Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
exp(x,0) |
→ s(0) |
| 2: |
|
exp(x,s(y)) |
→ x * exp(x,y) |
| 3: |
|
0 * y |
→ 0 |
| 4: |
|
s(x) * y |
→ y + (x * y) |
| 5: |
|
0 - y |
→ 0 |
| 6: |
|
x - 0 |
→ x |
| 7: |
|
s(x) - s(y) |
→ x - y |
|
There are 4 dependency pairs:
|
| 8: |
|
EXP(x,s(y)) |
→ x *# exp(x,y) |
| 9: |
|
EXP(x,s(y)) |
→ EXP(x,y) |
| 10: |
|
s(x) *# y |
→ x *# y |
| 11: |
|
s(x) -# s(y) |
→ x -# y |
|
The approximated dependency graph contains 3 SCCs:
{10},
{11}
and {9}.
-
Consider the SCC {10}.
There are no usable rules.
By taking the AF π with
π(*#) = 1 together with
the lexicographic path order with
empty precedence,
rule 10
is strictly decreasing.
-
Consider the SCC {11}.
There are no usable rules.
By taking the AF π with
π(-#) = 1 together with
the lexicographic path order with
empty precedence,
rule 11
is strictly decreasing.
-
Consider the SCC {9}.
There are no usable rules.
By taking the AF π with
π(EXP) = 2 together with
the lexicographic path order with
empty precedence,
rule 9
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 4, 2006