The rewrite relation of the following TRS is considered.
active(app(nil,YS)) |
→ |
mark(YS) |
(1) |
active(app(cons(X,XS),YS)) |
→ |
mark(cons(X,app(XS,YS))) |
(2) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(3) |
active(zWadr(nil,YS)) |
→ |
mark(nil) |
(4) |
active(zWadr(XS,nil)) |
→ |
mark(nil) |
(5) |
active(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(app(Y,cons(X,nil)),zWadr(XS,YS))) |
(6) |
active(prefix(L)) |
→ |
mark(cons(nil,zWadr(L,prefix(L)))) |
(7) |
active(app(X1,X2)) |
→ |
app(active(X1),X2) |
(8) |
active(app(X1,X2)) |
→ |
app(X1,active(X2)) |
(9) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(10) |
active(from(X)) |
→ |
from(active(X)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(12) |
active(zWadr(X1,X2)) |
→ |
zWadr(active(X1),X2) |
(13) |
active(zWadr(X1,X2)) |
→ |
zWadr(X1,active(X2)) |
(14) |
active(prefix(X)) |
→ |
prefix(active(X)) |
(15) |
app(mark(X1),X2) |
→ |
mark(app(X1,X2)) |
(16) |
app(X1,mark(X2)) |
→ |
mark(app(X1,X2)) |
(17) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(18) |
from(mark(X)) |
→ |
mark(from(X)) |
(19) |
s(mark(X)) |
→ |
mark(s(X)) |
(20) |
zWadr(mark(X1),X2) |
→ |
mark(zWadr(X1,X2)) |
(21) |
zWadr(X1,mark(X2)) |
→ |
mark(zWadr(X1,X2)) |
(22) |
prefix(mark(X)) |
→ |
mark(prefix(X)) |
(23) |
proper(app(X1,X2)) |
→ |
app(proper(X1),proper(X2)) |
(24) |
proper(nil) |
→ |
ok(nil) |
(25) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(26) |
proper(from(X)) |
→ |
from(proper(X)) |
(27) |
proper(s(X)) |
→ |
s(proper(X)) |
(28) |
proper(zWadr(X1,X2)) |
→ |
zWadr(proper(X1),proper(X2)) |
(29) |
proper(prefix(X)) |
→ |
prefix(proper(X)) |
(30) |
app(ok(X1),ok(X2)) |
→ |
ok(app(X1,X2)) |
(31) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(32) |
from(ok(X)) |
→ |
ok(from(X)) |
(33) |
s(ok(X)) |
→ |
ok(s(X)) |
(34) |
zWadr(ok(X1),ok(X2)) |
→ |
ok(zWadr(X1,X2)) |
(35) |
prefix(ok(X)) |
→ |
ok(prefix(X)) |
(36) |
top(mark(X)) |
→ |
top(proper(X)) |
(37) |
top(ok(X)) |
→ |
top(active(X)) |
(38) |
active#(app(cons(X,XS),YS)) |
→ |
app#(XS,YS) |
(39) |
active#(app(cons(X,XS),YS)) |
→ |
cons#(X,app(XS,YS)) |
(40) |
active#(from(X)) |
→ |
s#(X) |
(41) |
active#(from(X)) |
→ |
from#(s(X)) |
(42) |
active#(from(X)) |
→ |
cons#(X,from(s(X))) |
(43) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
zWadr#(XS,YS) |
(44) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
cons#(X,nil) |
(45) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
app#(Y,cons(X,nil)) |
(46) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
cons#(app(Y,cons(X,nil)),zWadr(XS,YS)) |
(47) |
active#(prefix(L)) |
→ |
zWadr#(L,prefix(L)) |
(48) |
active#(prefix(L)) |
→ |
cons#(nil,zWadr(L,prefix(L))) |
(49) |
active#(app(X1,X2)) |
→ |
active#(X1) |
(50) |
active#(app(X1,X2)) |
→ |
app#(active(X1),X2) |
(51) |
active#(app(X1,X2)) |
→ |
active#(X2) |
(52) |
active#(app(X1,X2)) |
→ |
app#(X1,active(X2)) |
(53) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(54) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(55) |
active#(from(X)) |
→ |
active#(X) |
(56) |
active#(from(X)) |
→ |
from#(active(X)) |
(57) |
active#(s(X)) |
→ |
active#(X) |
(58) |
active#(s(X)) |
→ |
s#(active(X)) |
(59) |
active#(zWadr(X1,X2)) |
→ |
active#(X1) |
(60) |
active#(zWadr(X1,X2)) |
→ |
zWadr#(active(X1),X2) |
(61) |
active#(zWadr(X1,X2)) |
→ |
active#(X2) |
(62) |
active#(zWadr(X1,X2)) |
→ |
zWadr#(X1,active(X2)) |
(63) |
active#(prefix(X)) |
→ |
active#(X) |
(64) |
active#(prefix(X)) |
→ |
prefix#(active(X)) |
(65) |
app#(mark(X1),X2) |
→ |
app#(X1,X2) |
(66) |
app#(X1,mark(X2)) |
→ |
app#(X1,X2) |
(67) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
from#(mark(X)) |
→ |
from#(X) |
(69) |
s#(mark(X)) |
→ |
s#(X) |
(70) |
zWadr#(mark(X1),X2) |
→ |
zWadr#(X1,X2) |
(71) |
zWadr#(X1,mark(X2)) |
→ |
zWadr#(X1,X2) |
(72) |
prefix#(mark(X)) |
→ |
prefix#(X) |
(73) |
proper#(app(X1,X2)) |
→ |
proper#(X2) |
(74) |
proper#(app(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(app(X1,X2)) |
→ |
app#(proper(X1),proper(X2)) |
(76) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(77) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(78) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(79) |
proper#(from(X)) |
→ |
proper#(X) |
(80) |
proper#(from(X)) |
→ |
from#(proper(X)) |
(81) |
proper#(s(X)) |
→ |
proper#(X) |
(82) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(83) |
proper#(zWadr(X1,X2)) |
→ |
proper#(X2) |
(84) |
proper#(zWadr(X1,X2)) |
→ |
proper#(X1) |
(85) |
proper#(zWadr(X1,X2)) |
→ |
zWadr#(proper(X1),proper(X2)) |
(86) |
proper#(prefix(X)) |
→ |
proper#(X) |
(87) |
proper#(prefix(X)) |
→ |
prefix#(proper(X)) |
(88) |
app#(ok(X1),ok(X2)) |
→ |
app#(X1,X2) |
(89) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(90) |
from#(ok(X)) |
→ |
from#(X) |
(91) |
s#(ok(X)) |
→ |
s#(X) |
(92) |
zWadr#(ok(X1),ok(X2)) |
→ |
zWadr#(X1,X2) |
(93) |
prefix#(ok(X)) |
→ |
prefix#(X) |
(94) |
top#(mark(X)) |
→ |
proper#(X) |
(95) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(96) |
top#(ok(X)) |
→ |
active#(X) |
(97) |
top#(ok(X)) |
→ |
top#(active(X)) |
(98) |
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(98) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(96) |
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(prefix) |
= |
7 |
|
stat(prefix) |
= |
lex
|
prec(zWadr) |
= |
3 |
|
stat(zWadr) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(from) |
= |
3 |
|
stat(from) |
= |
lex
|
prec(cons) |
= |
2 |
|
stat(cons) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(app) |
= |
2 |
|
stat(app) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
1 |
π(proper) |
= |
1 |
π(prefix) |
= |
[1] |
π(zWadr) |
= |
[1,2] |
π(s) |
= |
1 |
π(from) |
= |
[1] |
π(cons) |
= |
[1] |
π(mark) |
= |
[1] |
π(active) |
= |
1 |
π(app) |
= |
[1,2] |
π(nil) |
= |
[] |
together with the usable
rules
active(app(nil,YS)) |
→ |
mark(YS) |
(1) |
active(app(cons(X,XS),YS)) |
→ |
mark(cons(X,app(XS,YS))) |
(2) |
active(from(X)) |
→ |
mark(cons(X,from(s(X)))) |
(3) |
active(zWadr(nil,YS)) |
→ |
mark(nil) |
(4) |
active(zWadr(XS,nil)) |
→ |
mark(nil) |
(5) |
active(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(app(Y,cons(X,nil)),zWadr(XS,YS))) |
(6) |
active(prefix(L)) |
→ |
mark(cons(nil,zWadr(L,prefix(L)))) |
(7) |
active(app(X1,X2)) |
→ |
app(active(X1),X2) |
(8) |
active(app(X1,X2)) |
→ |
app(X1,active(X2)) |
(9) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(10) |
active(from(X)) |
→ |
from(active(X)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(12) |
active(zWadr(X1,X2)) |
→ |
zWadr(active(X1),X2) |
(13) |
active(zWadr(X1,X2)) |
→ |
zWadr(X1,active(X2)) |
(14) |
active(prefix(X)) |
→ |
prefix(active(X)) |
(15) |
app(mark(X1),X2) |
→ |
mark(app(X1,X2)) |
(16) |
app(X1,mark(X2)) |
→ |
mark(app(X1,X2)) |
(17) |
app(ok(X1),ok(X2)) |
→ |
ok(app(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(18) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(32) |
s(mark(X)) |
→ |
mark(s(X)) |
(20) |
s(ok(X)) |
→ |
ok(s(X)) |
(34) |
from(mark(X)) |
→ |
mark(from(X)) |
(19) |
from(ok(X)) |
→ |
ok(from(X)) |
(33) |
zWadr(mark(X1),X2) |
→ |
mark(zWadr(X1,X2)) |
(21) |
zWadr(X1,mark(X2)) |
→ |
mark(zWadr(X1,X2)) |
(22) |
zWadr(ok(X1),ok(X2)) |
→ |
ok(zWadr(X1,X2)) |
(35) |
prefix(mark(X)) |
→ |
mark(prefix(X)) |
(23) |
prefix(ok(X)) |
→ |
ok(prefix(X)) |
(36) |
proper(app(X1,X2)) |
→ |
app(proper(X1),proper(X2)) |
(24) |
proper(nil) |
→ |
ok(nil) |
(25) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(26) |
proper(from(X)) |
→ |
from(proper(X)) |
(27) |
proper(s(X)) |
→ |
s(proper(X)) |
(28) |
proper(zWadr(X1,X2)) |
→ |
zWadr(proper(X1),proper(X2)) |
(29) |
proper(prefix(X)) |
→ |
prefix(proper(X)) |
(30) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(96) |
could be deleted.
1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
π(top#)
|
= |
{
1
}
|
π(ok)
|
= |
{
1, 1, 1
}
|
π(prefix)
|
= |
{
1
}
|
π(zWadr)
|
= |
{
1, 2
}
|
π(s)
|
= |
{
1, 1
}
|
π(from)
|
= |
{
1
}
|
π(cons)
|
= |
{
2
}
|
π(mark)
|
= |
{
1
}
|
π(active)
|
= |
{
1, 1
}
|
π(app)
|
= |
{
2
}
|
to remove the pairs:
top#(ok(X)) |
→ |
top#(active(X)) |
(98) |
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
proper#(prefix(X)) |
→ |
proper#(X) |
(87) |
proper#(zWadr(X1,X2)) |
→ |
proper#(X1) |
(85) |
proper#(zWadr(X1,X2)) |
→ |
proper#(X2) |
(84) |
proper#(s(X)) |
→ |
proper#(X) |
(82) |
proper#(from(X)) |
→ |
proper#(X) |
(80) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(78) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(77) |
proper#(app(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(app(X1,X2)) |
→ |
proper#(X2) |
(74) |
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#(prefix(X)) |
→ |
proper#(X) |
(87) |
|
1 |
> |
1 |
proper#(zWadr(X1,X2)) |
→ |
proper#(X1) |
(85) |
|
1 |
> |
1 |
proper#(zWadr(X1,X2)) |
→ |
proper#(X2) |
(84) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(82) |
|
1 |
> |
1 |
proper#(from(X)) |
→ |
proper#(X) |
(80) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(78) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(77) |
|
1 |
> |
1 |
proper#(app(X1,X2)) |
→ |
proper#(X1) |
(75) |
|
1 |
> |
1 |
proper#(app(X1,X2)) |
→ |
proper#(X2) |
(74) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
active#(prefix(X)) |
→ |
active#(X) |
(64) |
active#(zWadr(X1,X2)) |
→ |
active#(X2) |
(62) |
active#(zWadr(X1,X2)) |
→ |
active#(X1) |
(60) |
active#(s(X)) |
→ |
active#(X) |
(58) |
active#(from(X)) |
→ |
active#(X) |
(56) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(54) |
active#(app(X1,X2)) |
→ |
active#(X2) |
(52) |
active#(app(X1,X2)) |
→ |
active#(X1) |
(50) |
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#(prefix(X)) |
→ |
active#(X) |
(64) |
|
1 |
> |
1 |
active#(zWadr(X1,X2)) |
→ |
active#(X2) |
(62) |
|
1 |
> |
1 |
active#(zWadr(X1,X2)) |
→ |
active#(X1) |
(60) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(58) |
|
1 |
> |
1 |
active#(from(X)) |
→ |
active#(X) |
(56) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(54) |
|
1 |
> |
1 |
active#(app(X1,X2)) |
→ |
active#(X2) |
(52) |
|
1 |
> |
1 |
active#(app(X1,X2)) |
→ |
active#(X1) |
(50) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
app#(ok(X1),ok(X2)) |
→ |
app#(X1,X2) |
(89) |
app#(X1,mark(X2)) |
→ |
app#(X1,X2) |
(67) |
app#(mark(X1),X2) |
→ |
app#(X1,X2) |
(66) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
app#(ok(X1),ok(X2)) |
→ |
app#(X1,X2) |
(89) |
|
2 |
> |
2 |
1 |
> |
1 |
app#(X1,mark(X2)) |
→ |
app#(X1,X2) |
(67) |
|
2 |
> |
2 |
1 |
≥ |
1 |
app#(mark(X1),X2) |
→ |
app#(X1,X2) |
(66) |
|
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
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(90) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
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) |
(90) |
|
2 |
> |
2 |
1 |
> |
1 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
|
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
from#(ok(X)) |
→ |
from#(X) |
(91) |
from#(mark(X)) |
→ |
from#(X) |
(69) |
1.1.6 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) |
(91) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(69) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(92) |
s#(mark(X)) |
→ |
s#(X) |
(70) |
1.1.7 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) |
(92) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(70) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
zWadr#(ok(X1),ok(X2)) |
→ |
zWadr#(X1,X2) |
(93) |
zWadr#(X1,mark(X2)) |
→ |
zWadr#(X1,X2) |
(72) |
zWadr#(mark(X1),X2) |
→ |
zWadr#(X1,X2) |
(71) |
1.1.8 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
zWadr#(ok(X1),ok(X2)) |
→ |
zWadr#(X1,X2) |
(93) |
|
2 |
> |
2 |
1 |
> |
1 |
zWadr#(X1,mark(X2)) |
→ |
zWadr#(X1,X2) |
(72) |
|
2 |
> |
2 |
1 |
≥ |
1 |
zWadr#(mark(X1),X2) |
→ |
zWadr#(X1,X2) |
(71) |
|
2 |
≥ |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
prefix#(ok(X)) |
→ |
prefix#(X) |
(94) |
prefix#(mark(X)) |
→ |
prefix#(X) |
(73) |
1.1.9 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
prefix#(ok(X)) |
→ |
prefix#(X) |
(94) |
|
1 |
> |
1 |
prefix#(mark(X)) |
→ |
prefix#(X) |
(73) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.