Certification Problem

Input (TPDB SRS_Standard/Zantema_06/02)

The rewrite relation of the following TRS is considered.

a(p(x1))	→	p(a(A(x1)))	(1)
a(A(x1))	→	A(a(x1))	(2)
p(A(A(x1)))	→	a(p(x1))	(3)

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.

p^#(A(A(x1)))	→	a^#(p(x1))	(4)
a^#(p(x1))	→	a^#(A(x1))	(5)
p^#(A(A(x1)))	→	p^#(x1)	(6)
a^#(A(x1))	→	a^#(x1)	(7)
a^#(p(x1))	→	p^#(a(A(x1)))	(8)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a^#(p(x1))	→	p^#(a(A(x1)))	(8)
a^#(A(x1))	→	a^#(x1)	(7)
p^#(A(A(x1)))	→	p^#(x1)	(6)
a^#(p(x1))	→	a^#(A(x1))	(5)
p^#(A(A(x1)))	→	a^#(p(x1))	(4)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[p^#(x₁)]	=	x₁ + 1
[p(x₁)]	=	x₁ + 1
[A(x₁)]	=	x₁ + 0
[a^#(x₁)]	=	x₁ + 0

together with the usable rules

a(p(x1))	→	p(a(A(x1)))	(1)
p(A(A(x1)))	→	a(p(x1))	(3)
a(A(x1))	→	A(a(x1))	(2)

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

a^#(p(x1))

→

a^#(A(x1))

(5)

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^#(p(x1))	→	p^#(a(A(x1)))	(8)
p^#(A(A(x1)))	→	p^#(x1)	(6)
p^#(A(A(x1)))	→	a^#(p(x1))	(4)
a^#(A(x1))	→	a^#(x1)	(7)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[p^#(x₁)]	=	1
[p(x₁)]	=	1
[A(x₁)]	=	x₁ + 32286
[a^#(x₁)]	=	x₁ + 0

together with the usable rules

a(p(x1))	→	p(a(A(x1)))	(1)
p(A(A(x1)))	→	a(p(x1))	(3)
a(A(x1))	→	A(a(x1))	(2)

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

a^#(A(x1))

→

a^#(x1)

(7)

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

a^#(p(x1))	→	p^#(a(A(x1)))	(8)
p^#(A(A(x1)))	→	p^#(x1)	(6)
p^#(A(A(x1)))	→	a^#(p(x1))	(4)

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)]

x₁ +

1912

[p^#(x₁)]

0	1
0	0

· x₁ +

20274

[p(x₁)]

0	1
0	0

· x₁ +

20274

[A(x₁)]

x₁ +

51071

1912

[a^#(x₁)]

x₁ +

1912

together with the usable rules

a(p(x1))	→	p(a(A(x1)))	(1)
p(A(A(x1)))	→	a(p(x1))	(3)
a(A(x1))	→	A(a(x1))	(2)

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

a^#(p(x1))	→	p^#(a(A(x1)))	(8)
p^#(A(A(x1)))	→	p^#(x1)	(6)
p^#(A(A(x1)))	→	a^#(p(x1))	(4)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.