Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
w(r(x)) |
→ r(w(x)) |
| 2: |
|
b(r(x)) |
→ r(b(x)) |
| 3: |
|
b(w(x)) |
→ w(b(x)) |
|
There are 4 dependency pairs:
|
| 4: |
|
W(r(x)) |
→ W(x) |
| 5: |
|
B(r(x)) |
→ B(x) |
| 6: |
|
B(w(x)) |
→ W(b(x)) |
| 7: |
|
B(w(x)) |
→ B(x) |
|
The approximated dependency graph contains 2 SCCs:
{4}
and {5,7}.
-
Consider the SCC {4}.
There are no usable rules.
By taking the AF π with
π(W) = 1 together with
the lexicographic path order with
empty precedence,
rule 4
is strictly decreasing.
-
Consider the SCC {5,7}.
There are no usable rules.
By taking the AF π with
π(B) = π(r) = 1 together with
the lexicographic path order with
empty precedence,
rule 5
is weakly decreasing and
rule 7
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {5}.
By taking the AF π with
π(B) = 1 together with
the lexicographic path order with
empty precedence,
rule 5
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 3, 2006