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