The rewrite relation of the following TRS is considered.
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[ok(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[0] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[p(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[ok(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[0] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[p(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[ok(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[0] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[p(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[ok(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[0] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[p(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
proper#(p(X)) |
→ |
proper#(X) |
(41) |
proper#(f(X)) |
→ |
proper#(X) |
(34) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(36) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(37) |
proper#(s(X)) |
→ |
proper#(X) |
(39) |
1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(p(X)) |
→ |
proper#(X) |
(41) |
|
1 |
> |
1 |
proper#(f(X)) |
→ |
proper#(X) |
(34) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(36) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(37) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(39) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(48) |
1.1.1.1.1.1.2 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1, 1
}
|
π(p)
|
= |
{
1, 1
}
|
π(mark)
|
= |
{
1
}
|
π(cons)
|
= |
{
2, 2
}
|
π(s)
|
= |
{
1, 1
}
|
π(active)
|
= |
{
1, 1
}
|
π(f)
|
= |
{
1, 1
}
|
to remove the pairs:
top#(ok(X)) |
→ |
top#(active(X)) |
(48) |
1.1.1.1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
active#(f(X)) |
→ |
active#(X) |
(23) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(25) |
active#(s(X)) |
→ |
active#(X) |
(27) |
active#(p(X)) |
→ |
active#(X) |
(29) |
1.1.1.1.1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(f(X)) |
→ |
active#(X) |
(23) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(25) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(27) |
|
1 |
> |
1 |
active#(p(X)) |
→ |
active#(X) |
(29) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
p#(mark(X)) |
→ |
p#(X) |
(33) |
p#(ok(X)) |
→ |
p#(X) |
(46) |
1.1.1.1.1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
p#(mark(X)) |
→ |
p#(X) |
(33) |
|
1 |
> |
1 |
p#(ok(X)) |
→ |
p#(X) |
(46) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
1.1.1.1.1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(45) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(32) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(44) |
1.1.1.1.1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(32) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(44) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
f#(mark(X)) |
→ |
f#(X) |
(31) |
f#(ok(X)) |
→ |
f#(X) |
(43) |
1.1.1.1.1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(mark(X)) |
→ |
f#(X) |
(31) |
|
1 |
> |
1 |
f#(ok(X)) |
→ |
f#(X) |
(43) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.