Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
O(0) |
→ 0 |
| 2: |
|
0 + x |
→ x |
| 3: |
|
x + 0 |
→ x |
| 4: |
|
O(x) + O(y) |
→ O(x + y) |
| 5: |
|
O(x) + I(y) |
→ I(x + y) |
| 6: |
|
I(x) + O(y) |
→ I(x + y) |
| 7: |
|
I(x) + I(y) |
→ O((x + y) + I(0)) |
| 8: |
|
0 * x |
→ 0 |
| 9: |
|
x * 0 |
→ 0 |
| 10: |
|
O(x) * y |
→ O(x * y) |
| 11: |
|
I(x) * y |
→ O(x * y) + y |
|
There are 12 dependency pairs:
|
| 12: |
|
O(x) +# O(y) |
→ O#(x + y) |
| 13: |
|
O(x) +# O(y) |
→ x +# y |
| 14: |
|
O(x) +# I(y) |
→ x +# y |
| 15: |
|
I(x) +# O(y) |
→ x +# y |
| 16: |
|
I(x) +# I(y) |
→ O#((x + y) + I(0)) |
| 17: |
|
I(x) +# I(y) |
→ (x + y) +# I(0) |
| 18: |
|
I(x) +# I(y) |
→ x +# y |
| 19: |
|
O(x) *# y |
→ O#(x * y) |
| 20: |
|
O(x) *# y |
→ x *# y |
| 21: |
|
I(x) *# y |
→ O(x * y) +# y |
| 22: |
|
I(x) *# y |
→ O#(x * y) |
| 23: |
|
I(x) *# y |
→ x *# y |
|
The approximated dependency graph contains 2 SCCs:
{13-15,17,18}
and {20,23}.
-
Consider the SCC {13-15,17,18}.
The usable rules are {1-7}.
The constraints could not be solved.
-
Consider the SCC {20,23}.
There are no usable rules.
By taking the AF π with
π(*#) = π(I) = 1 together with
the lexicographic path order with
empty precedence,
rule 23
is weakly decreasing and
rule 20
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {23}.
By taking the AF π with
π(*#) = 1 together with
the lexicographic path order with
empty precedence,
rule 23
is strictly decreasing.
Tyrolean Termination Tool (0.04 seconds)
--- May 4, 2006