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