The rewrite relation of the following TRS is considered.
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
mark#(2nd(X)) |
→ |
active#(2nd(mark(X))) |
(31) |
active#(2nd(cons1(X,cons(Y,Z)))) |
→ |
mark#(Y) |
(23) |
mark#(2nd(X)) |
→ |
mark#(X) |
(33) |
mark#(cons1(X1,X2)) |
→ |
active#(cons1(mark(X1),mark(X2))) |
(34) |
active#(2nd(cons(X,X1))) |
→ |
mark#(2nd(cons1(X,X1))) |
(24) |
mark#(cons1(X1,X2)) |
→ |
mark#(X1) |
(36) |
mark#(cons1(X1,X2)) |
→ |
mark#(X2) |
(37) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(38) |
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(27) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(40) |
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(41) |
mark#(from(X)) |
→ |
mark#(X) |
(43) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(44) |
mark#(s(X)) |
→ |
mark#(X) |
(46) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
1 |
|
stat(mark#) |
= |
lex
|
prec(2nd) |
= |
1 |
|
stat(2nd) |
= |
lex
|
prec(mark) |
= |
1 |
|
stat(mark) |
= |
lex
|
prec(cons1) |
= |
1 |
|
stat(cons1) |
= |
lex
|
prec(cons) |
= |
1 |
|
stat(cons) |
= |
lex
|
prec(from) |
= |
1 |
|
stat(from) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(active) |
= |
1 |
|
stat(active) |
= |
lex
|
π(mark#) |
= |
[] |
π(2nd) |
= |
[] |
π(active#) |
= |
1 |
π(mark) |
= |
[] |
π(cons1) |
= |
[] |
π(cons) |
= |
[] |
π(from) |
= |
[] |
π(s) |
= |
[] |
π(active) |
= |
[] |
together with the usable
rules
2nd(active(X)) |
→ |
2nd(X) |
(10) |
2nd(mark(X)) |
→ |
2nd(X) |
(9) |
cons1(X1,mark(X2)) |
→ |
cons1(X1,X2) |
(12) |
cons1(mark(X1),X2) |
→ |
cons1(X1,X2) |
(11) |
cons1(active(X1),X2) |
→ |
cons1(X1,X2) |
(13) |
cons1(X1,active(X2)) |
→ |
cons1(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(16) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(15) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(17) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(18) |
s(active(X)) |
→ |
s(X) |
(22) |
s(mark(X)) |
→ |
s(X) |
(21) |
from(active(X)) |
→ |
from(X) |
(20) |
from(mark(X)) |
→ |
from(X) |
(19) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(44) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[2nd(x1)] |
= |
2 |
[active#(x1)] |
= |
x1 |
[cons(x1, x2)] |
= |
1 |
[cons1(x1, x2)] |
= |
-2 |
[from(x1)] |
= |
2 |
[mark(x1)] |
= |
-1 + 2 · x1
|
[active(x1)] |
= |
-2 |
[s(x1)] |
= |
-2 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
2nd(active(X)) |
→ |
2nd(X) |
(10) |
2nd(mark(X)) |
→ |
2nd(X) |
(9) |
cons1(X1,mark(X2)) |
→ |
cons1(X1,X2) |
(12) |
cons1(mark(X1),X2) |
→ |
cons1(X1,X2) |
(11) |
cons1(active(X1),X2) |
→ |
cons1(X1,X2) |
(13) |
cons1(X1,active(X2)) |
→ |
cons1(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(16) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(15) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(17) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(18) |
from(active(X)) |
→ |
from(X) |
(20) |
from(mark(X)) |
→ |
from(X) |
(19) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons1(X1,X2)) |
→ |
active#(cons1(mark(X1),mark(X2))) |
(34) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(38) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[mark#(x1)] |
= |
+ · x1
|
[2nd(x1)] |
= |
+ · x1
|
[active#(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[cons1(x1, x2)] |
= |
+ · x1 + · x2
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[from(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
the
pair
active#(2nd(cons1(X,cons(Y,Z)))) |
→ |
mark#(Y) |
(23) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[mark#(x1)] |
= |
+ · x1
|
[2nd(x1)] |
= |
+ · x1
|
[active#(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[cons1(x1, x2)] |
= |
+ · x1 + · x2
|
[from(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
the
pair
mark#(from(X)) |
→ |
mark#(X) |
(43) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[mark#(x1)] |
= |
+ · x1
|
[2nd(x1)] |
= |
+ · x1
|
[active#(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[cons1(x1, x2)] |
= |
+ · x1 + · x2
|
[from(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
the
pair
mark#(2nd(X)) |
→ |
mark#(X) |
(33) |
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[mark#(x1)] |
= |
+ · x1
|
[2nd(x1)] |
= |
+ · x1
|
[active#(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[cons1(x1, x2)] |
= |
+ · x1 + · x2
|
[from(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
the
pairs
mark#(cons1(X1,X2)) |
→ |
mark#(X1) |
(36) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(40) |
could be deleted.
1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(2nd) |
= |
0 |
|
weight(2nd) |
= |
1 |
|
|
|
prec(from) |
= |
2 |
|
weight(from) |
= |
3 |
|
|
|
prec(cons) |
= |
1 |
|
weight(cons) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(2nd) |
= |
[] |
π(active#) |
= |
1 |
π(cons1) |
= |
2 |
π(from) |
= |
[] |
π(cons) |
= |
[] |
π(s) |
= |
1 |
together with the usable
rules
2nd(active(X)) |
→ |
2nd(X) |
(10) |
2nd(mark(X)) |
→ |
2nd(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(20) |
from(mark(X)) |
→ |
from(X) |
(19) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(16) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(15) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(17) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(18) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(from(X)) |
→ |
mark#(cons(X,from(s(X)))) |
(27) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(2nd) |
= |
1 |
|
weight(2nd) |
= |
1 |
|
|
|
prec(from) |
= |
2 |
|
weight(from) |
= |
2 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(2nd) |
= |
[] |
π(active#) |
= |
1 |
π(cons1) |
= |
2 |
π(from) |
= |
[] |
π(s) |
= |
[1] |
together with the usable
rules
2nd(active(X)) |
→ |
2nd(X) |
(10) |
2nd(mark(X)) |
→ |
2nd(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(20) |
from(mark(X)) |
→ |
from(X) |
(19) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(s(X)) |
→ |
mark#(X) |
(46) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(2nd) |
= |
0 |
|
weight(2nd) |
= |
1 |
|
|
|
prec(cons1) |
= |
1 |
|
weight(cons1) |
= |
1 |
|
|
|
prec(from) |
= |
2 |
|
weight(from) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(2nd) |
= |
[] |
π(active#) |
= |
1 |
π(cons1) |
= |
[2] |
π(from) |
= |
[] |
together with the usable
rules
2nd(active(X)) |
→ |
2nd(X) |
(10) |
2nd(mark(X)) |
→ |
2nd(X) |
(9) |
from(active(X)) |
→ |
from(X) |
(20) |
from(mark(X)) |
→ |
from(X) |
(19) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(cons1(X1,X2)) |
→ |
mark#(X2) |
(37) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[2nd(x1)] |
= |
1 |
[active#(x1)] |
= |
0 |
[from(x1)] |
= |
2 |
[mark(x1)] |
= |
2 |
[active(x1)] |
= |
2 |
[cons1(x1, x2)] |
= |
-2 + x1
|
[cons(x1, x2)] |
= |
2 |
[s(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 + 2 · x1
|
together with the usable
rules
2nd(active(X)) |
→ |
2nd(X) |
(10) |
2nd(mark(X)) |
→ |
2nd(X) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(from(X)) |
→ |
active#(from(mark(X))) |
(41) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the rationals with delta = 1/4
[mark#(x1)] |
= |
1/2 + 4 · x1
|
[2nd(x1)] |
= |
0 + 1 · x1
|
[active#(x1)] |
= |
1/4 + 4 · x1
|
[mark(x1)] |
= |
0 + 1 · x1
|
[cons(x1, x2)] |
= |
2 + 4 · x1 + 1/4 · x2
|
[cons1(x1, x2)] |
= |
1 + 0 · x1 + 1/4 · x2
|
[active(x1)] |
= |
0 + 1 · x1
|
[from(x1)] |
= |
4 + 4 · x1
|
[s(x1)] |
= |
1 + 0 · x1
|
the
pairs
mark#(2nd(X)) |
→ |
active#(2nd(mark(X))) |
(31) |
active#(2nd(cons(X,X1))) |
→ |
mark#(2nd(cons1(X,X1))) |
(24) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
2nd#(active(X)) |
→ |
2nd#(X) |
(48) |
2nd#(mark(X)) |
→ |
2nd#(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
|
[2nd#(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.
2nd#(active(X)) |
→ |
2nd#(X) |
(48) |
|
1 |
> |
1 |
2nd#(mark(X)) |
→ |
2nd#(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
cons1#(X1,mark(X2)) |
→ |
cons1#(X1,X2) |
(50) |
cons1#(mark(X1),X2) |
→ |
cons1#(X1,X2) |
(49) |
cons1#(active(X1),X2) |
→ |
cons1#(X1,X2) |
(51) |
cons1#(X1,active(X2)) |
→ |
cons1#(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
|
[cons1#(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.
cons1#(X1,mark(X2)) |
→ |
cons1#(X1,X2) |
(50) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons1#(mark(X1),X2) |
→ |
cons1#(X1,X2) |
(49) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons1#(active(X1),X2) |
→ |
cons1#(X1,X2) |
(51) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons1#(X1,active(X2)) |
→ |
cons1#(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
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(54) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(53) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(55) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(56) |
1.1.4 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.
1.1.4.1 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) |
(54) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(53) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(55) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(56) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
from#(active(X)) |
→ |
from#(X) |
(58) |
from#(mark(X)) |
→ |
from#(X) |
(57) |
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
|
[from#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 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) |
(58) |
|
1 |
> |
1 |
from#(mark(X)) |
→ |
from#(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
s#(active(X)) |
→ |
s#(X) |
(60) |
s#(mark(X)) |
→ |
s#(X) |
(59) |
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
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 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) |
(60) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(59) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.