Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
f(f(x)) |
→ f(g(f(x),x)) |
| 2: |
|
f(f(x)) |
→ f(h(f(x),f(x))) |
| 3: |
|
g(x,y) |
→ y |
| 4: |
|
h(x,x) |
→ g(x,0) |
|
There are 5 dependency pairs:
|
| 5: |
|
F(f(x)) |
→ F(g(f(x),x)) |
| 6: |
|
F(f(x)) |
→ G(f(x),x) |
| 7: |
|
F(f(x)) |
→ F(h(f(x),f(x))) |
| 8: |
|
F(f(x)) |
→ H(f(x),f(x)) |
| 9: |
|
H(x,x) |
→ G(x,0) |
|
The approximated dependency graph contains one SCC:
{5,7}.
-
Consider the SCC {5,7}.
By taking the AF π with
π(F) = 1,
π(g) = 2
and π(h) = [ ] together with
the lexicographic path order with
precedence f ≻ h ≻ 0,
rule 3
is weakly decreasing and
the rules in {1,2,4,5,7}
are strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.02 seconds)
--- May 4, 2006