Certification Problem
Input (TPDB TRS_Standard/Zantema_15/ex01)
The rewrite relation of the following TRS is considered.
|
a(L(x)) |
→ |
L(a(x)) |
(1) |
|
R(a(x)) |
→ |
a(R(x)) |
(2) |
|
b(L(x)) |
→ |
b(R(x)) |
(3) |
|
R(b(x)) |
→ |
L(a(b(x))) |
(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 R
(a)n b →+ b R
(a)n + 1 b
The derivation can be derived as follows.
-
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)
-
R b →+ L a b:
This is an original rule (OC1).
-
R
(a)n b →+
(a)n L a b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
R
(a)n →+
(a)n R
-
R b →+ L a b
-
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)
-
R
(a)n b →+ L
(a)n a b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 2.1)
-
R
(a)n b →+
(a)n L a b
-
(a)n L →+ L
(a)n
-
R
(a)n b →+ L
(a)n + 1 b:
The derivation pattern is equivalent to the following derivation pattern.
-
b L →+ b R:
This is an original rule (OC1).
-
b R
(a)n b →+ b R
(a)n + 1 b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
R
(a)n b →+ L
(a)n + 1 b
-
b L →+ b R