The rewrite relation of the following TRS is considered.
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(49) |
top#(ok(X)) |
→ |
top#(active(X)) |
(44) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(top) |
= |
1 |
π(top#) |
= |
1 |
π(tl#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(active) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(adx#) |
= |
0 |
|
status(adx#) |
= |
[] |
|
list-extension(adx#) |
= |
Lex |
prec(incr) |
= |
2 |
|
status(incr) |
= |
[1] |
|
list-extension(incr) |
= |
Lex |
prec(hd) |
= |
6 |
|
status(hd) |
= |
[1] |
|
list-extension(hd) |
= |
Lex |
prec(cons#) |
= |
0 |
|
status(cons#) |
= |
[] |
|
list-extension(cons#) |
= |
Lex |
prec(s) |
= |
2 |
|
status(s) |
= |
[] |
|
list-extension(s) |
= |
Lex |
prec(adx) |
= |
3 |
|
status(adx) |
= |
[1] |
|
list-extension(adx) |
= |
Lex |
prec(zeros) |
= |
4 |
|
status(zeros) |
= |
[] |
|
list-extension(zeros) |
= |
Lex |
prec(0) |
= |
4 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(tl) |
= |
6 |
|
status(tl) |
= |
[1] |
|
list-extension(tl) |
= |
Lex |
prec(mark) |
= |
1 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(incr#) |
= |
0 |
|
status(incr#) |
= |
[] |
|
list-extension(incr#) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(hd#) |
= |
0 |
|
status(hd#) |
= |
[] |
|
list-extension(hd#) |
= |
Lex |
prec(nats) |
= |
7 |
|
status(nats) |
= |
[] |
|
list-extension(nats) |
= |
Lex |
prec(cons) |
= |
3 |
|
status(cons) |
= |
[] |
|
list-extension(cons) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
and the following
Max-polynomial interpretation
[adx#(x1)] |
=
|
1 |
[incr(x1)] |
=
|
x1 + 0 |
[hd(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
x2 + 1 |
[s(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
0 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
1 |
[tl(x1)] |
=
|
x1 + 35495 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
1 |
[proper#(x1)] |
=
|
1 |
[hd#(x1)] |
=
|
1 |
[nats] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
1 |
together with the usable
rules
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(18) |
active(adx(cons(X,Y))) |
→ |
mark(incr(cons(X,adx(Y)))) |
(4) |
proper(nats) |
→ |
ok(nats) |
(15) |
active(incr(X)) |
→ |
incr(active(X)) |
(8) |
active(nats) |
→ |
mark(adx(zeros)) |
(1) |
active(incr(cons(X,Y))) |
→ |
mark(cons(s(X),incr(Y))) |
(3) |
proper(adx(X)) |
→ |
adx(proper(X)) |
(16) |
proper(s(X)) |
→ |
s(proper(X)) |
(21) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(26) |
proper(0) |
→ |
ok(0) |
(19) |
proper(zeros) |
→ |
ok(zeros) |
(17) |
s(ok(X)) |
→ |
ok(s(X)) |
(27) |
proper(hd(X)) |
→ |
hd(proper(X)) |
(22) |
hd(ok(X)) |
→ |
ok(hd(X)) |
(28) |
active(hd(cons(X,Y))) |
→ |
mark(X) |
(5) |
active(tl(X)) |
→ |
tl(active(X)) |
(10) |
active(adx(X)) |
→ |
adx(active(X)) |
(7) |
proper(incr(X)) |
→ |
incr(proper(X)) |
(20) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
tl(mark(X)) |
→ |
mark(tl(X)) |
(14) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(12) |
proper(tl(X)) |
→ |
tl(proper(X)) |
(23) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
active(hd(X)) |
→ |
hd(active(X)) |
(9) |
hd(mark(X)) |
→ |
mark(hd(X)) |
(13) |
active(tl(cons(X,Y))) |
→ |
mark(Y) |
(6) |
tl(ok(X)) |
→ |
ok(tl(X)) |
(29) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(49) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
proper#(incr(X)) |
→ |
proper#(X) |
(74) |
proper#(hd(X)) |
→ |
proper#(X) |
(51) |
proper#(adx(X)) |
→ |
proper#(X) |
(48) |
proper#(tl(X)) |
→ |
proper#(X) |
(43) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(36) |
proper#(s(X)) |
→ |
proper#(X) |
(34) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(33) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
1 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
14522 |
[ok(x1)] |
=
|
14523 |
[0] |
=
|
2805 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
49472 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
1 |
[active(x1)] |
=
|
49471 |
[cons(x1, x2)] |
=
|
x1 + x2 + 3024 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(incr(X)) |
→ |
proper#(X) |
(74) |
proper#(hd(X)) |
→ |
proper#(X) |
(51) |
proper#(adx(X)) |
→ |
proper#(X) |
(48) |
proper#(tl(X)) |
→ |
proper#(X) |
(43) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(36) |
proper#(s(X)) |
→ |
proper#(X) |
(34) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(33) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
active#(tl(X)) |
→ |
active#(X) |
(59) |
active#(hd(X)) |
→ |
active#(X) |
(57) |
active#(incr(X)) |
→ |
active#(X) |
(38) |
active#(adx(X)) |
→ |
active#(X) |
(52) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 11575 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
1 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
2 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
1 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(tl(X)) |
→ |
active#(X) |
(59) |
active#(hd(X)) |
→ |
active#(X) |
(57) |
active#(incr(X)) |
→ |
active#(X) |
(38) |
active#(adx(X)) |
→ |
active#(X) |
(52) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
hd#(ok(X)) |
→ |
hd#(X) |
(46) |
hd#(mark(X)) |
→ |
hd#(X) |
(69) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 16266 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 24324 |
[zeros] |
=
|
14923 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
47966 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 6812 |
[mark(x1)] |
=
|
x1 + 12064 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
x1 + 0 |
[nats] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
hd#(ok(X)) |
→ |
hd#(X) |
(46) |
hd#(mark(X)) |
→ |
hd#(X) |
(69) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
tl#(mark(X)) |
→ |
tl#(X) |
(42) |
tl#(ok(X)) |
→ |
tl#(X) |
(32) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 15047 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 25532 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
47040 |
[tl#(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
61407 |
[ok(x1)] |
=
|
x1 + 61408 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 26531 |
[mark(x1)] |
=
|
x1 + 12064 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
tl#(mark(X)) |
→ |
tl#(X) |
(42) |
tl#(ok(X)) |
→ |
tl#(X) |
(32) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 15047 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 9245 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
13825 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 3 |
[0] |
=
|
40748 |
[s#(x1)] |
=
|
x1 + 0 |
[tl(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 72167 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
0 |
[active(x1)] |
=
|
60104 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
incr#(ok(X)) |
→ |
incr#(X) |
(61) |
incr#(mark(X)) |
→ |
incr#(X) |
(55) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 10114 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
9246 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1184 |
[zeros] |
=
|
1 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
29503 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 12064 |
[incr#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
incr#(ok(X)) |
→ |
incr#(X) |
(61) |
incr#(mark(X)) |
→ |
incr#(X) |
(55) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
8th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(68) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 10114 |
[cons#(x1, x2)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 1 |
[zeros] |
=
|
1 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
57122 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 28256 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(68) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
9th
component contains the
pair
adx#(mark(X)) |
→ |
adx#(X) |
(40) |
adx#(ok(X)) |
→ |
adx#(X) |
(53) |
1.1.9 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[adx#(x1)] |
=
|
x1 + 0 |
[incr(x1)] |
=
|
x1 + 1 |
[hd(x1)] |
=
|
x1 + 32157 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[adx(x1)] |
=
|
x1 + 19912 |
[zeros] |
=
|
1 |
[tl#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 13477 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[tl(x1)] |
=
|
x1 + 6437 |
[mark(x1)] |
=
|
x1 + 28256 |
[incr#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[hd#(x1)] |
=
|
0 |
[nats] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(25) |
adx(ok(X)) |
→ |
ok(adx(X)) |
(24) |
adx(mark(X)) |
→ |
mark(adx(X)) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
adx#(mark(X)) |
→ |
adx#(X) |
(40) |
adx#(ok(X)) |
→ |
adx#(X) |
(53) |
could be deleted.
1.1.9.1 Dependency Graph Processor
The dependency pairs are split into 0
components.