Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/01-oppelt08)
The rewrite relation of the following TRS is considered.
|
a(b(c(x1))) |
→ |
c(b(a(x1))) |
(1) |
|
a(c(x1)) |
→ |
a(a(d(a(x1)))) |
(2) |
|
d(a(b(x1))) |
→ |
b(a(d(x1))) |
(3) |
|
d(a(c(x1))) |
→ |
b(c(c(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.
a
(a b)n + 1 c c →+ a
(a b)n + 2 c c
The derivation can be derived as follows.
-
a b c →+ c b a:
This is an original rule (OC1).
-
(a b)n c →+ c
(b a)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
(a b)n + 1 c →+ c
(b a)n + 1:
The derivation pattern is obtained from lifting the following derivation pattern.
-
(a b)n + 1 c →+ c b
(a b)n a:
The derivation pattern is equivalent to the following derivation pattern.
-
(a b)n + 1 c →+ c
(b a)n + 1
-
a c →+ a a d a:
This is an original rule (OC1).
-
a
(a b)n + 1 c →+ a a d a b
(a b)n a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
(a b)n + 1 c →+ c b
(a b)n a
-
a c →+ a a d a
-
a
(a b)n + 1 c →+ a a d
(a b)n + 1 a:
The derivation pattern is equivalent to the following derivation pattern.
-
a
(a b)n + 1 c →+ a a d a b
(a b)n a
-
d a b →+ b a d:
This is an original rule (OC1).
-
d
(a b)n →+
(b a)n d:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
d
(a b)n + 1 →+
(b a)n + 1 d:
The derivation pattern is obtained from lifting the following derivation pattern.
-
a
(a b)n + 1 c →+ a a
(b a)n + 1 d a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 1.1)
-
a
(a b)n + 1 c →+ a a d
(a b)n + 1 a
-
d
(a b)n + 1 →+
(b a)n + 1 d
-
d a c →+ b c c:
This is an original rule (OC1).
-
a
(a b)n + 1 c c →+ a a
(b a)n + 1 b c c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
a
(a b)n + 1 c →+ a a
(b a)n + 1 d a
-
d a c →+ b c c
-
a
(a b)n + 1 c c →+ a
(a b)n + 2 c c:
The derivation pattern is equivalent to the following derivation pattern.
-
a
(a b)n + 1 c c →+ a a
(b a)n + 1 b c c