The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(59) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(60) |
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.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(59) |
|
2 |
> |
2 |
1 |
≥ |
1 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(60) |
|
2 |
> |
2 |
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
mark#(tl(X)) |
→ |
mark#(X) |
(54) |
mark#(hd(X)) |
→ |
mark#(X) |
(51) |
mark#(incr(X)) |
→ |
mark#(X) |
(47) |
mark#(adx(X)) |
→ |
active#(adx(mark(X))) |
(43) |
active#(adx(cons(X,Y))) |
→ |
mark#(incr(cons(X,adx(Y)))) |
(39) |
mark#(adx(X)) |
→ |
mark#(X) |
(41) |
1.1.1.1.1.1.2 Subterm Criterion Processor
We use the projection to multisets
π(mark#)
|
= |
{
1, 1
}
|
π(active#)
|
= |
{
1, 1
}
|
π(tl)
|
= |
{
1
}
|
π(hd)
|
= |
{
1, 1
}
|
π(incr)
|
= |
{
1
}
|
π(cons)
|
= |
{
2
}
|
π(mark)
|
= |
{
1
}
|
π(adx)
|
= |
{
1, 1
}
|
π(active)
|
= |
{
1
}
|
to remove the pairs:
mark#(hd(X)) |
→ |
mark#(X) |
(51) |
mark#(adx(X)) |
→ |
mark#(X) |
(41) |
1.1.1.1.1.1.2.1 Subterm Criterion Processor
We use the projection to multisets
π(mark#)
|
= |
{
1
}
|
π(active#)
|
= |
{
1
}
|
π(tl)
|
= |
{
1, 1
}
|
π(hd)
|
= |
{
1, 1
}
|
π(s)
|
= |
{
1, 1
}
|
π(incr)
|
= |
{
1
}
|
π(cons)
|
= |
{
2
}
|
π(mark)
|
= |
{
1
}
|
π(adx)
|
= |
{
1
}
|
π(active)
|
= |
{
1
}
|
to remove the pairs:
mark#(tl(X)) |
→ |
mark#(X) |
(54) |
1.1.1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
0 |
|
stat(mark#) |
= |
lex
|
prec(active#) |
= |
0 |
|
stat(active#) |
= |
lex
|
prec(tl) |
= |
0 |
|
stat(tl) |
= |
lex
|
prec(hd) |
= |
0 |
|
stat(hd) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(incr) |
= |
0 |
|
stat(incr) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(adx) |
= |
0 |
|
stat(adx) |
= |
lex
|
prec(zeros) |
= |
1 |
|
stat(zeros) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(nats) |
= |
0 |
|
stat(nats) |
= |
lex
|
π(mark#) |
= |
1 |
π(active#) |
= |
1 |
π(tl) |
= |
[] |
π(hd) |
= |
1 |
π(s) |
= |
1 |
π(incr) |
= |
1 |
π(cons) |
= |
1 |
π(0) |
= |
[] |
π(mark) |
= |
1 |
π(adx) |
= |
[1] |
π(zeros) |
= |
[] |
π(active) |
= |
1 |
π(nats) |
= |
[] |
together with the usable
rules
active(zeros) |
→ |
mark(cons(0,zeros)) |
(2) |
active(incr(cons(X,Y))) |
→ |
mark(cons(s(X),incr(Y))) |
(3) |
active(adx(cons(X,Y))) |
→ |
mark(incr(cons(X,adx(Y)))) |
(4) |
mark(nats) |
→ |
active(nats) |
(7) |
mark(adx(X)) |
→ |
active(adx(mark(X))) |
(8) |
mark(zeros) |
→ |
active(zeros) |
(9) |
mark(cons(X1,X2)) |
→ |
active(cons(X1,X2)) |
(10) |
mark(0) |
→ |
active(0) |
(11) |
mark(incr(X)) |
→ |
active(incr(mark(X))) |
(12) |
mark(s(X)) |
→ |
active(s(X)) |
(13) |
mark(hd(X)) |
→ |
active(hd(mark(X))) |
(14) |
mark(tl(X)) |
→ |
active(tl(mark(X))) |
(15) |
adx(mark(X)) |
→ |
adx(X) |
(16) |
adx(active(X)) |
→ |
adx(X) |
(17) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(19) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(21) |
incr(mark(X)) |
→ |
incr(X) |
(22) |
incr(active(X)) |
→ |
incr(X) |
(23) |
s(mark(X)) |
→ |
s(X) |
(24) |
s(active(X)) |
→ |
s(X) |
(25) |
hd(mark(X)) |
→ |
hd(X) |
(26) |
hd(active(X)) |
→ |
hd(X) |
(27) |
tl(mark(X)) |
→ |
tl(X) |
(28) |
tl(active(X)) |
→ |
tl(X) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(adx(cons(X,Y))) |
→ |
mark#(incr(cons(X,adx(Y)))) |
(39) |
could be deleted.
1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(incr(X)) |
→ |
mark#(X) |
(47) |
1.1.1.1.1.1.2.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(incr(X)) |
→ |
mark#(X) |
(47) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
incr#(mark(X)) |
→ |
incr#(X) |
(61) |
incr#(active(X)) |
→ |
incr#(X) |
(62) |
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.
incr#(mark(X)) |
→ |
incr#(X) |
(61) |
|
1 |
> |
1 |
incr#(active(X)) |
→ |
incr#(X) |
(62) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(63) |
s#(active(X)) |
→ |
s#(X) |
(64) |
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.
s#(mark(X)) |
→ |
s#(X) |
(63) |
|
1 |
> |
1 |
s#(active(X)) |
→ |
s#(X) |
(64) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
adx#(mark(X)) |
→ |
adx#(X) |
(57) |
adx#(active(X)) |
→ |
adx#(X) |
(58) |
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.
adx#(mark(X)) |
→ |
adx#(X) |
(57) |
|
1 |
> |
1 |
adx#(active(X)) |
→ |
adx#(X) |
(58) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
hd#(mark(X)) |
→ |
hd#(X) |
(65) |
hd#(active(X)) |
→ |
hd#(X) |
(66) |
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.
hd#(mark(X)) |
→ |
hd#(X) |
(65) |
|
1 |
> |
1 |
hd#(active(X)) |
→ |
hd#(X) |
(66) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
tl#(mark(X)) |
→ |
tl#(X) |
(67) |
tl#(active(X)) |
→ |
tl#(X) |
(68) |
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.
tl#(mark(X)) |
→ |
tl#(X) |
(67) |
|
1 |
> |
1 |
tl#(active(X)) |
→ |
tl#(X) |
(68) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.