Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/04-oppelt08)
The rewrite relation of the following TRS is considered.
|
b(b(x1)) |
→ |
b(a(x1)) |
(1) |
|
a(b(x1)) |
→ |
b(c(c(x1))) |
(2) |
|
a(c(x1)) |
→ |
a(a(x1)) |
(3) |
|
c(a(x1)) |
→ |
b(a(a(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.
b a
(c)n b a →+ b a
(c)2n + 1 b a a
The derivation can be derived as follows.
-
a c →+ a a:
This is an original rule (OC1).
-
a
(c)n →+
(a)n a:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
a b →+ b c c:
This is an original rule (OC1).
-
c a →+ b a a:
This is an original rule (OC1).
-
a b a →+ b c b a a:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b →+ b c c
-
c a →+ b a a
-
a
(c)n b a →+
(a)n b c b a a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
a
(c)n →+
(a)n a
-
a b a →+ b c b a a
-
(a)n b →+ b
(c c)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
a
(c)n b a →+ b
(c c)n c b a a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 2.1)
-
a
(c)n b a →+
(a)n b c b a a
-
(a)n b →+ b
(c c)n
-
a
(c)n b a →+ b
(c)2n + 1 b a a:
The derivation pattern is equivalent to the following derivation pattern.
-
a
(c)n b a →+ b
(c c)n c b a a
-
b b →+ b a:
This is an original rule (OC1).
-
b a
(c)n b a →+ b a
(c)2n + 1 b a a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
a
(c)n b a →+ b
(c)2n + 1 b a a
-
b b →+ b a