Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup15)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(a(x1))	→	b^#(b(b(b(b(x1)))))	(5)
a^#(a(x1))	→	b^#(x1)	(6)
b^#(b(d(d(b(b(x1))))))	→	c^#(c(x1))	(7)
a^#(a(d(d(x1))))	→	b^#(b(x1))	(8)
a^#(a(d(d(x1))))	→	b^#(x1)	(9)
b^#(b(d(d(b(b(x1))))))	→	a^#(a(c(c(x1))))	(10)
b^#(b(d(d(b(b(x1))))))	→	c^#(x1)	(11)
a^#(a(x1))	→	b^#(b(x1))	(12)
a^#(a(x1))	→	b^#(b(b(b(x1))))	(13)
a^#(a(x1))	→	b^#(b(b(b(b(b(x1))))))	(14)
b^#(b(d(d(b(b(x1))))))	→	a^#(c(c(x1)))	(15)
a^#(a(x1))	→	b^#(b(b(x1)))	(16)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(a(x1))	→	b^#(b(b(x1)))	(16)
a^#(a(d(d(x1))))	→	b^#(x1)	(9)
b^#(b(d(d(b(b(x1))))))	→	a^#(c(c(x1)))	(15)
a^#(a(x1))	→	b^#(b(b(b(b(b(x1))))))	(14)
a^#(a(x1))	→	b^#(b(b(b(x1))))	(13)
a^#(a(d(d(x1))))	→	b^#(b(x1))	(8)
a^#(a(x1))	→	b^#(x1)	(6)
a^#(a(x1))	→	b^#(b(x1))	(12)
b^#(b(d(d(b(b(x1))))))	→	a^#(a(c(c(x1))))	(10)
a^#(a(x1))	→	b^#(b(b(b(b(x1)))))	(5)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	15923
[d(x₁)]	=	1
[b(x₁)]	=	15923
[c(x₁)]	=	1
[c^#(x₁)]	=	0
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	15923

together with the usable rules

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

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

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

→

a^#(c(c(x1)))

(15)

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

a^#(a(d(d(x1))))	→	b^#(x1)	(9)
a^#(a(d(d(x1))))	→	b^#(b(x1))	(8)
b^#(b(d(d(b(b(x1))))))	→	a^#(a(c(c(x1))))	(10)
a^#(a(x1))	→	b^#(x1)	(6)
a^#(a(x1))	→	b^#(b(x1))	(12)
a^#(a(x1))	→	b^#(b(b(x1)))	(16)
a^#(a(x1))	→	b^#(b(b(b(x1))))	(13)
a^#(a(x1))	→	b^#(b(b(b(b(x1)))))	(5)
a^#(a(x1))	→	b^#(b(b(b(b(b(x1))))))	(14)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[d(x₁)]	=	x₁ + 52868
[b(x₁)]	=	x₁ + 0
[c(x₁)]	=	x₁ + 52868
[c^#(x₁)]	=	0
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

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

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

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

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

b^#(b(d(d(b(b(x1))))))	→	a^#(a(c(c(x1))))	(10)
a^#(a(x1))	→	b^#(x1)	(6)
a^#(a(x1))	→	b^#(b(x1))	(12)
a^#(a(x1))	→	b^#(b(b(x1)))	(16)
a^#(a(x1))	→	b^#(b(b(b(x1))))	(13)
a^#(a(x1))	→	b^#(b(b(b(b(x1)))))	(5)
a^#(a(x1))	→	b^#(b(b(b(b(b(x1))))))	(14)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 2241
[d(x₁)]	=	0
[b(x₁)]	=	4482
[c(x₁)]	=	0
[c^#(x₁)]	=	0
[a^#(x₁)]	=	x₁ + 2241
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

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

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

a^#(a(x1))

→

b^#(x1)

(6)

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

b^#(b(d(d(b(b(x1))))))	→	a^#(a(c(c(x1))))	(10)
a^#(a(x1))	→	b^#(b(x1))	(12)
a^#(a(x1))	→	b^#(b(b(x1)))	(16)
a^#(a(x1))	→	b^#(b(b(b(x1))))	(13)
a^#(a(x1))	→	b^#(b(b(b(b(x1)))))	(5)
a^#(a(x1))	→	b^#(b(b(b(b(b(x1))))))	(14)

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

[a(x₁)]

0	1
1	0

· x₁ +

[d(x₁)]

1	1
1	1

· x₁ +

[b(x₁)]

x₁ +

[c(x₁)]

1	1
1	1

· x₁ +

[c^#(x₁)]

[a^#(x₁)]

1	1
1	1

· x₁ +

[b^#(x₁)]

1	1
1	1

· x₁ +

together with the usable rules

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

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

b^#(b(d(d(b(b(x1))))))	→	a^#(a(c(c(x1))))	(10)
a^#(a(x1))	→	b^#(b(x1))	(12)
a^#(a(x1))	→	b^#(b(b(x1)))	(16)
a^#(a(x1))	→	b^#(b(b(b(x1))))	(13)
a^#(a(x1))	→	b^#(b(b(b(b(x1)))))	(5)
a^#(a(x1))	→	b^#(b(b(b(b(b(x1))))))	(14)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.