Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size11/size-11-alpha-3-num-18)
The rewrite relation of the following TRS is considered.
a(x1) |
→ |
x1 |
(1) |
a(b(x1)) |
→ |
b(a(c(b(a(x1))))) |
(2) |
b(x1) |
→ |
x1 |
(3) |
c(c(x1)) |
→ |
x1 |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
No.Proof (by AProVE @ termCOMP 2023)
1 Looping derivation
There is a looping derivation.
a b b b c b →+ ε a b b b c b ε
The derivation can be derived as follows.
-
a b →+ b a c b a:
This is an original rule (OC1).
-
b →+ ε:
This is an original rule (OC1).
-
a b →+ b a c a:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a →+ ε:
This is an original rule (OC1).
-
a b →+ b c b a:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a b b →+ b a c b c b a:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b →+ b a c a
-
a b →+ b c b a
-
a b b →+ b a c c b a:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a b b →+ b a c b c b a
-
b →+ ε
-
c c →+ ε:
This is an original rule (OC1).
-
a b b →+ b a b a:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a b b →+ b a c c b a
-
c c →+ ε
-
a b →+ b a c b:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b b b →+ b a b b a c b:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b b →+ b a b a
-
a b →+ b a c b
-
a b b b →+ b a b b a c:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b b b →+ b a b b a c b
-
b →+ ε
-
a b b b c →+ b a b b a:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b b b →+ b a b b a c
-
c c →+ ε
-
a b b b c →+ a b b a:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a b b b c →+ b a b b a
-
b →+ ε
-
a b b b c b →+ a b b b a c b:
The overlap closure is obtained from the following two overlap closures (OC2).
-
a b b b c →+ a b b a
-
a b →+ b a c b
-
a b b b c b →+ a b b b c b:
The overlap closure is obtained from the following two overlap closures (OC3).
-
a b b b c b →+ a b b b a c b
-
a →+ ε