The rewrite relation of the following TRS is considered.
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(74) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(72) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[nats] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
1 · x1
|
[pairs] |
= |
0 |
[odds] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[head(x1)] |
= |
1 · x1
|
[tail(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
proper(nats) |
→ |
ok(nats) |
(17) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(18) |
proper(0) |
→ |
ok(0) |
(19) |
proper(incr(X)) |
→ |
incr(proper(X)) |
(20) |
proper(pairs) |
→ |
ok(pairs) |
(21) |
proper(odds) |
→ |
ok(odds) |
(22) |
proper(s(X)) |
→ |
s(proper(X)) |
(23) |
proper(head(X)) |
→ |
head(proper(X)) |
(24) |
proper(tail(X)) |
→ |
tail(proper(X)) |
(25) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
s(mark(X)) |
→ |
mark(s(X)) |
(14) |
s(ok(X)) |
→ |
ok(s(X)) |
(28) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
active(head(X)) |
→ |
head(active(X)) |
(10) |
active(tail(X)) |
→ |
tail(active(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(mark) |
= |
1 |
|
stat(mark) |
= |
lex
|
prec(nats) |
= |
4 |
|
stat(nats) |
= |
lex
|
prec(cons) |
= |
1 |
|
stat(cons) |
= |
lex
|
prec(0) |
= |
2 |
|
stat(0) |
= |
lex
|
prec(incr) |
= |
3 |
|
stat(incr) |
= |
lex
|
prec(pairs) |
= |
5 |
|
stat(pairs) |
= |
lex
|
prec(odds) |
= |
6 |
|
stat(odds) |
= |
lex
|
prec(s) |
= |
3 |
|
stat(s) |
= |
lex
|
prec(head) |
= |
7 |
|
stat(head) |
= |
lex
|
π(top#) |
= |
[1] |
π(ok) |
= |
1 |
π(active) |
= |
1 |
π(mark) |
= |
[1] |
π(proper) |
= |
1 |
π(nats) |
= |
[] |
π(cons) |
= |
[1] |
π(0) |
= |
[] |
π(incr) |
= |
[1] |
π(pairs) |
= |
[] |
π(odds) |
= |
[] |
π(s) |
= |
[1] |
π(head) |
= |
[1] |
π(tail) |
= |
1 |
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(72) |
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
[active(x1)] |
= |
1 · x1
|
[nats] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
1 · x1
|
[pairs] |
= |
0 |
[odds] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[head(x1)] |
= |
1 · x1
|
[tail(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
active(head(X)) |
→ |
head(active(X)) |
(10) |
active(tail(X)) |
→ |
tail(active(X)) |
(11) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
s(mark(X)) |
→ |
mark(s(X)) |
(14) |
s(ok(X)) |
→ |
ok(s(X)) |
(28) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(nats) |
= |
8 |
|
weight(nats) |
= |
1 |
|
|
|
prec(0) |
= |
5 |
|
weight(0) |
= |
1 |
|
|
|
prec(pairs) |
= |
7 |
|
weight(pairs) |
= |
3 |
|
|
|
prec(odds) |
= |
2 |
|
weight(odds) |
= |
1 |
|
|
|
prec(active) |
= |
11 |
|
weight(active) |
= |
5 |
|
|
|
prec(mark) |
= |
0 |
|
weight(mark) |
= |
1 |
|
|
|
prec(incr) |
= |
6 |
|
weight(incr) |
= |
2 |
|
|
|
prec(s) |
= |
9 |
|
weight(s) |
= |
3 |
|
|
|
prec(head) |
= |
4 |
|
weight(head) |
= |
7 |
|
|
|
prec(tail) |
= |
10 |
|
weight(tail) |
= |
7 |
|
|
|
prec(ok) |
= |
3 |
|
weight(ok) |
= |
6 |
|
|
|
prec(top#) |
= |
12 |
|
weight(top#) |
= |
1 |
|
|
|
prec(cons) |
= |
1 |
|
weight(cons) |
= |
0 |
|
|
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(74) |
and
the
rules
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
active(head(X)) |
→ |
head(active(X)) |
(10) |
active(tail(X)) |
→ |
tail(active(X)) |
(11) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
s(mark(X)) |
→ |
mark(s(X)) |
(14) |
s(ok(X)) |
→ |
ok(s(X)) |
(28) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
could be deleted.
1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(incr(X)) |
→ |
active#(X) |
(44) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(42) |
active#(s(X)) |
→ |
active#(X) |
(46) |
active#(head(X)) |
→ |
active#(X) |
(48) |
active#(tail(X)) |
→ |
active#(X) |
(50) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[incr(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[head(x1)] |
= |
1 · x1
|
[tail(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(incr(X)) |
→ |
active#(X) |
(44) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(42) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(46) |
|
1 |
> |
1 |
active#(head(X)) |
→ |
active#(X) |
(48) |
|
1 |
> |
1 |
active#(tail(X)) |
→ |
active#(X) |
(50) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(58) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(57) |
proper#(incr(X)) |
→ |
proper#(X) |
(60) |
proper#(s(X)) |
→ |
proper#(X) |
(62) |
proper#(head(X)) |
→ |
proper#(X) |
(64) |
proper#(tail(X)) |
→ |
proper#(X) |
(66) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[incr(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[head(x1)] |
= |
1 · x1
|
[tail(x1)] |
= |
1 · x1
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(58) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(57) |
|
1 |
> |
1 |
proper#(incr(X)) |
→ |
proper#(X) |
(60) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(62) |
|
1 |
> |
1 |
proper#(head(X)) |
→ |
proper#(X) |
(64) |
|
1 |
> |
1 |
proper#(tail(X)) |
→ |
proper#(X) |
(66) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(67) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(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.
1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(67) |
|
1 |
> |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(51) |
|
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
incr#(ok(X)) |
→ |
incr#(X) |
(68) |
incr#(mark(X)) |
→ |
incr#(X) |
(52) |
1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
incr#(ok(X)) |
→ |
incr#(X) |
(68) |
|
1 |
> |
1 |
incr#(mark(X)) |
→ |
incr#(X) |
(52) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(69) |
s#(mark(X)) |
→ |
s#(X) |
(53) |
1.1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(69) |
|
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
7th
component contains the
pair
head#(ok(X)) |
→ |
head#(X) |
(70) |
head#(mark(X)) |
→ |
head#(X) |
(54) |
1.1.1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
head#(ok(X)) |
→ |
head#(X) |
(70) |
|
1 |
> |
1 |
head#(mark(X)) |
→ |
head#(X) |
(54) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
tail#(ok(X)) |
→ |
tail#(X) |
(71) |
tail#(mark(X)) |
→ |
tail#(X) |
(55) |
1.1.1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tail#(ok(X)) |
→ |
tail#(X) |
(71) |
|
1 |
> |
1 |
tail#(mark(X)) |
→ |
tail#(X) |
(55) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.