The rewrite relation of the following TRS is considered.
[tt] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[nil] |
= |
|
[U11(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[__(x1, x2)] |
= |
· x1 + · x2 +
|
[isNePal(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[U12(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[tt] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[nil] |
= |
|
[U11(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[__(x1, x2)] |
= |
· x1 + · x2 +
|
[isNePal(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[U12(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[tt] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[nil] |
= |
|
[U11(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[__(x1, x2)] |
= |
· x1 + · x2 +
|
[isNePal(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[U12(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
[tt] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
[nil] |
= |
|
[U11(x1)] |
= |
· x1 +
|
[top(x1)] |
= |
· x1 +
|
[__(x1, x2)] |
= |
· x1 + · x2 +
|
[isNePal(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[U12(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(62) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(60) |
1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(proper) |
= |
0 |
|
stat(proper) |
= |
lex
|
prec(isNePal) |
= |
0 |
|
stat(isNePal) |
= |
lex
|
prec(U12) |
= |
0 |
|
stat(U12) |
= |
lex
|
prec(U11) |
= |
8 |
|
stat(U11) |
= |
lex
|
prec(tt) |
= |
0 |
|
stat(tt) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(__) |
= |
1 |
|
stat(__) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
1 |
π(proper) |
= |
1 |
π(isNePal) |
= |
1 |
π(U12) |
= |
1 |
π(U11) |
= |
[1] |
π(tt) |
= |
[] |
π(nil) |
= |
[] |
π(mark) |
= |
[1] |
π(active) |
= |
1 |
π(__) |
= |
[1,2] |
together with the usable
rules
active(__(__(X,Y),Z)) |
→ |
mark(__(X,__(Y,Z))) |
(1) |
active(__(X1,X2)) |
→ |
__(active(X1),X2) |
(7) |
active(__(X1,X2)) |
→ |
__(X1,active(X2)) |
(8) |
active(U11(X)) |
→ |
U11(active(X)) |
(9) |
active(U12(X)) |
→ |
U12(active(X)) |
(10) |
active(isNePal(X)) |
→ |
isNePal(active(X)) |
(11) |
__(mark(X1),X2) |
→ |
mark(__(X1,X2)) |
(12) |
__(X1,mark(X2)) |
→ |
mark(__(X1,X2)) |
(13) |
__(ok(X1),ok(X2)) |
→ |
ok(__(X1,X2)) |
(23) |
U11(mark(X)) |
→ |
mark(U11(X)) |
(14) |
U11(ok(X)) |
→ |
ok(U11(X)) |
(24) |
U12(mark(X)) |
→ |
mark(U12(X)) |
(15) |
U12(ok(X)) |
→ |
ok(U12(X)) |
(25) |
isNePal(mark(X)) |
→ |
mark(isNePal(X)) |
(16) |
isNePal(ok(X)) |
→ |
ok(isNePal(X)) |
(26) |
proper(__(X1,X2)) |
→ |
__(proper(X1),proper(X2)) |
(17) |
proper(nil) |
→ |
ok(nil) |
(18) |
proper(U11(X)) |
→ |
U11(proper(X)) |
(19) |
proper(tt) |
→ |
ok(tt) |
(20) |
proper(U12(X)) |
→ |
U12(proper(X)) |
(21) |
proper(isNePal(X)) |
→ |
isNePal(proper(X)) |
(22) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(60) |
could be deleted.
1.1.1.1.1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1, 1
}
|
π(isNePal)
|
= |
{
1, 1
}
|
π(U12)
|
= |
{
1, 1
}
|
π(U11)
|
= |
{
1, 1
}
|
π(mark)
|
= |
{
1
}
|
π(active)
|
= |
{
1, 1
}
|
π(__)
|
= |
{
2
}
|
to remove the pairs:
top#(ok(X)) |
→ |
top#(active(X)) |
(62) |
1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(__(X1,X2)) |
→ |
active#(X1) |
(31) |
active#(__(X1,X2)) |
→ |
active#(X2) |
(33) |
active#(U11(X)) |
→ |
active#(X) |
(35) |
active#(U12(X)) |
→ |
active#(X) |
(37) |
active#(isNePal(X)) |
→ |
active#(X) |
(39) |
1.1.1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(__(X1,X2)) |
→ |
active#(X1) |
(31) |
|
1 |
> |
1 |
active#(__(X1,X2)) |
→ |
active#(X2) |
(33) |
|
1 |
> |
1 |
active#(U11(X)) |
→ |
active#(X) |
(35) |
|
1 |
> |
1 |
active#(U12(X)) |
→ |
active#(X) |
(37) |
|
1 |
> |
1 |
active#(isNePal(X)) |
→ |
active#(X) |
(39) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(__(X1,X2)) |
→ |
proper#(X2) |
(46) |
proper#(__(X1,X2)) |
→ |
proper#(X1) |
(47) |
proper#(U11(X)) |
→ |
proper#(X) |
(49) |
proper#(U12(X)) |
→ |
proper#(X) |
(51) |
proper#(isNePal(X)) |
→ |
proper#(X) |
(53) |
1.1.1.1.1.1.3 Subterm Criterion Processor
We use the projection
and remove the pairs:
proper#(__(X1,X2)) |
→ |
proper#(X2) |
(46) |
proper#(__(X1,X2)) |
→ |
proper#(X1) |
(47) |
proper#(U11(X)) |
→ |
proper#(X) |
(49) |
proper#(U12(X)) |
→ |
proper#(X) |
(51) |
proper#(isNePal(X)) |
→ |
proper#(X) |
(53) |
1.1.1.1.1.1.3.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
isNePal#(mark(X)) |
→ |
isNePal#(X) |
(45) |
isNePal#(ok(X)) |
→ |
isNePal#(X) |
(58) |
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.
isNePal#(mark(X)) |
→ |
isNePal#(X) |
(45) |
|
1 |
> |
1 |
isNePal#(ok(X)) |
→ |
isNePal#(X) |
(58) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
U12#(mark(X)) |
→ |
U12#(X) |
(44) |
U12#(ok(X)) |
→ |
U12#(X) |
(57) |
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.
U12#(mark(X)) |
→ |
U12#(X) |
(44) |
|
1 |
> |
1 |
U12#(ok(X)) |
→ |
U12#(X) |
(57) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
U11#(mark(X)) |
→ |
U11#(X) |
(43) |
U11#(ok(X)) |
→ |
U11#(X) |
(56) |
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.
U11#(mark(X)) |
→ |
U11#(X) |
(43) |
|
1 |
> |
1 |
U11#(ok(X)) |
→ |
U11#(X) |
(56) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
__#(mark(X1),X2) |
→ |
__#(X1,X2) |
(41) |
__#(X1,mark(X2)) |
→ |
__#(X1,X2) |
(42) |
__#(ok(X1),ok(X2)) |
→ |
__#(X1,X2) |
(55) |
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.
__#(mark(X1),X2) |
→ |
__#(X1,X2) |
(41) |
|
2 |
≥ |
2 |
1 |
> |
1 |
__#(X1,mark(X2)) |
→ |
__#(X1,X2) |
(42) |
|
2 |
> |
2 |
1 |
≥ |
1 |
__#(ok(X1),ok(X2)) |
→ |
__#(X1,X2) |
(55) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.