Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/02-oppelt08)
The rewrite relation of the following TRS is considered.
|
c(b(x1)) |
→ |
b(a(x1)) |
(1) |
|
a(c(x1)) |
→ |
a(b(c(x1))) |
(2) |
|
b(b(x1)) |
→ |
b(c(x1)) |
(3) |
|
c(a(x1)) |
→ |
c(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.
b
(c)n + 1 b c →+ b
(c)2n + 3 b c
The derivation can be derived as follows.
-
c b →+ b a:
This is an original rule (OC1).
-
(c)n b →+ b
(a)n:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 1)
-
(c)n + 1 b →+ b
(a)n + 1:
The derivation pattern is obtained from lifting the following derivation pattern.
-
b b →+ b c:
This is an original rule (OC1).
-
b
(c)n + 1 b →+ b c
(a)n + 1:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 1.2)
-
(c)n + 1 b →+ b
(a)n + 1
-
b b →+ b c
-
b
(c)n + 1 b →+ b c
(a)n a:
The derivation pattern is equivalent to the following derivation pattern.
-
b
(c)n + 1 b →+ b c
(a)n + 1
-
a c →+ a b c:
This is an original rule (OC1).
-
b
(c)n + 1 b c →+ b c
(a)n a b c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP OC 3.2)
-
b
(c)n + 1 b →+ b c
(a)n a
-
a c →+ a b c
-
b
(c)n + 1 b c →+ b c
(a)n + 1 b c:
The derivation pattern is equivalent to the following derivation pattern.
-
b
(c)n + 1 b c →+ b c
(a)n a b c
-
c a →+ c c c:
This is an original rule (OC1).
-
c
(a)n →+
(c c)n c:
The derivation pattern is obtained from the following self-overlapping overlap closure (type 2)
-
c
(a)n + 1 →+
(c c)n + 1 c:
The derivation pattern is obtained from lifting the following derivation pattern.
-
b
(c)n + 1 b c →+ b
(c c)n + 1 c b c:
The derivation pattern is obtained from overlapping the following two derivation patterns (DP DP 1.1)
-
b
(c)n + 1 b c →+ b c
(a)n + 1 b c
-
c
(a)n + 1 →+
(c c)n + 1 c
-
b
(c)n + 1 b c →+ b
(c)2n + 3 b c:
The derivation pattern is equivalent to the following derivation pattern.
-
b
(c)n + 1 b c →+ b
(c c)n + 1 c b c