Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
f(0) |
→ 1 |
| 2: |
|
f(s(x)) |
→ g(x,s(x)) |
| 3: |
|
g(0,y) |
→ y |
| 4: |
|
g(s(x),y) |
→ g(x,y + s(x)) |
| 5: |
|
x + 0 |
→ x |
| 6: |
|
x + s(y) |
→ s(x + y) |
| 7: |
|
g(s(x),y) |
→ g(x,s(y + x)) |
|
There are 6 dependency pairs:
|
| 8: |
|
F(s(x)) |
→ G(x,s(x)) |
| 9: |
|
G(s(x),y) |
→ G(x,y + s(x)) |
| 10: |
|
G(s(x),y) |
→ y +# s(x) |
| 11: |
|
x +# s(y) |
→ x +# y |
| 12: |
|
G(s(x),y) |
→ G(x,s(y + x)) |
| 13: |
|
G(s(x),y) |
→ y +# x |
|
The approximated dependency graph contains 2 SCCs:
{11}
and {9,12}.
-
Consider the SCC {11}.
There are no usable rules.
By taking the AF π with
π(+#) = 2 together with
the lexicographic path order with
empty precedence,
rule 11
is strictly decreasing.
-
Consider the SCC {9,12}.
The usable rules are {5,6}.
By taking the AF π with
π(G) = 1 together with
the lexicographic path order with
precedence + ≻ s,
the rules in {5,6,9,12}
are strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.01 seconds)
--- May 3, 2006