Certification Problem
Input (TPDB TRS_Standard/Zantema_15/ex04)
The rewrite relation of the following TRS is considered.
|
a(L(x)) |
→ |
L(a(x)) |
(1) |
|
R(a(x)) |
→ |
a(R(x)) |
(2) |
|
a(L(L(x))) |
→ |
a(L(R(x))) |
(3) |
|
R(R(a(x))) |
→ |
L(a(R(a(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.
a L R
(a)n R a →+ a L R
(a)n + 1 R a
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 R a →+ L a R a:
This is an original rule (OC1).
-
R
(a)n R a →+
(a)n L a R a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
R
(a)n →+
(a)n R
-
R R a →+ L a R a
-
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 R a →+ L
(a)n a R a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 2.1)
-
R
(a)n R a →+
(a)n L a R a
-
(a)n L →+ L
(a)n
-
R
(a)n R a →+ L
(a)n + 1 R a:
The derivation pattern is equivalent to the following derivation pattern.
-
R
(a)n R a →+ L
(a)n a R a
-
a L L →+ a L R:
This is an original rule (OC1).
-
a L R
(a)n R a →+ a L R
(a)n + 1 R a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
R
(a)n R a →+ L
(a)n + 1 R a
-
a L L →+ a L R