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