Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
f(a,a) |
→ f(a,b) |
| 2: |
|
f(a,b) |
→ f(s(a),c) |
| 3: |
|
f(s(X),c) |
→ f(X,c) |
| 4: |
|
f(c,c) |
→ f(a,a) |
|
There are 4 dependency pairs:
|
| 5: |
|
F(a,a) |
→ F(a,b) |
| 6: |
|
F(a,b) |
→ F(s(a),c) |
| 7: |
|
F(s(X),c) |
→ F(X,c) |
| 8: |
|
F(c,c) |
→ F(a,a) |
|
The approximated dependency graph contains one SCC:
{5-8}.
-
Consider the SCC {5-8}.
There are no usable rules.
By taking the AF π with
π(F) = π(s) = 1 together with
the lexicographic path order with
precedence c ≻ a,
the rules in {5-7}
are weakly decreasing and
rule 8
is strictly decreasing.
There is one new SCC.
-
Consider the SCC {7}.
By taking the AF π with
π(F) = 1 together with
the lexicographic path order with
empty precedence,
rule 7
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.02 seconds)
--- May 3, 2006