The rewrite relation of the following TRS is considered.
[mark(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[adx(x1)] |
= |
· x1 +
|
[0] |
= |
|
[tl(x1)] |
= |
· x1 +
|
[nats] |
= |
|
[incr(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[hd(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[zeros] |
= |
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[adx(x1)] |
= |
· x1 +
|
[0] |
= |
|
[tl(x1)] |
= |
· x1 +
|
[nats] |
= |
|
[incr(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[hd(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[zeros] |
= |
|
all of the following rules can be deleted.
[mark(x1)] |
= |
· x1 +
|
[s(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[adx(x1)] |
= |
· x1 +
|
[0] |
= |
|
[tl(x1)] |
= |
· x1 +
|
[nats] |
= |
|
[incr(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
· x1 +
|
[hd(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[cons(x1, x2)] |
= |
· x1 + · x2 +
|
[zeros] |
= |
|
all of the following rules can be deleted.
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(72) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(70) |
1.1.1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1
}
|
π(proper)
|
= |
{
1, 1
}
|
π(tl)
|
= |
{
1, 1
}
|
π(hd)
|
= |
{
1, 1
}
|
π(s)
|
= |
{
1
}
|
π(incr)
|
= |
{
1, 1
}
|
π(cons)
|
= |
{
1, 1
}
|
π(mark)
|
= |
{
1, 1, 1
}
|
π(adx)
|
= |
{
1, 1, 1
}
|
π(active)
|
= |
{
1, 1
}
|
to remove the pairs:
top#(mark(X)) |
→ |
top#(proper(X)) |
(70) |
1.1.1.1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1, 1
}
|
π(tl)
|
= |
{
1, 1, 1
}
|
π(hd)
|
= |
{
1, 1
}
|
π(s)
|
= |
{
1
}
|
π(incr)
|
= |
{
1
}
|
π(cons)
|
= |
{
1, 1, 1
}
|
π(mark)
|
= |
{
1, 1
}
|
π(adx)
|
= |
{
1
}
|
π(active)
|
= |
{
1, 1
}
|
to remove the pairs:
top#(ok(X)) |
→ |
top#(active(X)) |
(72) |
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
proper#(tl(X)) |
→ |
proper#(X) |
(61) |
proper#(hd(X)) |
→ |
proper#(X) |
(59) |
proper#(s(X)) |
→ |
proper#(X) |
(57) |
proper#(incr(X)) |
→ |
proper#(X) |
(55) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(53) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(52) |
proper#(adx(X)) |
→ |
proper#(X) |
(50) |
1.1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(tl(X)) |
→ |
proper#(X) |
(61) |
|
1 |
> |
1 |
proper#(hd(X)) |
→ |
proper#(X) |
(59) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(57) |
|
1 |
> |
1 |
proper#(incr(X)) |
→ |
proper#(X) |
(55) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(53) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(52) |
|
1 |
> |
1 |
proper#(adx(X)) |
→ |
proper#(X) |
(50) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
active#(tl(X)) |
→ |
active#(X) |
(44) |
active#(hd(X)) |
→ |
active#(X) |
(42) |
active#(incr(X)) |
→ |
active#(X) |
(40) |
active#(adx(X)) |
→ |
active#(X) |
(38) |
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#(tl(X)) |
→ |
active#(X) |
(44) |
|
1 |
> |
1 |
active#(hd(X)) |
→ |
active#(X) |
(42) |
|
1 |
> |
1 |
active#(incr(X)) |
→ |
active#(X) |
(40) |
|
1 |
> |
1 |
active#(adx(X)) |
→ |
active#(X) |
(38) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
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.
s#(ok(X)) |
→ |
s#(X) |
(66) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(64) |
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.
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(64) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
adx#(ok(X)) |
→ |
adx#(X) |
(63) |
adx#(mark(X)) |
→ |
adx#(X) |
(46) |
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.
adx#(ok(X)) |
→ |
adx#(X) |
(63) |
|
1 |
> |
1 |
adx#(mark(X)) |
→ |
adx#(X) |
(46) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
incr#(ok(X)) |
→ |
incr#(X) |
(65) |
incr#(mark(X)) |
→ |
incr#(X) |
(47) |
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.
incr#(ok(X)) |
→ |
incr#(X) |
(65) |
|
1 |
> |
1 |
incr#(mark(X)) |
→ |
incr#(X) |
(47) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
hd#(ok(X)) |
→ |
hd#(X) |
(67) |
hd#(mark(X)) |
→ |
hd#(X) |
(48) |
1.1.1.1.1.8 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
hd#(ok(X)) |
→ |
hd#(X) |
(67) |
|
1 |
> |
1 |
hd#(mark(X)) |
→ |
hd#(X) |
(48) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
tl#(ok(X)) |
→ |
tl#(X) |
(68) |
tl#(mark(X)) |
→ |
tl#(X) |
(49) |
1.1.1.1.1.9 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tl#(ok(X)) |
→ |
tl#(X) |
(68) |
|
1 |
> |
1 |
tl#(mark(X)) |
→ |
tl#(X) |
(49) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.