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 |
| 12: |
|
x - 0 |
→ x |
| 13: |
|
0 - x |
→ 0 |
| 14: |
|
O(x) - O(y) |
→ O(x - y) |
| 15: |
|
O(x) - I(y) |
→ I((x - y) - I(1)) |
| 16: |
|
I(x) - O(y) |
→ I(x - y) |
| 17: |
|
I(x) - I(y) |
→ O(x - y) |
|
There are 19 dependency pairs:
|
| 18: |
|
O(x) +# O(y) |
→ O#(x + y) |
| 19: |
|
O(x) +# O(y) |
→ x +# y |
| 20: |
|
O(x) +# I(y) |
→ x +# y |
| 21: |
|
I(x) +# O(y) |
→ x +# y |
| 22: |
|
I(x) +# I(y) |
→ O#((x + y) + I(0)) |
| 23: |
|
I(x) +# I(y) |
→ (x + y) +# I(0) |
| 24: |
|
I(x) +# I(y) |
→ x +# y |
| 25: |
|
O(x) *# y |
→ O#(x * y) |
| 26: |
|
O(x) *# y |
→ x *# y |
| 27: |
|
I(x) *# y |
→ O(x * y) +# y |
| 28: |
|
I(x) *# y |
→ O#(x * y) |
| 29: |
|
I(x) *# y |
→ x *# y |
| 30: |
|
O(x) -# O(y) |
→ O#(x - y) |
| 31: |
|
O(x) -# O(y) |
→ x -# y |
| 32: |
|
O(x) -# I(y) |
→ (x - y) -# I(1) |
| 33: |
|
O(x) -# I(y) |
→ x -# y |
| 34: |
|
I(x) -# O(y) |
→ x -# y |
| 35: |
|
I(x) -# I(y) |
→ O#(x - y) |
| 36: |
|
I(x) -# I(y) |
→ x -# y |
|
The approximated dependency graph contains 3 SCCs:
{19-21,23,24},
{26,29}
and {31-34,36}.
-
Consider the SCC {19-21,23,24}.
The usable rules are {1-7}.
The constraints could not be solved.
-
Consider the SCC {26,29}.
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 29
is weakly decreasing and
rule 26
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {29}.
By taking the AF π with
π(*#) = 1 together with
the lexicographic path order with
empty precedence,
rule 29
is strictly decreasing.
-
Consider the SCC {31-34,36}.
The usable rules are {1,12-17}.
By taking the AF π with
π(-) = π(-#) = 1 together with
the lexicographic path order with
precedence I ≈ O,
the rules in {12-17}
are weakly decreasing and
the rules in {1,31-34,36}
are strictly decreasing.
Tyrolean Termination Tool (0.09 seconds)
--- May 4, 2006