Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z125)

The rewrite relation of the following TRS is considered.

f(x1)	→	n(c(n(a(x1))))	(1)
c(f(x1))	→	f(n(a(c(x1))))	(2)
n(a(x1))	→	c(x1)	(3)
c(c(x1))	→	c(x1)	(4)
n(s(x1))	→	f(s(s(x1)))	(5)
n(f(x1))	→	f(n(x1))	(6)

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.

c^#(f(x1))	→	n^#(a(c(x1)))	(7)
n^#(s(x1))	→	f^#(s(s(x1)))	(8)
f^#(x1)	→	c^#(n(a(x1)))	(9)
n^#(a(x1))	→	c^#(x1)	(10)
f^#(x1)	→	n^#(a(x1))	(11)
n^#(f(x1))	→	n^#(x1)	(12)
c^#(f(x1))	→	c^#(x1)	(13)
c^#(f(x1))	→	f^#(n(a(c(x1))))	(14)
f^#(x1)	→	n^#(c(n(a(x1))))	(15)
n^#(f(x1))	→	f^#(n(x1))	(16)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

n^#(f(x1))	→	f^#(n(x1))	(16)
f^#(x1)	→	n^#(c(n(a(x1))))	(15)
c^#(f(x1))	→	f^#(n(a(c(x1))))	(14)
c^#(f(x1))	→	c^#(x1)	(13)
f^#(x1)	→	c^#(n(a(x1)))	(9)
n^#(s(x1))	→	f^#(s(s(x1)))	(8)
n^#(f(x1))	→	n^#(x1)	(12)
f^#(x1)	→	n^#(a(x1))	(11)
n^#(a(x1))	→	c^#(x1)	(10)
c^#(f(x1))	→	n^#(a(c(x1)))	(7)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	0
[s(x₁)]	=	1
[n(x₁)]	=	x₁ + 0
[c(x₁)]	=	0
[f(x₁)]	=	x₁ + 0
[f^#(x₁)]	=	0
[c^#(x₁)]	=	0
[n^#(x₁)]	=	x₁ + 0

together with the usable rules

c(c(x1))	→	c(x1)	(4)
f(x1)	→	n(c(n(a(x1))))	(1)
n(a(x1))	→	c(x1)	(3)
n(s(x1))	→	f(s(s(x1)))	(5)
n(f(x1))	→	f(n(x1))	(6)
c(f(x1))	→	f(n(a(c(x1))))	(2)

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

n^#(s(x1))

→

f^#(s(s(x1)))

(8)

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

f^#(x1)	→	n^#(a(x1))	(11)
f^#(x1)	→	c^#(n(a(x1)))	(9)
f^#(x1)	→	n^#(c(n(a(x1))))	(15)
n^#(a(x1))	→	c^#(x1)	(10)
n^#(f(x1))	→	n^#(x1)	(12)
n^#(f(x1))	→	f^#(n(x1))	(16)
c^#(f(x1))	→	c^#(x1)	(13)
c^#(f(x1))	→	n^#(a(c(x1)))	(7)
c^#(f(x1))	→	f^#(n(a(c(x1))))	(14)

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	0
0	1

· x₁ +

[s(x₁)]

0	0
0	1

· x₁ +

5604

[n(x₁)]

1	0
1	1

· x₁ +

[c(x₁)]

0	0
0	1

· x₁ +

[f(x₁)]

x₁ +

[f^#(x₁)]

0	1
0	0

· x₁ +

[c^#(x₁)]

0	1
0	0

· x₁ +

[n^#(x₁)]

1	1
0	0

· x₁ +

together with the usable rules

c(c(x1))	→	c(x1)	(4)
f(x1)	→	n(c(n(a(x1))))	(1)
n(a(x1))	→	c(x1)	(3)
n(s(x1))	→	f(s(s(x1)))	(5)
n(f(x1))	→	f(n(x1))	(6)
c(f(x1))	→	f(n(a(c(x1))))	(2)

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

f^#(x1)	→	n^#(a(x1))	(11)
f^#(x1)	→	c^#(n(a(x1)))	(9)
f^#(x1)	→	n^#(c(n(a(x1))))	(15)
n^#(a(x1))	→	c^#(x1)	(10)
n^#(f(x1))	→	n^#(x1)	(12)
n^#(f(x1))	→	f^#(n(x1))	(16)
c^#(f(x1))	→	c^#(x1)	(13)
c^#(f(x1))	→	n^#(a(c(x1)))	(7)
c^#(f(x1))	→	f^#(n(a(c(x1))))	(14)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.