Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/06-oppelt08)
The rewrite relation of the following TRS is considered.
|
b(c(x1)) |
→ |
d(c(x1)) |
(1) |
|
b(d(x1)) |
→ |
d(b(x1)) |
(2) |
|
a(d(x1)) |
→ |
a(b(b(x1))) |
(3) |
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
(b)n d c →+ a
(b)n + 1 d c
The derivation can be derived as follows.
-
b d →+ d b:
This is an original rule (OC1).
-
(b)n d →+ d
(b)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
a d →+ a b b:
This is an original rule (OC1).
-
a
(b)n d →+ a b b
(b)n:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
(b)n d →+ d
(b)n
-
a d →+ a b b
-
a
(b)n d →+ a
(b)n + 2:
The derivation pattern is equivalent to the following derivation pattern.
-
a
(b)n d →+ a
(b)n + 1 b:
The derivation pattern is equivalent to the following derivation pattern.
-
b c →+ d c:
This is an original rule (OC1).
-
a
(b)n d c →+ a
(b)n + 1 d c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
a
(b)n d →+ a
(b)n + 1 b
-
b c →+ d c