The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(45) |
active#(incr(cons(X,Y))) |
→ |
mark#(cons(s(X),incr(Y))) |
(32) |
mark#(adx(X)) |
→ |
active#(adx(mark(X))) |
(41) |
active#(adx(cons(X,Y))) |
→ |
mark#(incr(cons(X,adx(Y)))) |
(36) |
mark#(adx(X)) |
→ |
mark#(X) |
(43) |
mark#(zeros) |
→ |
active#(zeros) |
(44) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(30) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(47) |
mark#(incr(X)) |
→ |
mark#(X) |
(49) |
mark#(s(X)) |
→ |
active#(s(X)) |
(50) |
mark#(hd(X)) |
→ |
active#(hd(mark(X))) |
(51) |
mark#(hd(X)) |
→ |
mark#(X) |
(53) |
mark#(tl(X)) |
→ |
active#(tl(mark(X))) |
(54) |
mark#(tl(X)) |
→ |
mark#(X) |
(56) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[adx(x1)] |
= |
1 · x1
|
[nats] |
= |
0 |
[hd(x1)] |
= |
2 + 2 · x1
|
[tl(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(hd(X)) |
→ |
mark#(X) |
(53) |
and
no rules
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[adx(x1)] |
= |
1 · x1
|
[nats] |
= |
0 |
[hd(x1)] |
= |
1 · x1
|
[tl(x1)] |
= |
1 + 1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
the
pair
mark#(tl(X)) |
→ |
mark#(X) |
(56) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[adx(x1)] |
= |
1 + 2 · x1
|
[nats] |
= |
0 |
[hd(x1)] |
= |
1 · x1
|
[tl(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(adx(X)) |
→ |
mark#(X) |
(43) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
3 |
|
stat(mark#) |
= |
lex
|
prec(cons) |
= |
3 |
|
stat(cons) |
= |
lex
|
prec(incr) |
= |
3 |
|
stat(incr) |
= |
lex
|
prec(s) |
= |
3 |
|
stat(s) |
= |
lex
|
prec(adx) |
= |
3 |
|
stat(adx) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(zeros) |
= |
3 |
|
stat(zeros) |
= |
lex
|
prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
prec(hd) |
= |
3 |
|
stat(hd) |
= |
lex
|
prec(tl) |
= |
2 |
|
stat(tl) |
= |
lex
|
prec(nats) |
= |
4 |
|
stat(nats) |
= |
lex
|
π(mark#) |
= |
[] |
π(cons) |
= |
[] |
π(active#) |
= |
1 |
π(incr) |
= |
[] |
π(s) |
= |
[] |
π(adx) |
= |
[] |
π(mark) |
= |
[] |
π(zeros) |
= |
[] |
π(0) |
= |
[] |
π(hd) |
= |
[] |
π(tl) |
= |
[] |
π(active) |
= |
1 |
π(nats) |
= |
[] |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(19) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(18) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(20) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(21) |
s(active(X)) |
→ |
s(X) |
(25) |
s(mark(X)) |
→ |
s(X) |
(24) |
incr(active(X)) |
→ |
incr(X) |
(23) |
incr(mark(X)) |
→ |
incr(X) |
(22) |
adx(active(X)) |
→ |
adx(X) |
(17) |
adx(mark(X)) |
→ |
adx(X) |
(16) |
hd(active(X)) |
→ |
hd(X) |
(27) |
hd(mark(X)) |
→ |
hd(X) |
(26) |
tl(active(X)) |
→ |
tl(X) |
(29) |
tl(mark(X)) |
→ |
tl(X) |
(28) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(tl(X)) |
→ |
active#(tl(mark(X))) |
(54) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(cons) |
= |
6 |
|
weight(cons) |
= |
2 |
|
|
|
prec(adx) |
= |
7 |
|
weight(adx) |
= |
3 |
|
|
|
prec(zeros) |
= |
4 |
|
weight(zeros) |
= |
3 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(hd) |
= |
2 |
|
weight(hd) |
= |
2 |
|
|
|
prec(nats) |
= |
1 |
|
weight(nats) |
= |
2 |
|
|
|
prec(tl) |
= |
3 |
|
weight(tl) |
= |
2 |
|
|
|
prec(0) |
= |
5 |
|
weight(0) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(cons) |
= |
[] |
π(active#) |
= |
1 |
π(incr) |
= |
1 |
π(adx) |
= |
[] |
π(zeros) |
= |
[] |
π(mark) |
= |
1 |
π(s) |
= |
[] |
π(hd) |
= |
[] |
π(active) |
= |
1 |
π(nats) |
= |
[] |
π(tl) |
= |
[] |
π(0) |
= |
[] |
the
pairs
active#(adx(cons(X,Y))) |
→ |
mark#(incr(cons(X,adx(Y)))) |
(36) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(30) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
active#(incr(cons(X,Y))) |
→ |
mark#(cons(s(X),incr(Y))) |
(32) |
mark#(adx(X)) |
→ |
active#(adx(mark(X))) |
(41) |
mark#(cons(X1,X2)) |
→ |
active#(cons(X1,X2)) |
(45) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(47) |
mark#(incr(X)) |
→ |
mark#(X) |
(49) |
mark#(s(X)) |
→ |
active#(s(X)) |
(50) |
mark#(hd(X)) |
→ |
active#(hd(mark(X))) |
(51) |
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
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-1 + 2 · x1
|
[cons(x1, x2)] |
= |
0 |
[s(x1)] |
= |
2 + 2 · x1
|
[active(x1)] |
= |
1 |
[mark(x1)] |
= |
1 + x1
|
[incr(x1)] |
= |
1 + x1
|
[adx(x1)] |
= |
1 |
[hd(x1)] |
= |
2 + 2 · x1
|
[nats] |
= |
2 |
[zeros] |
= |
1 |
[0] |
= |
2 |
[tl(x1)] |
= |
2 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(19) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(18) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(20) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(21) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(adx(X)) |
→ |
active#(adx(mark(X))) |
(41) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(47) |
mark#(incr(X)) |
→ |
mark#(X) |
(49) |
mark#(s(X)) |
→ |
active#(s(X)) |
(50) |
mark#(hd(X)) |
→ |
active#(hd(mark(X))) |
(51) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 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
|
[active(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[incr(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(19) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(18) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(20) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(21) |
s(active(X)) |
→ |
s(X) |
(25) |
s(mark(X)) |
→ |
s(X) |
(24) |
incr(active(X)) |
→ |
incr(X) |
(23) |
incr(mark(X)) |
→ |
incr(X) |
(22) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 · x1
|
[s(x1)] |
= |
1 · x1
|
[incr(x1)] |
= |
1 + 2 · x1
|
[active#(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
together with the usable
rules
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(18) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(19) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(20) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(21) |
s(mark(X)) |
→ |
s(X) |
(24) |
s(active(X)) |
→ |
s(X) |
(25) |
incr(mark(X)) |
→ |
incr(X) |
(22) |
incr(active(X)) |
→ |
incr(X) |
(23) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(incr(cons(X,Y))) |
→ |
mark#(cons(s(X),incr(Y))) |
(32) |
and
the
rules
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(18) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(19) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(20) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(21) |
s(mark(X)) |
→ |
s(X) |
(24) |
s(active(X)) |
→ |
s(X) |
(25) |
incr(mark(X)) |
→ |
incr(X) |
(22) |
incr(active(X)) |
→ |
incr(X) |
(23) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 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)) |
→ |
active#(cons(X1,X2)) |
(45) |
|
1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
adx#(active(X)) |
→ |
adx#(X) |
(58) |
adx#(mark(X)) |
→ |
adx#(X) |
(57) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[adx#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
adx#(active(X)) |
→ |
adx#(X) |
(58) |
|
1 |
> |
1 |
adx#(mark(X)) |
→ |
adx#(X) |
(57) |
|
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) |
(60) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(59) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(61) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(62) |
1.1.1.1.1.3 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.
1.1.1.1.1.3.1 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) |
(60) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(59) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(61) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(62) |
|
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
incr#(active(X)) |
→ |
incr#(X) |
(64) |
incr#(mark(X)) |
→ |
incr#(X) |
(63) |
1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[incr#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
incr#(active(X)) |
→ |
incr#(X) |
(64) |
|
1 |
> |
1 |
incr#(mark(X)) |
→ |
incr#(X) |
(63) |
|
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) |
(66) |
s#(mark(X)) |
→ |
s#(X) |
(65) |
1.1.1.1.1.5 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.
1.1.1.1.1.5.1 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) |
(66) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(65) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
hd#(active(X)) |
→ |
hd#(X) |
(68) |
hd#(mark(X)) |
→ |
hd#(X) |
(67) |
1.1.1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[hd#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
hd#(active(X)) |
→ |
hd#(X) |
(68) |
|
1 |
> |
1 |
hd#(mark(X)) |
→ |
hd#(X) |
(67) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
tl#(active(X)) |
→ |
tl#(X) |
(70) |
tl#(mark(X)) |
→ |
tl#(X) |
(69) |
1.1.1.1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[tl#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tl#(active(X)) |
→ |
tl#(X) |
(70) |
|
1 |
> |
1 |
tl#(mark(X)) |
→ |
tl#(X) |
(69) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.