Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
norm(nil) |
→ 0 |
| 2: |
|
norm(g(x,y)) |
→ s(norm(x)) |
| 3: |
|
f(x,nil) |
→ g(nil,x) |
| 4: |
|
f(x,g(y,z)) |
→ g(f(x,y),z) |
| 5: |
|
rem(nil,y) |
→ nil |
| 6: |
|
rem(g(x,y),0) |
→ g(x,y) |
| 7: |
|
rem(g(x,y),s(z)) |
→ rem(x,z) |
|
There are 3 dependency pairs:
|
| 8: |
|
NORM(g(x,y)) |
→ NORM(x) |
| 9: |
|
F(x,g(y,z)) |
→ F(x,y) |
| 10: |
|
REM(g(x,y),s(z)) |
→ REM(x,z) |
|
The approximated dependency graph contains 3 SCCs:
{9},
{8}
and {10}.
-
Consider the SCC {9}.
There are no usable rules.
By taking the AF π with
π(F) = 2
and π(g) = [1] together with
the lexicographic path order with
empty precedence,
rule 9
is strictly decreasing.
-
Consider the SCC {8}.
There are no usable rules.
By taking the AF π with
π(NORM) = 1
and π(g) = [1] together with
the lexicographic path order with
empty precedence,
rule 8
is strictly decreasing.
-
Consider the SCC {10}.
There are no usable rules.
By taking the AF π with
π(REM) = 1
and π(g) = [1] together with
the lexicographic path order with
empty precedence,
rule 10
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.00 seconds)
--- May 4, 2006