YES
Termination Proof
Termination Proof
by ttt2 (version ttt2 1.15)
Input
The rewrite relation of the following TRS is considered.
|
0(*(x0)) |
→ |
*(1(x0)) |
|
1(*(x0)) |
→ |
0(#(x0)) |
|
#(0(x0)) |
→ |
0(#(x0)) |
|
#(1(x0)) |
→ |
1(#(x0)) |
|
#($(x0)) |
→ |
*($(x0)) |
|
#(#(x0)) |
→ |
#(x0) |
|
#(*(x0)) |
→ |
*(x0) |
Proof
1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [$(x1)] |
= |
4 ·
x1 +
-∞
|
| [0(x1)] |
= |
8 ·
x1 +
-∞
|
| [#(x1)] |
= |
4 ·
x1 +
-∞
|
| [*(x1)] |
= |
4 ·
x1 +
-∞
|
| [1(x1)] |
= |
8 ·
x1 +
-∞
|
the
rules
|
0(*(x0)) |
→ |
*(1(x0)) |
|
1(*(x0)) |
→ |
0(#(x0)) |
|
#(0(x0)) |
→ |
0(#(x0)) |
|
#(1(x0)) |
→ |
1(#(x0)) |
|
#($(x0)) |
→ |
*($(x0)) |
remain.
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight function
| prec($) |
= |
4 |
|
weight($) |
= |
0 |
|
|
|
| prec(#) |
= |
3 |
|
weight(#) |
= |
1 |
|
|
|
| prec(1) |
= |
2 |
|
weight(1) |
= |
1 |
|
|
|
| prec(0) |
= |
1 |
|
weight(0) |
= |
1 |
|
|
|
| prec(*) |
= |
0 |
|
weight(*) |
= |
1 |
|
|
|
all rules could be removed.
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.