Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z119)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a^#(b(x1))	→	b^#(d(x1))	(7)
a^#(b(x1))	→	d^#(x1)	(8)
a^#(c(x1))	→	d^#(d(d(x1)))	(9)
a^#(c(x1))	→	d^#(d(x1))	(10)
a^#(c(x1))	→	d^#(x1)	(11)
b^#(d(x1))	→	a^#(c(b(x1)))	(12)
b^#(d(x1))	→	c^#(b(x1))	(13)
b^#(d(x1))	→	b^#(x1)	(14)
c^#(f(x1))	→	d^#(d(c(x1)))	(15)
c^#(f(x1))	→	d^#(c(x1))	(16)
c^#(f(x1))	→	c^#(x1)	(17)
d^#(d(x1))	→	f^#(x1)	(18)
f^#(f(x1))	→	a^#(x1)	(19)

1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[a^#(x₁)]	=	1 · x₁
[b(x₁)]	=	1 + 1 · x₁
[b^#(x₁)]	=	1 + 1 · x₁
[d(x₁)]	=	1 · x₁
[d^#(x₁)]	=	1 · x₁
[c(x₁)]	=	1 · x₁
[c^#(x₁)]	=	1 · x₁
[f(x₁)]	=	1 · x₁
[f^#(x₁)]	=	1 · x₁
[a(x₁)]	=	1 · x₁

the pair

a^#(b(x1))

→

d^#(x1)

(8)

could be deleted.

1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[a(x₁)]	=	3 + 1 · x₁
[b(x₁)]	=	3 · x₁
[d(x₁)]	=	1 + 1 · x₁
[c(x₁)]	=	1 · x₁
[f(x₁)]	=	2 + 1 · x₁
[a^#(x₁)]	=	3 + 1 · x₁
[b^#(x₁)]	=	3 · x₁
[d^#(x₁)]	=	1 · x₁
[c^#(x₁)]	=	2 + 1 · x₁
[f^#(x₁)]	=	1 + 1 · x₁

the pairs

a^#(c(x1))	→	d^#(d(d(x1)))	(9)
a^#(c(x1))	→	d^#(d(x1))	(10)
a^#(c(x1))	→	d^#(x1)	(11)
b^#(d(x1))	→	c^#(b(x1))	(13)
b^#(d(x1))	→	b^#(x1)	(14)
c^#(f(x1))	→	d^#(d(c(x1)))	(15)
c^#(f(x1))	→	d^#(c(x1))	(16)
c^#(f(x1))	→	c^#(x1)	(17)

and the rule

f(f(x1))

→

a(x1)

(6)

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

b^#(d(x1)) → a^#(c(b(x1))) (12)

a^#(b(x1)) → b^#(d(x1)) (7)

1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[b^#(x₁)] = 1

[d(x₁)] = 0

[a^#(x₁)] = 1 + 1 · x₁

[c(x₁)] = 0

[b(x₁)] = 1

[a(x₁)] = 0

[f(x₁)] = 0

together with the usable rules

c(f(x1)) → d(d(c(x1))) (4)

d(d(x1)) → f(x1) (5)

(w.r.t. the implicit argument filter of the reduction pair), the pair
a^#(b(x1)) → b^#(d(x1)) (7)
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[a(x₁)] = 1 · x₁

[b(x₁)] = 2 · x₁

[d(x₁)] = 1 · x₁

[c(x₁)] = 1 · x₁

[f(x₁)] = 1 · x₁

[b^#(x₁)] = 3 + 3 · x₁

[a^#(x₁)] = 1 · x₁

the pair
b^#(d(x1)) → a^#(c(b(x1))) (12)
and no rules could be deleted.
1.1.1.1.1.1.1 P is empty

There are no pairs anymore.