The rewrite relation of the following TRS is considered.
The dependency pairs are split into 8
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(50) |
top#(ok(X)) |
→ |
top#(active(X)) |
(59) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(s) |
= |
1 |
π(head#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(proper#) |
= |
1 |
π(active) |
= |
1 |
π(head) |
= |
1 |
π(active#) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(incr) |
= |
2 |
|
status(incr) |
= |
[1] |
|
list-extension(incr) |
= |
Lex |
prec(cons#) |
= |
0 |
|
status(cons#) |
= |
[1, 2] |
|
list-extension(cons#) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(pairs) |
= |
8 |
|
status(pairs) |
= |
[] |
|
list-extension(pairs) |
= |
Lex |
prec(top#) |
= |
0 |
|
status(top#) |
= |
[1] |
|
list-extension(top#) |
= |
Lex |
prec(tail) |
= |
7 |
|
status(tail) |
= |
[1] |
|
list-extension(tail) |
= |
Lex |
prec(0) |
= |
6 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(odds) |
= |
9 |
|
status(odds) |
= |
[] |
|
list-extension(odds) |
= |
Lex |
prec(tail#) |
= |
0 |
|
status(tail#) |
= |
[] |
|
list-extension(tail#) |
= |
Lex |
prec(mark) |
= |
2 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(incr#) |
= |
0 |
|
status(incr#) |
= |
[] |
|
list-extension(incr#) |
= |
Lex |
prec(nats) |
= |
5 |
|
status(nats) |
= |
[] |
|
list-extension(nats) |
= |
Lex |
prec(cons) |
= |
4 |
|
status(cons) |
= |
[1] |
|
list-extension(cons) |
= |
Lex |
and the following
Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
1 |
[pairs] |
=
|
19265 |
[top#(x1)] |
=
|
x1 + 1 |
[tail(x1)] |
=
|
x1 + 29260 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
1 |
[odds] |
=
|
19265 |
[tail#(x1)] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
1 |
[nats] |
=
|
36251 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
together with the usable
rules
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(18) |
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(4) |
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
proper(pairs) |
→ |
ok(pairs) |
(21) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
s(ok(X)) |
→ |
ok(s(X)) |
(28) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
active(head(X)) |
→ |
head(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
proper(incr(X)) |
→ |
incr(proper(X)) |
(20) |
proper(tail(X)) |
→ |
tail(proper(X)) |
(25) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
s(mark(X)) |
→ |
mark(s(X)) |
(14) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
proper(s(X)) |
→ |
s(proper(X)) |
(23) |
proper(head(X)) |
→ |
head(proper(X)) |
(24) |
active(tail(X)) |
→ |
tail(active(X)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(50) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
active#(head(X)) |
→ |
active#(X) |
(52) |
active#(tail(X)) |
→ |
active#(X) |
(72) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(47) |
active#(incr(X)) |
→ |
active#(X) |
(67) |
active#(s(X)) |
→ |
active#(X) |
(41) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
1 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
3 |
[0] |
=
|
2 |
[s#(x1)] |
=
|
0 |
[odds] |
=
|
21654 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 32742 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[nats] |
=
|
2 |
[active(x1)] |
=
|
x1 + 54399 |
[head(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(4) |
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
s(ok(X)) |
→ |
ok(s(X)) |
(28) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
active(head(X)) |
→ |
head(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
s(mark(X)) |
→ |
mark(s(X)) |
(14) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
active(tail(X)) |
→ |
tail(active(X)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(head(X)) |
→ |
active#(X) |
(52) |
active#(tail(X)) |
→ |
active#(X) |
(72) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(47) |
active#(incr(X)) |
→ |
active#(X) |
(67) |
active#(s(X)) |
→ |
active#(X) |
(41) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
proper#(s(X)) |
→ |
proper#(X) |
(48) |
proper#(incr(X)) |
→ |
proper#(X) |
(68) |
proper#(tail(X)) |
→ |
proper#(X) |
(65) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(37) |
proper#(head(X)) |
→ |
proper#(X) |
(56) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(34) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 2 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
1 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 29824 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
3 |
[0] |
=
|
3154 |
[s#(x1)] |
=
|
0 |
[odds] |
=
|
16675 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[nats] |
=
|
2 |
[active(x1)] |
=
|
x1 + 19831 |
[head(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(4) |
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
s(ok(X)) |
→ |
ok(s(X)) |
(28) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
active(head(X)) |
→ |
head(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(7) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
s(mark(X)) |
→ |
mark(s(X)) |
(14) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
active(tail(X)) |
→ |
tail(active(X)) |
(11) |
active(s(X)) |
→ |
s(active(X)) |
(9) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(s(X)) |
→ |
proper#(X) |
(48) |
proper#(incr(X)) |
→ |
proper#(X) |
(68) |
proper#(tail(X)) |
→ |
proper#(X) |
(65) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(37) |
proper#(head(X)) |
→ |
proper#(X) |
(56) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(34) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
head#(ok(X)) |
→ |
head#(X) |
(70) |
head#(mark(X)) |
→ |
head#(X) |
(39) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
2090 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
23638 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
x1 + 0 |
[tail(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
54887 |
[ok(x1)] |
=
|
x1 + 31250 |
[0] |
=
|
23637 |
[s#(x1)] |
=
|
0 |
[odds] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[nats] |
=
|
15620 |
[active(x1)] |
=
|
x1 + 23638 |
[head(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
head#(ok(X)) |
→ |
head#(X) |
(70) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(69) |
s#(ok(X)) |
→ |
s#(X) |
(66) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
2 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
x1 + 0 |
[odds] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[nats] |
=
|
1 |
[active(x1)] |
=
|
x1 + 3 |
[head(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(mark(X)) |
→ |
s#(X) |
(69) |
s#(ok(X)) |
→ |
s#(X) |
(66) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
tail#(mark(X)) |
→ |
tail#(X) |
(49) |
tail#(ok(X)) |
→ |
tail#(X) |
(63) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
2 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[odds] |
=
|
1 |
[tail#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[nats] |
=
|
1 |
[active(x1)] |
=
|
x1 + 3 |
[head(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
tail#(mark(X)) |
→ |
tail#(X) |
(49) |
tail#(ok(X)) |
→ |
tail#(X) |
(63) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(51) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(43) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
x2 + 0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
2 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[odds] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[nats] |
=
|
1 |
[active(x1)] |
=
|
x1 + 3 |
[head(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(51) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
8th
component contains the
pair
incr#(mark(X)) |
→ |
incr#(X) |
(74) |
incr#(ok(X)) |
→ |
incr#(X) |
(62) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[pairs] |
=
|
2 |
[top#(x1)] |
=
|
0 |
[head#(x1)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
2 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[odds] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[nats] |
=
|
1 |
[active(x1)] |
=
|
x1 + 3 |
[head(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
head(mark(X)) |
→ |
mark(head(X)) |
(15) |
active(nats) |
→ |
mark(cons(0,incr(nats))) |
(1) |
active(odds) |
→ |
mark(incr(pairs)) |
(3) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(16) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(nats) |
→ |
ok(nats) |
(17) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(27) |
proper(odds) |
→ |
ok(odds) |
(22) |
active(head(cons(X,XS))) |
→ |
mark(X) |
(5) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(30) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(12) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(13) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(6) |
head(ok(X)) |
→ |
ok(head(X)) |
(29) |
active(pairs) |
→ |
mark(cons(0,incr(odds))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
incr#(mark(X)) |
→ |
incr#(X) |
(74) |
incr#(ok(X)) |
→ |
incr#(X) |
(62) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.