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) |
mark(app(X1,X2)) |
→ |
active(app(mark(X1),mark(X2))) |
(8) |
mark(nil) |
→ |
active(nil) |
(9) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(10) |
mark(from(X)) |
→ |
active(from(mark(X))) |
(11) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(12) |
mark(zWadr(X1,X2)) |
→ |
active(zWadr(mark(X1),mark(X2))) |
(13) |
mark(prefix(X)) |
→ |
active(prefix(mark(X))) |
(14) |
app(mark(X1),X2) |
→ |
app(X1,X2) |
(15) |
app(X1,mark(X2)) |
→ |
app(X1,X2) |
(16) |
app(active(X1),X2) |
→ |
app(X1,X2) |
(17) |
app(X1,active(X2)) |
→ |
app(X1,X2) |
(18) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(19) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(20) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(21) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(22) |
from(mark(X)) |
→ |
from(X) |
(23) |
from(active(X)) |
→ |
from(X) |
(24) |
s(mark(X)) |
→ |
s(X) |
(25) |
s(active(X)) |
→ |
s(X) |
(26) |
zWadr(mark(X1),X2) |
→ |
zWadr(X1,X2) |
(27) |
zWadr(X1,mark(X2)) |
→ |
zWadr(X1,X2) |
(28) |
zWadr(active(X1),X2) |
→ |
zWadr(X1,X2) |
(29) |
zWadr(X1,active(X2)) |
→ |
zWadr(X1,X2) |
(30) |
prefix(mark(X)) |
→ |
prefix(X) |
(31) |
prefix(active(X)) |
→ |
prefix(X) |
(32) |
active#(app(nil,YS)) |
→ |
mark#(YS) |
(33) |
active#(app(cons(X,XS),YS)) |
→ |
mark#(cons(X,app(XS,YS))) |
(34) |
active#(app(cons(X,XS),YS)) |
→ |
cons#(X,app(XS,YS)) |
(35) |
active#(app(cons(X,XS),YS)) |
→ |
app#(XS,YS) |
(36) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(37) |
active#(from(X)) |
→ |
cons#(X,from(s(X))) |
(38) |
active#(from(X)) |
→ |
from#(s(X)) |
(39) |
active#(from(X)) |
→ |
s#(X) |
(40) |
active#(zWadr(nil,YS)) |
→ |
mark#(nil) |
(41) |
active#(zWadr(XS,nil)) |
→ |
mark#(nil) |
(42) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
mark#(cons(app(Y,cons(X,nil)),zWadr(XS,YS))) |
(43) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
cons#(app(Y,cons(X,nil)),zWadr(XS,YS)) |
(44) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
app#(Y,cons(X,nil)) |
(45) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
cons#(X,nil) |
(46) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
zWadr#(XS,YS) |
(47) |
active#(prefix(L)) |
→ |
mark#(cons(nil,zWadr(L,prefix(L)))) |
(48) |
active#(prefix(L)) |
→ |
cons#(nil,zWadr(L,prefix(L))) |
(49) |
active#(prefix(L)) |
→ |
zWadr#(L,prefix(L)) |
(50) |
mark#(app(X1,X2)) |
→ |
active#(app(mark(X1),mark(X2))) |
(51) |
mark#(app(X1,X2)) |
→ |
app#(mark(X1),mark(X2)) |
(52) |
mark#(app(X1,X2)) |
→ |
mark#(X1) |
(53) |
mark#(app(X1,X2)) |
→ |
mark#(X2) |
(54) |
mark#(nil) |
→ |
active#(nil) |
(55) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(56) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(57) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(58) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(59) |
mark#(from(X)) |
→ |
from#(mark(X)) |
(60) |
mark#(from(X)) |
→ |
mark#(X) |
(61) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(62) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(63) |
mark#(s(X)) |
→ |
mark#(X) |
(64) |
mark#(zWadr(X1,X2)) |
→ |
active#(zWadr(mark(X1),mark(X2))) |
(65) |
mark#(zWadr(X1,X2)) |
→ |
zWadr#(mark(X1),mark(X2)) |
(66) |
mark#(zWadr(X1,X2)) |
→ |
mark#(X1) |
(67) |
mark#(zWadr(X1,X2)) |
→ |
mark#(X2) |
(68) |
mark#(prefix(X)) |
→ |
active#(prefix(mark(X))) |
(69) |
mark#(prefix(X)) |
→ |
prefix#(mark(X)) |
(70) |
mark#(prefix(X)) |
→ |
mark#(X) |
(71) |
app#(mark(X1),X2) |
→ |
app#(X1,X2) |
(72) |
app#(X1,mark(X2)) |
→ |
app#(X1,X2) |
(73) |
app#(active(X1),X2) |
→ |
app#(X1,X2) |
(74) |
app#(X1,active(X2)) |
→ |
app#(X1,X2) |
(75) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(76) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(77) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(78) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(79) |
from#(mark(X)) |
→ |
from#(X) |
(80) |
from#(active(X)) |
→ |
from#(X) |
(81) |
s#(mark(X)) |
→ |
s#(X) |
(82) |
s#(active(X)) |
→ |
s#(X) |
(83) |
zWadr#(mark(X1),X2) |
→ |
zWadr#(X1,X2) |
(84) |
zWadr#(X1,mark(X2)) |
→ |
zWadr#(X1,X2) |
(85) |
zWadr#(active(X1),X2) |
→ |
zWadr#(X1,X2) |
(86) |
zWadr#(X1,active(X2)) |
→ |
zWadr#(X1,X2) |
(87) |
prefix#(mark(X)) |
→ |
prefix#(X) |
(88) |
prefix#(active(X)) |
→ |
prefix#(X) |
(89) |
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
mark#(app(X1,X2)) |
→ |
active#(app(mark(X1),mark(X2))) |
(51) |
active#(app(nil,YS)) |
→ |
mark#(YS) |
(33) |
mark#(app(X1,X2)) |
→ |
mark#(X1) |
(53) |
mark#(app(X1,X2)) |
→ |
mark#(X2) |
(54) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(56) |
active#(app(cons(X,XS),YS)) |
→ |
mark#(cons(X,app(XS,YS))) |
(34) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(58) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(59) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(37) |
mark#(from(X)) |
→ |
mark#(X) |
(61) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(62) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
mark#(cons(app(Y,cons(X,nil)),zWadr(XS,YS))) |
(43) |
mark#(s(X)) |
→ |
mark#(X) |
(64) |
mark#(zWadr(X1,X2)) |
→ |
active#(zWadr(mark(X1),mark(X2))) |
(65) |
active#(prefix(L)) |
→ |
mark#(cons(nil,zWadr(L,prefix(L)))) |
(48) |
mark#(zWadr(X1,X2)) |
→ |
mark#(X1) |
(67) |
mark#(zWadr(X1,X2)) |
→ |
mark#(X2) |
(68) |
mark#(prefix(X)) |
→ |
active#(prefix(mark(X))) |
(69) |
mark#(prefix(X)) |
→ |
mark#(X) |
(71) |
1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(app) |
= |
1 |
|
weight(app) |
= |
1 |
|
|
|
prec(nil) |
= |
2 |
|
weight(nil) |
= |
1 |
|
|
|
prec(zWadr) |
= |
0 |
|
weight(zWadr) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(app) |
= |
[1,2] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(nil) |
= |
[] |
π(cons) |
= |
1 |
π(from) |
= |
1 |
π(s) |
= |
1 |
π(zWadr) |
= |
[1,2] |
π(prefix) |
= |
1 |
π(active) |
= |
1 |
the
pairs
active#(app(nil,YS)) |
→ |
mark#(YS) |
(33) |
mark#(app(X1,X2)) |
→ |
mark#(X1) |
(53) |
mark#(app(X1,X2)) |
→ |
mark#(X2) |
(54) |
active#(app(cons(X,XS),YS)) |
→ |
mark#(cons(X,app(XS,YS))) |
(34) |
active#(zWadr(cons(X,XS),cons(Y,YS))) |
→ |
mark#(cons(app(Y,cons(X,nil)),zWadr(XS,YS))) |
(43) |
mark#(zWadr(X1,X2)) |
→ |
mark#(X1) |
(67) |
mark#(zWadr(X1,X2)) |
→ |
mark#(X2) |
(68) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(app) |
= |
1 |
|
weight(app) |
= |
1 |
|
|
|
prec(zWadr) |
= |
0 |
|
weight(zWadr) |
= |
2 |
|
|
|
prec(prefix) |
= |
2 |
|
weight(prefix) |
= |
1 |
|
|
|
prec(nil) |
= |
3 |
|
weight(nil) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(app) |
= |
[1,2] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
1 |
π(from) |
= |
1 |
π(s) |
= |
1 |
π(zWadr) |
= |
[1,2] |
π(prefix) |
= |
[1] |
π(nil) |
= |
[] |
π(active) |
= |
1 |
the
pairs
active#(prefix(L)) |
→ |
mark#(cons(nil,zWadr(L,prefix(L)))) |
(48) |
mark#(prefix(X)) |
→ |
mark#(X) |
(71) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(app) |
= |
2 |
|
weight(app) |
= |
1 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(zWadr) |
= |
1 |
|
weight(zWadr) |
= |
2 |
|
|
|
prec(prefix) |
= |
4 |
|
weight(prefix) |
= |
1 |
|
|
|
prec(nil) |
= |
3 |
|
weight(nil) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(app) |
= |
[1,2] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
1 |
π(from) |
= |
1 |
π(s) |
= |
[1] |
π(zWadr) |
= |
[1,2] |
π(prefix) |
= |
[] |
π(active) |
= |
1 |
π(nil) |
= |
[] |
the
pair
mark#(s(X)) |
→ |
mark#(X) |
(64) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(app) |
= |
2 |
|
weight(app) |
= |
1 |
|
|
|
prec(from) |
= |
4 |
|
weight(from) |
= |
1 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(zWadr) |
= |
1 |
|
weight(zWadr) |
= |
2 |
|
|
|
prec(prefix) |
= |
3 |
|
weight(prefix) |
= |
3 |
|
|
|
prec(nil) |
= |
5 |
|
weight(nil) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(app) |
= |
[1,2] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
1 |
π(from) |
= |
[1] |
π(s) |
= |
[] |
π(zWadr) |
= |
[1,2] |
π(prefix) |
= |
[] |
π(active) |
= |
1 |
π(nil) |
= |
[] |
the
pairs
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(37) |
mark#(from(X)) |
→ |
mark#(X) |
(61) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(58) |
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(58) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
app#(X1,mark(X2)) |
→ |
app#(X1,X2) |
(73) |
app#(mark(X1),X2) |
→ |
app#(X1,X2) |
(72) |
app#(active(X1),X2) |
→ |
app#(X1,X2) |
(74) |
app#(X1,active(X2)) |
→ |
app#(X1,X2) |
(75) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[app#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
app#(X1,mark(X2)) |
→ |
app#(X1,X2) |
(73) |
|
1 |
≥ |
1 |
2 |
> |
2 |
app#(mark(X1),X2) |
→ |
app#(X1,X2) |
(72) |
|
1 |
> |
1 |
2 |
≥ |
2 |
app#(active(X1),X2) |
→ |
app#(X1,X2) |
(74) |
|
1 |
> |
1 |
2 |
≥ |
2 |
app#(X1,active(X2)) |
→ |
app#(X1,X2) |
(75) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(77) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(76) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(78) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(79) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 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) |
(77) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(76) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(78) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(79) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
from#(active(X)) |
→ |
from#(X) |
(81) |
from#(mark(X)) |
→ |
from#(X) |
(80) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[from#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
from#(active(X)) |
→ |
from#(X) |
(81) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(X) |
(80) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(83) |
s#(mark(X)) |
→ |
s#(X) |
(82) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(83) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(82) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
zWadr#(X1,mark(X2)) |
→ |
zWadr#(X1,X2) |
(85) |
zWadr#(mark(X1),X2) |
→ |
zWadr#(X1,X2) |
(84) |
zWadr#(active(X1),X2) |
→ |
zWadr#(X1,X2) |
(86) |
zWadr#(X1,active(X2)) |
→ |
zWadr#(X1,X2) |
(87) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[zWadr#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
zWadr#(X1,mark(X2)) |
→ |
zWadr#(X1,X2) |
(85) |
|
1 |
≥ |
1 |
2 |
> |
2 |
zWadr#(mark(X1),X2) |
→ |
zWadr#(X1,X2) |
(84) |
|
1 |
> |
1 |
2 |
≥ |
2 |
zWadr#(active(X1),X2) |
→ |
zWadr#(X1,X2) |
(86) |
|
1 |
> |
1 |
2 |
≥ |
2 |
zWadr#(X1,active(X2)) |
→ |
zWadr#(X1,X2) |
(87) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
prefix#(active(X)) |
→ |
prefix#(X) |
(89) |
prefix#(mark(X)) |
→ |
prefix#(X) |
(88) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[prefix#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
prefix#(active(X)) |
→ |
prefix#(X) |
(89) |
|
1 |
> |
1 |
prefix#(mark(X)) |
→ |
prefix#(X) |
(88) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.