The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
active#(incr(cons(X,L))) |
→ |
mark#(cons(s(X),incr(L))) |
(34) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(44) |
active#(adx(cons(X,L))) |
→ |
mark#(incr(cons(X,adx(L)))) |
(38) |
mark#(incr(X)) |
→ |
mark#(X) |
(46) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(48) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(50) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(51) |
mark#(s(X)) |
→ |
mark#(X) |
(53) |
mark#(adx(X)) |
→ |
active#(adx(mark(X))) |
(54) |
mark#(adx(X)) |
→ |
mark#(X) |
(56) |
mark#(zeros) |
→ |
active#(zeros) |
(58) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(42) |
mark#(head(X)) |
→ |
active#(head(mark(X))) |
(60) |
mark#(head(X)) |
→ |
mark#(X) |
(62) |
mark#(tail(X)) |
→ |
active#(tail(mark(X))) |
(63) |
mark#(tail(X)) |
→ |
mark#(X) |
(65) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[incr(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[s(x1)] |
= |
1 · x1
|
[adx(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[nats] |
= |
0 |
[head(x1)] |
= |
2 + 1 · x1
|
[tail(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
the
pair
mark#(head(X)) |
→ |
mark#(X) |
(62) |
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
|
[incr(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[s(x1)] |
= |
1 · x1
|
[adx(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[nats] |
= |
0 |
[head(x1)] |
= |
1 · x1
|
[tail(x1)] |
= |
2 + 2 · x1
|
[active#(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
the
pair
mark#(tail(X)) |
→ |
mark#(X) |
(65) |
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
|
[incr(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[adx(x1)] |
= |
1 + 1 · x1
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[nats] |
= |
0 |
[head(x1)] |
= |
1 · x1
|
[tail(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
the
pair
mark#(adx(X)) |
→ |
mark#(X) |
(56) |
and
no rules
could be deleted.
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(zeros) |
= |
2 |
|
weight(zeros) |
= |
3 |
|
|
|
prec(0) |
= |
3 |
|
weight(0) |
= |
2 |
|
|
|
prec(head) |
= |
4 |
|
weight(head) |
= |
2 |
|
|
|
prec(tail) |
= |
0 |
|
weight(tail) |
= |
1 |
|
|
|
prec(nil) |
= |
5 |
|
weight(nil) |
= |
2 |
|
|
|
prec(nats) |
= |
1 |
|
weight(nats) |
= |
2 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(incr) |
= |
1 |
π(cons) |
= |
1 |
π(mark#) |
= |
1 |
π(s) |
= |
1 |
π(mark) |
= |
1 |
π(adx) |
= |
1 |
π(zeros) |
= |
[] |
π(0) |
= |
[] |
π(head) |
= |
[] |
π(tail) |
= |
[] |
π(active) |
= |
1 |
π(nil) |
= |
[] |
π(nats) |
= |
[] |
the
pair
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(42) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
incr#(active(X)) |
→ |
incr#(X) |
(67) |
incr#(mark(X)) |
→ |
incr#(X) |
(66) |
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
|
[incr#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 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) |
(67) |
|
1 |
> |
1 |
incr#(mark(X)) |
→ |
incr#(X) |
(66) |
|
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) |
(69) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(70) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(71) |
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) |
(69) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(68) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(70) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(71) |
|
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) |
(73) |
s#(mark(X)) |
→ |
s#(X) |
(72) |
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
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.4.1 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) |
(73) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(72) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
adx#(active(X)) |
→ |
adx#(X) |
(75) |
adx#(mark(X)) |
→ |
adx#(X) |
(74) |
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
|
[adx#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.5.1 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) |
(75) |
|
1 |
> |
1 |
adx#(mark(X)) |
→ |
adx#(X) |
(74) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
head#(active(X)) |
→ |
head#(X) |
(77) |
head#(mark(X)) |
→ |
head#(X) |
(76) |
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
|
[head#(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.
head#(active(X)) |
→ |
head#(X) |
(77) |
|
1 |
> |
1 |
head#(mark(X)) |
→ |
head#(X) |
(76) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
tail#(active(X)) |
→ |
tail#(X) |
(79) |
tail#(mark(X)) |
→ |
tail#(X) |
(78) |
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
|
[tail#(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.
tail#(active(X)) |
→ |
tail#(X) |
(79) |
|
1 |
> |
1 |
tail#(mark(X)) |
→ |
tail#(X) |
(78) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.