Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/05-oppelt08)
The rewrite relation of the following TRS is considered.
|
b(a(x1)) |
→ |
a(b(x1)) |
(1) |
|
c(c(x1)) |
→ |
a(b(c(x1))) |
(2) |
|
c(b(x1)) |
→ |
b(c(x1)) |
(3) |
|
a(a(x1)) |
→ |
b(a(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 c
(b)n c →+ b a c
(b)n + 1 c
The derivation can be derived as follows.
-
c b →+ b c:
This is an original rule (OC1).
-
c
(b)n →+
(b)n c:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
c c →+ a b c:
This is an original rule (OC1).
-
c
(b)n c →+
(b)n a b c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
c
(b)n →+
(b)n c
-
c c →+ a b c
-
b a →+ a b:
This is an original rule (OC1).
-
(b)n a →+ a
(b)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
c
(b)n c →+ a
(b)n b c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 2.1)
-
c
(b)n c →+
(b)n a b c
-
(b)n a →+ a
(b)n
-
c
(b)n c →+ a
(b)n + 1 c:
The derivation pattern is equivalent to the following derivation pattern.
-
a a →+ b a c:
This is an original rule (OC1).
-
a c
(b)n c →+ b a c
(b)n + 1 c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
c
(b)n c →+ a
(b)n + 1 c
-
a a →+ b a c