x₁ +

together with the usable rules

a(a(x1))	→	a(b(a(b(a(x1)))))	(1)
a(c(x1))	→	c(a(a(b(a(x1)))))	(9)
b(b(b(x1)))	→	b(a(x1))	(10)
b(c(x1))	→	c(a(a(x1)))	(11)
d(d(x1))	→	d(b(d(b(d(x1)))))	(6)
c(c(x1))	→	c(d(c(x1)))	(7)
a(a(a(x1)))	→	b(b(a(x1)))	(12)

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

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

→

a^#(x1)

(23)

and no rules could be deleted.

1.1.1.1.1.3.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.