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