Termination Proof Script
Consider the TRS R consisting of the rewrite rules
|
| 1: |
|
x * (y * z) |
→ otimes(x,y) * z |
| 2: |
|
1 * y |
→ y |
| 3: |
|
(x + y) * z |
→ oplus(x * z,y * z) |
| 4: |
|
x * oplus(y,z) |
→ oplus(x * y,x * z) |
|
There are 5 dependency pairs:
|
| 5: |
|
x *# (y * z) |
→ otimes(x,y) *# z |
| 6: |
|
(x + y) *# z |
→ x *# z |
| 7: |
|
(x + y) *# z |
→ y *# z |
| 8: |
|
x *# oplus(y,z) |
→ x *# y |
| 9: |
|
x *# oplus(y,z) |
→ x *# z |
|
The approximated dependency graph contains one SCC:
{5-9}.
-
Consider the SCC {5-9}.
There are no usable rules.
By taking the AF π with
π(*#) = π(otimes) = 1 together with
the lexicographic path order with
empty precedence,
the rules in {5,8,9}
are weakly decreasing and
the rules in {6,7}
are strictly decreasing.
There is one new SCC.
-
Consider the SCC {5,8,9}.
By taking the AF π with
π(*) = π(*#) = 2 together with
the lexicographic path order with
empty precedence,
rule 5
is weakly decreasing and
the rules in {8,9}
are strictly decreasing.
There is one new SCC.
-
Consider the SCC {5}.
By taking the AF π with
π(*#) = 2
and π(*) = [2] together with
the lexicographic path order with
empty precedence,
rule 5
is strictly decreasing.
Hence the TRS is terminating.
Tyrolean Termination Tool (0.02 seconds)
--- May 3, 2006