Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
a(c(d(x))) |
→ c(x) |
| 2: |
|
u(b(d(d(x)))) |
→ b(x) |
| 3: |
|
v(a(a(x))) |
→ u(v(x)) |
| 4: |
|
v(a(c(x))) |
→ u(b(d(x))) |
| 5: |
|
v(c(x)) |
→ b(x) |
| 6: |
|
w(a(a(x))) |
→ u(w(x)) |
| 7: |
|
w(a(c(x))) |
→ u(b(d(x))) |
| 8: |
|
w(c(x)) |
→ b(x) |
|
There are 6 dependency pairs:
|
| 9: |
|
V(a(a(x))) |
→ U(v(x)) |
| 10: |
|
V(a(a(x))) |
→ V(x) |
| 11: |
|
V(a(c(x))) |
→ U(b(d(x))) |
| 12: |
|
W(a(a(x))) |
→ U(w(x)) |
| 13: |
|
W(a(a(x))) |
→ W(x) |
| 14: |
|
W(a(c(x))) |
→ U(b(d(x))) |
|
The approximated dependency graph contains 2 SCCs:
{10}
and {13}.
-
Consider the SCC {10}.
There are no usable rules.
By taking the AF π with
π(V) = 1 together with
the lexicographic path order with
empty precedence,
rule 10
is strictly decreasing.
-
Consider the SCC {13}.
There are no usable rules.
By taking the AF π with
π(W) = 1 together with
the lexicographic path order with
empty precedence,
rule 13
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 3, 2006