Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
p(s(x)) |
→ x |
| 2: |
|
s(p(x)) |
→ x |
| 3: |
|
0 + y |
→ y |
| 4: |
|
s(x) + y |
→ s(x + y) |
| 5: |
|
p(x) + y |
→ p(x + y) |
| 6: |
|
minus(0) |
→ 0 |
| 7: |
|
minus(s(x)) |
→ p(minus(x)) |
| 8: |
|
minus(p(x)) |
→ s(minus(x)) |
| 9: |
|
0 * y |
→ 0 |
| 10: |
|
s(x) * y |
→ (x * y) + y |
| 11: |
|
p(x) * y |
→ (x * y) + minus(y) |
|
There are 13 dependency pairs:
|
| 12: |
|
s(x) +# y |
→ S(x + y) |
| 13: |
|
s(x) +# y |
→ x +# y |
| 14: |
|
p(x) +# y |
→ P(x + y) |
| 15: |
|
p(x) +# y |
→ x +# y |
| 16: |
|
MINUS(s(x)) |
→ P(minus(x)) |
| 17: |
|
MINUS(s(x)) |
→ MINUS(x) |
| 18: |
|
MINUS(p(x)) |
→ S(minus(x)) |
| 19: |
|
MINUS(p(x)) |
→ MINUS(x) |
| 20: |
|
s(x) *# y |
→ (x * y) +# y |
| 21: |
|
s(x) *# y |
→ x *# y |
| 22: |
|
p(x) *# y |
→ (x * y) +# minus(y) |
| 23: |
|
p(x) *# y |
→ x *# y |
| 24: |
|
p(x) *# y |
→ MINUS(y) |
|
The approximated dependency graph contains 3 SCCs:
{13,15},
{17,19}
and {21,23}.
-
Consider the SCC {13,15}.
There are no usable rules.
By taking the AF π with
π(+#) = π(p) = 1 together with
the lexicographic path order with
empty precedence,
rule 15
is weakly decreasing and
rule 13
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {15}.
By taking the AF π with
π(+#) = 1 together with
the lexicographic path order with
empty precedence,
rule 15
is strictly decreasing.
-
Consider the SCC {17,19}.
There are no usable rules.
By taking the AF π with
π(MINUS) = π(p) = 1 together with
the lexicographic path order with
empty precedence,
rule 19
is weakly decreasing and
rule 17
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {19}.
By taking the AF π with
π(MINUS) = 1 together with
the lexicographic path order with
empty precedence,
rule 19
is strictly decreasing.
-
Consider the SCC {21,23}.
There are no usable rules.
By taking the AF π with
π(*#) = π(p) = 1 together with
the lexicographic path order with
empty precedence,
rule 23
is weakly decreasing and
rule 21
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.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.02 seconds)
--- May 4, 2006