Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z073)
The rewrite relation of the following TRS is considered.
|
a(l(x1)) |
→ |
l(a(x1)) |
(1) |
|
r(a(x1)) |
→ |
a(r(x1)) |
(2) |
|
b(l(x1)) |
→ |
b(a(r(x1))) |
(3) |
|
r(b(x1)) |
→ |
l(b(x1)) |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
No.Proof (by AProVE @ termCOMP 2023)
1 Infinite derivation
There is a self-embedding derivation structure which implies nontermination.
b
(a)n l b →+ b
(a)n + 1 l b
The derivation can be derived as follows.
-
a l →+ l a:
This is an original rule (OC1).
-
(a)n l →+ l
(a)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
b l →+ b a r:
This is an original rule (OC1).
-
b
(a)n l →+ b a r
(a)n:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
(a)n l →+ l
(a)n
-
b l →+ b a r
-
r a →+ a r:
This is an original rule (OC1).
-
r
(a)n →+
(a)n r:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
b
(a)n l →+ b a
(a)n r:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 1.2)
-
b
(a)n l →+ b a r
(a)n
-
r
(a)n →+
(a)n r
-
b
(a)n l →+ b
(a)n + 1 r:
The derivation pattern is equivalent to the following derivation pattern.
-
r b →+ l b:
This is an original rule (OC1).
-
b
(a)n l b →+ b
(a)n + 1 l b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
b
(a)n l →+ b
(a)n + 1 r
-
r b →+ l b