Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-135)
The rewrite relation of the following TRS is considered.
a(x1) |
→ |
x1 |
(1) |
a(b(x1)) |
→ |
x1 |
(2) |
b(b(x1)) |
→ |
c(x1) |
(3) |
c(a(x1)) |
→ |
a(a(b(c(x1)))) |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
No.Proof (by AProVE @ termCOMP 2023)
1 Looping derivation
There is a looping derivation.
c a a a a a →+ a c a a a a a b c b c
The derivation can be derived as follows.
-
c a →+ a a b c:
This is an original rule (OC1).
-
a →+ ε:
This is an original rule (OC1).
-
c a →+ a b c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a →+ a b a b c:
The overlap closure is obtained from the following two overlap closures (OC2).
-
c a →+ a b c
-
c a →+ a b c
-
c a a →+ a b b c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a →+ a b a b c
-
a →+ ε
-
b b →+ c:
This is an original rule (OC1).
-
c a a →+ a c c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a →+ a b b c
-
b b →+ c
-
c a a →+ a a b a b c:
The overlap closure is obtained from the following two overlap closures (OC2).
-
c a →+ a a b c
-
c a →+ a b c
-
c a a →+ a a b b c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a →+ a a b a b c
-
a →+ ε
-
c a a →+ a a c c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a →+ a a b b c
-
b b →+ c
-
c a a a →+ a a c a a b c:
The overlap closure is obtained from the following two overlap closures (OC2).
-
c a a →+ a a c c
-
c a →+ a a b c
-
a b →+ ε:
This is an original rule (OC1).
-
c a →+ a c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a a →+ a a a c a b c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a a →+ a a c a a b c
-
c a →+ a c
-
c a a a →+ a a a a a b c b c:
The overlap closure is obtained from the following two overlap closures (OC3).
-
c a a a →+ a a a c a b c
-
c a →+ a a b c
-
c a a a a a →+ a c a a a a a b c b c:
The overlap closure is obtained from the following two overlap closures (OC2).
-
c a a →+ a c c
-
c a a a →+ a a a a a b c b c