YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
b(b(x0)) |
→ |
b(a(b(a(a(b(x0)))))) |
|
b(b(a(b(x0)))) |
→ |
b(a(b(b(x0)))) |
|
b(a(a(a(b(a(a(b(x0)))))))) |
→ |
b(a(a(a(b(b(x0)))))) |
|
b(b(a(a(b(x0))))) |
→ |
b(a(a(b(b(x0))))) |
Proof
1 Rule Removal
Using the
linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1
over the naturals
| [a(x1)] |
= |
| 1 |
1 |
0 |
0 |
| 0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
| [b(x1)] |
= |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
1 |
0 |
1 |
·
x1 +
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
the
rules
|
b(b(x0)) |
→ |
b(a(b(a(a(b(x0)))))) |
|
b(b(a(b(x0)))) |
→ |
b(a(b(b(x0)))) |
|
b(b(a(a(b(x0))))) |
→ |
b(a(a(b(b(x0))))) |
remain.
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b(b(x0)) |
→ |
b(a(a(b(a(b(x0)))))) |
|
b(a(b(b(x0)))) |
→ |
b(b(a(b(x0)))) |
|
b(a(a(b(b(x0))))) |
→ |
b(b(a(a(b(x0))))) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [a(x1)] |
= |
·
x1 +
|
| [b(x1)] |
= |
·
x1 +
|
the
rules
|
b(a(b(b(x0)))) |
→ |
b(b(a(b(x0)))) |
|
b(a(a(b(b(x0))))) |
→ |
b(b(a(a(b(x0))))) |
remain.
1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight function
| prec(a) |
= |
1 |
|
weight(a) |
= |
0 |
|
|
|
| prec(b) |
= |
0 |
|
weight(b) |
= |
1 |
|
|
|
all rules could be removed.
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.