The rewrite relation of the following TRS is considered.
[mark(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[cons1(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
[from(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[2nd(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[cons1(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
[from(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[2nd(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[cons1(x1, x2)] |
= |
· x1 + · x2 +
|
[active(x1)] |
= |
· x1 +
|
[from(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[2nd(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
active#(s(X)) |
→ |
active#(X) |
(35) |
active#(cons1(X1,X2)) |
→ |
active#(X2) |
(39) |
active#(cons1(X1,X2)) |
→ |
active#(X1) |
(37) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(33) |
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(s(X)) |
→ |
active#(X) |
(35) |
|
1 |
> |
1 |
active#(cons1(X1,X2)) |
→ |
active#(X2) |
(39) |
|
1 |
> |
1 |
active#(cons1(X1,X2)) |
→ |
active#(X1) |
(37) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(33) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
proper#(s(X)) |
→ |
proper#(X) |
(54) |
proper#(cons1(X1,X2)) |
→ |
proper#(X1) |
(57) |
proper#(cons1(X1,X2)) |
→ |
proper#(X2) |
(56) |
proper#(from(X)) |
→ |
proper#(X) |
(52) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(50) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(49) |
proper#(2nd(X)) |
→ |
proper#(X) |
(47) |
1.1.1.1.1.2 Subterm Criterion Processor
We use the projection
and remove the pairs:
proper#(s(X)) |
→ |
proper#(X) |
(54) |
proper#(cons1(X1,X2)) |
→ |
proper#(X1) |
(57) |
proper#(cons1(X1,X2)) |
→ |
proper#(X2) |
(56) |
proper#(from(X)) |
→ |
proper#(X) |
(52) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(50) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(49) |
proper#(2nd(X)) |
→ |
proper#(X) |
(47) |
1.1.1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
2nd#(ok(X)) |
→ |
2nd#(X) |
(59) |
2nd#(mark(X)) |
→ |
2nd#(X) |
(41) |
1.1.1.1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
2nd#(ok(X)) |
→ |
2nd#(X) |
(59) |
|
1 |
> |
1 |
2nd#(mark(X)) |
→ |
2nd#(X) |
(41) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(42) |
1.1.1.1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(42) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
from#(ok(X)) |
→ |
from#(X) |
(60) |
from#(mark(X)) |
→ |
from#(X) |
(43) |
1.1.1.1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
from#(ok(X)) |
→ |
from#(X) |
(60) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(43) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(61) |
s#(mark(X)) |
→ |
s#(X) |
(44) |
1.1.1.1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(61) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(44) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
cons1#(X1,mark(X2)) |
→ |
cons1#(X1,X2) |
(46) |
cons1#(mark(X1),X2) |
→ |
cons1#(X1,X2) |
(45) |
1.1.1.1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons1#(X1,mark(X2)) |
→ |
cons1#(X1,X2) |
(46) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons1#(mark(X1),X2) |
→ |
cons1#(X1,X2) |
(45) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.