The rewrite relation of the following TRS is considered.
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(34) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(24) |
mark#(from(X)) |
→ |
mark#(X) |
(36) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(37) |
active#(length(cons(X,Y))) |
→ |
mark#(s(length1(Y))) |
(29) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(39) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(40) |
active#(length1(X)) |
→ |
mark#(length(X)) |
(32) |
mark#(s(X)) |
→ |
mark#(X) |
(42) |
mark#(length(X)) |
→ |
active#(length(X)) |
(43) |
mark#(length1(X)) |
→ |
active#(length1(X)) |
(46) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
1 |
|
stat(mark#) |
= |
lex
|
prec(from) |
= |
1 |
|
stat(from) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(s) |
= |
1 |
|
stat(s) |
= |
lex
|
prec(length) |
= |
1 |
|
stat(length) |
= |
lex
|
prec(length1) |
= |
1 |
|
stat(length1) |
= |
lex
|
prec(nil) |
= |
2 |
|
stat(nil) |
= |
lex
|
prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
π(mark#) |
= |
[] |
π(from) |
= |
[] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
[] |
π(s) |
= |
[] |
π(length) |
= |
[] |
π(length1) |
= |
[] |
π(active) |
= |
1 |
π(nil) |
= |
[] |
π(0) |
= |
[] |
together with the usable
rules
from(active(X)) |
→ |
from(X) |
(13) |
from(mark(X)) |
→ |
from(X) |
(12) |
s(active(X)) |
→ |
s(X) |
(19) |
s(mark(X)) |
→ |
s(X) |
(18) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(15) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(14) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(16) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(17) |
length1(active(X)) |
→ |
length1(X) |
(23) |
length1(mark(X)) |
→ |
length1(X) |
(22) |
length(active(X)) |
→ |
length(X) |
(21) |
length(mark(X)) |
→ |
length(X) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(37) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[from(x1)] |
= |
1 |
[s(x1)] |
= |
-2 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
-2 |
[cons(x1, x2)] |
= |
-2 + x1
|
[length(x1)] |
= |
1 |
[length1(x1)] |
= |
1 |
[nil] |
= |
0 |
[0] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
from(active(X)) |
→ |
from(X) |
(13) |
from(mark(X)) |
→ |
from(X) |
(12) |
s(active(X)) |
→ |
s(X) |
(19) |
s(mark(X)) |
→ |
s(X) |
(18) |
length1(active(X)) |
→ |
length1(X) |
(23) |
length1(mark(X)) |
→ |
length1(X) |
(22) |
length(active(X)) |
→ |
length(X) |
(21) |
length(mark(X)) |
→ |
length(X) |
(20) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(40) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 + 2 · x1
|
[from(x1)] |
= |
2 + x1
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[cons(x1, x2)] |
= |
x1 |
[s(x1)] |
= |
2 · x1
|
[length(x1)] |
= |
0 |
[length1(x1)] |
= |
0 |
[nil] |
= |
0 |
[0] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
the
pairs
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(34) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(24) |
mark#(from(X)) |
→ |
mark#(X) |
(36) |
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[length1(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
length1(active(X)) |
→ |
length1(X) |
(23) |
length1(mark(X)) |
→ |
length1(X) |
(22) |
length(active(X)) |
→ |
length(X) |
(21) |
length(mark(X)) |
→ |
length(X) |
(20) |
s(active(X)) |
→ |
s(X) |
(19) |
s(mark(X)) |
→ |
s(X) |
(18) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[length1(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 · x1
|
[length(x1)] |
= |
2 · x1
|
[s(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
[mark#(x1)] |
= |
1 · x1
|
together with the usable
rules
length1(mark(X)) |
→ |
length1(X) |
(22) |
length1(active(X)) |
→ |
length1(X) |
(23) |
length(mark(X)) |
→ |
length(X) |
(20) |
length(active(X)) |
→ |
length(X) |
(21) |
s(mark(X)) |
→ |
s(X) |
(18) |
s(active(X)) |
→ |
s(X) |
(19) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(length(cons(X,Y))) |
→ |
mark#(s(length1(Y))) |
(29) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(39) |
and
the
rules
length1(mark(X)) |
→ |
length1(X) |
(22) |
length1(active(X)) |
→ |
length1(X) |
(23) |
length(mark(X)) |
→ |
length(X) |
(20) |
length(active(X)) |
→ |
length(X) |
(21) |
s(mark(X)) |
→ |
s(X) |
(18) |
s(active(X)) |
→ |
s(X) |
(19) |
could be deleted.
1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(s(X)) |
→ |
mark#(X) |
(42) |
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#(s(X)) |
→ |
mark#(X) |
(42) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
from#(active(X)) |
→ |
from#(X) |
(48) |
from#(mark(X)) |
→ |
from#(X) |
(47) |
1.1.2 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.
1.1.2.1 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) |
(48) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(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
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(50) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(49) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(52) |
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) |
(50) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(49) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(52) |
|
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
s#(active(X)) |
→ |
s#(X) |
(54) |
s#(mark(X)) |
→ |
s#(X) |
(53) |
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
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 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) |
(54) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(53) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(56) |
length#(mark(X)) |
→ |
length#(X) |
(55) |
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
|
[length#(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.
length#(active(X)) |
→ |
length#(X) |
(56) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
length1#(active(X)) |
→ |
length1#(X) |
(58) |
length1#(mark(X)) |
→ |
length1#(X) |
(57) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[length1#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length1#(active(X)) |
→ |
length1#(X) |
(58) |
|
1 |
> |
1 |
length1#(mark(X)) |
→ |
length1#(X) |
(57) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.