Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/03-oppelt08)
The rewrite relation of the following TRS is considered.
|
c(c(x1)) |
→ |
b(b(x1)) |
(1) |
|
a(b(x1)) |
→ |
a(c(b(x1))) |
(2) |
|
b(a(x1)) |
→ |
b(b(x1)) |
(3) |
|
b(c(x1)) |
→ |
c(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.
a b b
(a)n c b →+ a b b
(a)2n + 4 c b
The derivation can be derived as follows.
-
b a →+ b b:
This is an original rule (OC1).
-
b
(a)n →+
(b)n b:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
b c →+ c a a:
This is an original rule (OC1).
-
a b →+ a c b:
This is an original rule (OC1).
-
b c b →+ c a a c b:
The overlap closure is obtained from the following two overlap closures (OC2).
-
b c →+ c a a
-
a b →+ a c b
-
b
(a)n c b →+
(b)n c a a c b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
b
(a)n →+
(b)n b
-
b c b →+ c a a c b
-
(b)n c →+ c
(a a)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
b
(a)n c b →+ c
(a a)n a a c b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 2.1)
-
b
(a)n c b →+
(b)n c a a c b
-
(b)n c →+ c
(a a)n
-
b
(a)n c b →+ c
(a)2n + 2 c b:
The derivation pattern is equivalent to the following derivation pattern.
-
b
(a)n c b →+ c
(a a)n a a c b
-
a b c →+ a c c a a:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b →+ a c b
-
b c →+ c a a
-
c c →+ b b:
This is an original rule (OC1).
-
a b c →+ a b b a a:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a b c →+ a c c a a
-
c c →+ b b
-
a b b
(a)n c b →+ a b b a a
(a)2n + 2 c b:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
b
(a)n c b →+ c
(a)2n + 2 c b
-
a b c →+ a b b a a
-
a b b
(a)n c b →+ a b b
(a)2n + 4 c b:
The derivation pattern is equivalent to the following derivation pattern.
-
a b b
(a)n c b →+ a b b a a
(a)2n + 2 c b