Certification Problem
Input (TPDB SRS_Standard/Waldmann_06_SRS/uni-3)
The rewrite relation of the following TRS is considered.
|
a(a(a(x1))) |
→ |
b(b(c(a(a(x1))))) |
(1) |
|
b(b(x1)) |
→ |
c(a(b(x1))) |
(2) |
|
c(x1) |
→ |
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.
(b)n + 3 →+
(b)2n a a b
The derivation can be derived as follows.
-
b b →+ c a b:
This is an original rule (OC1).
-
b
(b)n →+
(c a)n b:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
b
(b)n + 1 →+
(c a)n + 1 b:
The derivation pattern is obtained from lifting the following derivation pattern.
-
b
(b)n + 2 →+
(c a)n + 2 b:
The derivation pattern is obtained from lifting the following derivation pattern.
-
b
(b)n + 1 →+
(c a)n + 1 b
-
c →+ ε:
This is an original rule (OC1).
-
b
(b)n + 2 →+
(a)n + 2 b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 2)
-
b
(b)n + 2 →+
(c a)n + 2 b
-
c →+ ε
-
(b)n + 3 →+
(a)n + 2 b:
The derivation pattern is equivalent to the following derivation pattern.
-
a a a →+ b b c a a:
This is an original rule (OC1).
-
a a
(a)n →+
(b b c)n a a:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
a a
(a)n →+
(b b)n a a:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 2)
-
a a
(a)n →+
(b b c)n a a
-
c →+ ε
-
(a)n + 2 →+
(b)2n a a:
The derivation pattern is equivalent to the following derivation pattern.
-
(b)n + 3 →+
(b)2n a a b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 2.1)
-
(b)n + 3 →+
(a)n + 2 b
-
(a)n + 2 →+
(b)2n a a