The rewrite relation of the following TRS is considered.
active(pairNs) |
→ |
mark(cons(0,incr(oddNs))) |
(1) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(3) |
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(take(s(N),cons(X,XS))) |
→ |
mark(cons(X,take(N,XS))) |
(5) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
active(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(pair(X,Y),zip(XS,YS))) |
(8) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(repItems(cons(X,XS))) |
→ |
mark(cons(X,cons(X,repItems(XS)))) |
(11) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(12) |
active(incr(X)) |
→ |
incr(active(X)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(14) |
active(take(X1,X2)) |
→ |
take(active(X1),X2) |
(15) |
active(take(X1,X2)) |
→ |
take(X1,active(X2)) |
(16) |
active(zip(X1,X2)) |
→ |
zip(active(X1),X2) |
(17) |
active(zip(X1,X2)) |
→ |
zip(X1,active(X2)) |
(18) |
active(pair(X1,X2)) |
→ |
pair(active(X1),X2) |
(19) |
active(pair(X1,X2)) |
→ |
pair(X1,active(X2)) |
(20) |
active(tail(X)) |
→ |
tail(active(X)) |
(21) |
active(repItems(X)) |
→ |
repItems(active(X)) |
(22) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
s(mark(X)) |
→ |
mark(s(X)) |
(25) |
take(mark(X1),X2) |
→ |
mark(take(X1,X2)) |
(26) |
take(X1,mark(X2)) |
→ |
mark(take(X1,X2)) |
(27) |
zip(mark(X1),X2) |
→ |
mark(zip(X1,X2)) |
(28) |
zip(X1,mark(X2)) |
→ |
mark(zip(X1,X2)) |
(29) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(32) |
repItems(mark(X)) |
→ |
mark(repItems(X)) |
(33) |
proper(pairNs) |
→ |
ok(pairNs) |
(34) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(35) |
proper(0) |
→ |
ok(0) |
(36) |
proper(incr(X)) |
→ |
incr(proper(X)) |
(37) |
proper(oddNs) |
→ |
ok(oddNs) |
(38) |
proper(s(X)) |
→ |
s(proper(X)) |
(39) |
proper(take(X1,X2)) |
→ |
take(proper(X1),proper(X2)) |
(40) |
proper(nil) |
→ |
ok(nil) |
(41) |
proper(zip(X1,X2)) |
→ |
zip(proper(X1),proper(X2)) |
(42) |
proper(pair(X1,X2)) |
→ |
pair(proper(X1),proper(X2)) |
(43) |
proper(tail(X)) |
→ |
tail(proper(X)) |
(44) |
proper(repItems(X)) |
→ |
repItems(proper(X)) |
(45) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
s(ok(X)) |
→ |
ok(s(X)) |
(48) |
take(ok(X1),ok(X2)) |
→ |
ok(take(X1,X2)) |
(49) |
zip(ok(X1),ok(X2)) |
→ |
ok(zip(X1,X2)) |
(50) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(52) |
repItems(ok(X)) |
→ |
ok(repItems(X)) |
(53) |
top(mark(X)) |
→ |
top(proper(X)) |
(54) |
top(ok(X)) |
→ |
top(active(X)) |
(55) |
proper#(pair(X1,X2)) |
→ |
pair#(proper(X1),proper(X2)) |
(56) |
active#(pair(X1,X2)) |
→ |
active#(X2) |
(57) |
incr#(mark(X)) |
→ |
incr#(X) |
(58) |
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(59) |
active#(incr(cons(X,XS))) |
→ |
s#(X) |
(60) |
repItems#(mark(X)) |
→ |
repItems#(X) |
(61) |
active#(pair(X1,X2)) |
→ |
pair#(X1,active(X2)) |
(62) |
s#(mark(X)) |
→ |
s#(X) |
(63) |
active#(s(X)) |
→ |
active#(X) |
(64) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
pair#(X,Y) |
(65) |
active#(take(s(N),cons(X,XS))) |
→ |
cons#(X,take(N,XS)) |
(66) |
active#(repItems(X)) |
→ |
active#(X) |
(67) |
top#(mark(X)) |
→ |
proper#(X) |
(68) |
active#(take(s(N),cons(X,XS))) |
→ |
take#(N,XS) |
(69) |
active#(take(X1,X2)) |
→ |
take#(active(X1),X2) |
(70) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
zip#(XS,YS) |
(71) |
active#(pair(X1,X2)) |
→ |
pair#(active(X1),X2) |
(72) |
active#(tail(X)) |
→ |
tail#(active(X)) |
(73) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(74) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(75) |
active#(pair(X1,X2)) |
→ |
active#(X1) |
(76) |
active#(zip(X1,X2)) |
→ |
zip#(active(X1),X2) |
(77) |
active#(incr(cons(X,XS))) |
→ |
incr#(XS) |
(78) |
take#(ok(X1),ok(X2)) |
→ |
take#(X1,X2) |
(79) |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(80) |
proper#(incr(X)) |
→ |
incr#(proper(X)) |
(81) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(82) |
proper#(zip(X1,X2)) |
→ |
proper#(X1) |
(83) |
tail#(mark(X)) |
→ |
tail#(X) |
(84) |
active#(take(X1,X2)) |
→ |
active#(X1) |
(85) |
active#(incr(X)) |
→ |
incr#(active(X)) |
(86) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(87) |
pair#(ok(X1),ok(X2)) |
→ |
pair#(X1,X2) |
(88) |
active#(zip(X1,X2)) |
→ |
zip#(X1,active(X2)) |
(89) |
proper#(s(X)) |
→ |
proper#(X) |
(90) |
active#(pairNs) |
→ |
incr#(oddNs) |
(91) |
proper#(zip(X1,X2)) |
→ |
proper#(X2) |
(92) |
proper#(repItems(X)) |
→ |
repItems#(proper(X)) |
(93) |
active#(zip(X1,X2)) |
→ |
active#(X2) |
(94) |
active#(zip(X1,X2)) |
→ |
active#(X1) |
(95) |
proper#(incr(X)) |
→ |
proper#(X) |
(96) |
active#(tail(X)) |
→ |
active#(X) |
(97) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(98) |
active#(repItems(X)) |
→ |
repItems#(active(X)) |
(99) |
proper#(tail(X)) |
→ |
tail#(proper(X)) |
(100) |
active#(incr(X)) |
→ |
active#(X) |
(101) |
proper#(take(X1,X2)) |
→ |
take#(proper(X1),proper(X2)) |
(102) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(103) |
tail#(ok(X)) |
→ |
tail#(X) |
(104) |
proper#(zip(X1,X2)) |
→ |
zip#(proper(X1),proper(X2)) |
(105) |
active#(take(X1,X2)) |
→ |
active#(X2) |
(106) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(107) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
cons#(pair(X,Y),zip(XS,YS)) |
(108) |
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(109) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(110) |
active#(incr(cons(X,XS))) |
→ |
cons#(s(X),incr(XS)) |
(111) |
active#(repItems(cons(X,XS))) |
→ |
repItems#(XS) |
(112) |
proper#(take(X1,X2)) |
→ |
proper#(X2) |
(113) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(114) |
active#(repItems(cons(X,XS))) |
→ |
cons#(X,repItems(XS)) |
(115) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(116) |
s#(ok(X)) |
→ |
s#(X) |
(117) |
proper#(repItems(X)) |
→ |
proper#(X) |
(118) |
top#(ok(X)) |
→ |
active#(X) |
(119) |
active#(take(X1,X2)) |
→ |
take#(X1,active(X2)) |
(120) |
active#(repItems(cons(X,XS))) |
→ |
cons#(X,cons(X,repItems(XS))) |
(121) |
active#(pairNs) |
→ |
cons#(0,incr(oddNs)) |
(122) |
active#(s(X)) |
→ |
s#(active(X)) |
(123) |
proper#(take(X1,X2)) |
→ |
proper#(X1) |
(124) |
repItems#(ok(X)) |
→ |
repItems#(X) |
(125) |
proper#(pair(X1,X2)) |
→ |
proper#(X1) |
(126) |
zip#(ok(X1),ok(X2)) |
→ |
zip#(X1,X2) |
(127) |
active#(oddNs) |
→ |
incr#(pairNs) |
(128) |
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(129) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(130) |
incr#(ok(X)) |
→ |
incr#(X) |
(131) |
proper#(pair(X1,X2)) |
→ |
proper#(X2) |
(132) |
proper#(tail(X)) |
→ |
proper#(X) |
(133) |
top#(ok(X)) |
→ |
top#(active(X)) |
(134) |
The dependency pairs are split into 11
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(134) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(87) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(cons#) |
= |
2 |
π(top#) |
= |
1 |
π(zip#) |
= |
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(repItems) |
= |
5 |
|
status(repItems) |
= |
[1] |
|
list-extension(repItems) |
= |
Lex |
prec(incr) |
= |
2 |
|
status(incr) |
= |
[1] |
|
list-extension(incr) |
= |
Lex |
prec(s) |
= |
4 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(take#) |
= |
0 |
|
status(take#) |
= |
[2, 1] |
|
list-extension(take#) |
= |
Lex |
prec(take) |
= |
2 |
|
status(take) |
= |
[2, 1] |
|
list-extension(take) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(pair) |
= |
2 |
|
status(pair) |
= |
[1, 2] |
|
list-extension(pair) |
= |
Lex |
prec(tail) |
= |
1 |
|
status(tail) |
= |
[1] |
|
list-extension(tail) |
= |
Lex |
prec(0) |
= |
3 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(nil) |
= |
5 |
|
status(nil) |
= |
[] |
|
list-extension(nil) |
= |
Lex |
prec(tail#) |
= |
0 |
|
status(tail#) |
= |
[] |
|
list-extension(tail#) |
= |
Lex |
prec(mark) |
= |
0 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(incr#) |
= |
0 |
|
status(incr#) |
= |
[] |
|
list-extension(incr#) |
= |
Lex |
prec(pairNs) |
= |
6 |
|
status(pairNs) |
= |
[] |
|
list-extension(pairNs) |
= |
Lex |
prec(oddNs) |
= |
7 |
|
status(oddNs) |
= |
[] |
|
list-extension(oddNs) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(repItems#) |
= |
0 |
|
status(repItems#) |
= |
[] |
|
list-extension(repItems#) |
= |
Lex |
prec(cons) |
= |
2 |
|
status(cons) |
= |
[1] |
|
list-extension(cons) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
prec(pair#) |
= |
0 |
|
status(pair#) |
= |
[] |
|
list-extension(pair#) |
= |
Lex |
prec(zip) |
= |
5 |
|
status(zip) |
= |
[1, 2] |
|
list-extension(zip) |
= |
Lex |
and the following
Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
x1 + 0 |
[take#(x1, x2)] |
=
|
x1 + x2 + 1 |
[take(x1, x2)] |
=
|
x1 + x2 + 2 |
[top(x1)] |
=
|
1 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[tail(x1)] |
=
|
x1 + 1 |
[0] |
=
|
31122 |
[s#(x1)] |
=
|
1 |
[nil] |
=
|
31123 |
[tail#(x1)] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
1 |
[pairNs] |
=
|
31126 |
[oddNs] |
=
|
31126 |
[proper#(x1)] |
=
|
1 |
[repItems#(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
max(x1 + 3, x2 + 0, 0) |
[active#(x1)] |
=
|
1 |
[pair#(x1, x2)] |
=
|
x2 + 1 |
[zip(x1, x2)] |
=
|
x1 + x2 + 2 |
together with the usable
rules
active(zip(X1,X2)) |
→ |
zip(X1,active(X2)) |
(18) |
zip(ok(X1),ok(X2)) |
→ |
ok(zip(X1,X2)) |
(50) |
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(take(X1,X2)) |
→ |
take(active(X1),X2) |
(15) |
active(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(pair(X,Y),zip(XS,YS))) |
(8) |
active(pairNs) |
→ |
mark(cons(0,incr(oddNs))) |
(1) |
active(incr(cons(X,XS))) |
→ |
mark(cons(s(X),incr(XS))) |
(3) |
active(take(X1,X2)) |
→ |
take(X1,active(X2)) |
(16) |
active(tail(X)) |
→ |
tail(active(X)) |
(21) |
proper(0) |
→ |
ok(0) |
(36) |
take(mark(X1),X2) |
→ |
mark(take(X1,X2)) |
(26) |
active(pair(X1,X2)) |
→ |
pair(active(X1),X2) |
(19) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(32) |
active(zip(X1,X2)) |
→ |
zip(active(X1),X2) |
(17) |
take(X1,mark(X2)) |
→ |
mark(take(X1,X2)) |
(27) |
proper(pairNs) |
→ |
ok(pairNs) |
(34) |
active(repItems(X)) |
→ |
repItems(active(X)) |
(22) |
zip(mark(X1),X2) |
→ |
mark(zip(X1,X2)) |
(28) |
proper(tail(X)) |
→ |
tail(proper(X)) |
(44) |
active(take(s(N),cons(X,XS))) |
→ |
mark(cons(X,take(N,XS))) |
(5) |
repItems(mark(X)) |
→ |
mark(repItems(X)) |
(33) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
proper(s(X)) |
→ |
s(proper(X)) |
(39) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
active(pair(X1,X2)) |
→ |
pair(X1,active(X2)) |
(20) |
s(mark(X)) |
→ |
mark(s(X)) |
(25) |
take(ok(X1),ok(X2)) |
→ |
ok(take(X1,X2)) |
(49) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(52) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
active(s(X)) |
→ |
s(active(X)) |
(14) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(12) |
proper(repItems(X)) |
→ |
repItems(proper(X)) |
(45) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(repItems(cons(X,XS))) |
→ |
mark(cons(X,cons(X,repItems(XS)))) |
(11) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
active(incr(X)) |
→ |
incr(active(X)) |
(13) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
proper(take(X1,X2)) |
→ |
take(proper(X1),proper(X2)) |
(40) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
proper(oddNs) |
→ |
ok(oddNs) |
(38) |
s(ok(X)) |
→ |
ok(s(X)) |
(48) |
repItems(ok(X)) |
→ |
ok(repItems(X)) |
(53) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
proper(incr(X)) |
→ |
incr(proper(X)) |
(37) |
proper(nil) |
→ |
ok(nil) |
(41) |
proper(zip(X1,X2)) |
→ |
zip(proper(X1),proper(X2)) |
(42) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(35) |
zip(X1,mark(X2)) |
→ |
mark(zip(X1,X2)) |
(29) |
proper(pair(X1,X2)) |
→ |
pair(proper(X1),proper(X2)) |
(43) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(87) |
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#(tail(X)) |
→ |
proper#(X) |
(133) |
proper#(pair(X1,X2)) |
→ |
proper#(X2) |
(132) |
proper#(incr(X)) |
→ |
proper#(X) |
(96) |
proper#(zip(X1,X2)) |
→ |
proper#(X2) |
(92) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(130) |
proper#(s(X)) |
→ |
proper#(X) |
(90) |
proper#(pair(X1,X2)) |
→ |
proper#(X1) |
(126) |
proper#(take(X1,X2)) |
→ |
proper#(X1) |
(124) |
proper#(zip(X1,X2)) |
→ |
proper#(X1) |
(83) |
proper#(repItems(X)) |
→ |
proper#(X) |
(118) |
proper#(take(X1,X2)) |
→ |
proper#(X2) |
(113) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(103) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 57785 |
[ok(x1)] |
=
|
x1 + 57786 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
36828 |
[oddNs] |
=
|
36827 |
[proper#(x1)] |
=
|
x1 + 0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 2 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(tail(X)) |
→ |
proper#(X) |
(133) |
proper#(pair(X1,X2)) |
→ |
proper#(X2) |
(132) |
proper#(incr(X)) |
→ |
proper#(X) |
(96) |
proper#(zip(X1,X2)) |
→ |
proper#(X2) |
(92) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(130) |
proper#(s(X)) |
→ |
proper#(X) |
(90) |
proper#(pair(X1,X2)) |
→ |
proper#(X1) |
(126) |
proper#(take(X1,X2)) |
→ |
proper#(X1) |
(124) |
proper#(zip(X1,X2)) |
→ |
proper#(X1) |
(83) |
proper#(repItems(X)) |
→ |
proper#(X) |
(118) |
proper#(take(X1,X2)) |
→ |
proper#(X2) |
(113) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(103) |
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#(incr(X)) |
→ |
active#(X) |
(101) |
active#(tail(X)) |
→ |
active#(X) |
(97) |
active#(zip(X1,X2)) |
→ |
active#(X2) |
(94) |
active#(zip(X1,X2)) |
→ |
active#(X1) |
(95) |
active#(take(X1,X2)) |
→ |
active#(X1) |
(85) |
active#(pair(X1,X2)) |
→ |
active#(X1) |
(76) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(114) |
active#(repItems(X)) |
→ |
active#(X) |
(67) |
active#(s(X)) |
→ |
active#(X) |
(64) |
active#(take(X1,X2)) |
→ |
active#(X2) |
(106) |
active#(pair(X1,X2)) |
→ |
active#(X2) |
(57) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 32132 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
32002 |
[oddNs] |
=
|
51373 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 19374 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(incr(X)) |
→ |
active#(X) |
(101) |
active#(tail(X)) |
→ |
active#(X) |
(97) |
active#(zip(X1,X2)) |
→ |
active#(X2) |
(94) |
active#(zip(X1,X2)) |
→ |
active#(X1) |
(95) |
active#(take(X1,X2)) |
→ |
active#(X1) |
(85) |
active#(pair(X1,X2)) |
→ |
active#(X1) |
(76) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(114) |
active#(repItems(X)) |
→ |
active#(X) |
(67) |
active#(s(X)) |
→ |
active#(X) |
(64) |
active#(take(X1,X2)) |
→ |
active#(X2) |
(106) |
active#(pair(X1,X2)) |
→ |
active#(X2) |
(57) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
repItems#(ok(X)) |
→ |
repItems#(X) |
(125) |
repItems#(mark(X)) |
→ |
repItems#(X) |
(61) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 39229 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 27646 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
2 |
[oddNs] |
=
|
1 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 39230 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
repItems#(ok(X)) |
→ |
repItems#(X) |
(125) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
tail#(mark(X)) |
→ |
tail#(X) |
(84) |
tail#(ok(X)) |
→ |
tail#(X) |
(104) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 6835 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
1 |
[oddNs] |
=
|
1 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 4 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
tail#(mark(X)) |
→ |
tail#(X) |
(84) |
tail#(ok(X)) |
→ |
tail#(X) |
(104) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(117) |
s#(mark(X)) |
→ |
s#(X) |
(63) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 2089 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 25118 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 57205 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
x1 + 0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
43648 |
[oddNs] |
=
|
43647 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(117) |
s#(mark(X)) |
→ |
s#(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
incr#(ok(X)) |
→ |
incr#(X) |
(131) |
incr#(mark(X)) |
→ |
incr#(X) |
(58) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
x1 + 0 |
[pairNs] |
=
|
2 |
[oddNs] |
=
|
1 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
incr#(ok(X)) |
→ |
incr#(X) |
(131) |
incr#(mark(X)) |
→ |
incr#(X) |
(58) |
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#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(98) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(82) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
x1 + x2 + 0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
4 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
16610 |
[oddNs] |
=
|
16609 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(98) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(82) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
9th
component contains the
pair
zip#(ok(X1),ok(X2)) |
→ |
zip#(X1,X2) |
(127) |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(80) |
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(109) |
1.1.9 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 1 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 2 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 10109 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
x1 + x2 + 1 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
x2 + 0 |
[tail(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
35973 |
[oddNs] |
=
|
35972 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 3 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(tail(cons(X,XS))) |
→ |
mark(XS) |
(9) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
zip#(ok(X1),ok(X2)) |
→ |
zip#(X1,X2) |
(127) |
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(109) |
could be deleted.
1.1.9.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
10th
component contains the
pair
pair#(ok(X1),ok(X2)) |
→ |
pair#(X1,X2) |
(88) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(74) |
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(59) |
1.1.10 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
x1 + 0 |
[incr(x1)] |
=
|
x1 + 0 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[take#(x1, x2)] |
=
|
0 |
[take(x1, x2)] |
=
|
x1 + x2 + 1 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
3 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
1 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
1 |
[oddNs] |
=
|
38491 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 2 |
[cons(x1, x2)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
x1 + 0 |
[zip(x1, x2)] |
=
|
x1 + x2 + 1 |
together with the usable
rules
active(take(0,XS)) |
→ |
mark(nil) |
(4) |
active(repItems(nil)) |
→ |
mark(nil) |
(10) |
active(zip(X,nil)) |
→ |
mark(nil) |
(7) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
active(zip(nil,XS)) |
→ |
mark(nil) |
(6) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
active(oddNs) |
→ |
mark(incr(pairNs)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
pair#(ok(X1),ok(X2)) |
→ |
pair#(X1,X2) |
(88) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(74) |
could be deleted.
1.1.10.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
11th
component contains the
pair
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(129) |
take#(ok(X1),ok(X2)) |
→ |
take#(X1,X2) |
(79) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(116) |
1.1.11 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[repItems(x1)] |
=
|
12893 |
[incr(x1)] |
=
|
x1 + 1 |
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
31371 |
[take#(x1, x2)] |
=
|
x1 + x2 + 0 |
[take(x1, x2)] |
=
|
31609 |
[top(x1)] |
=
|
0 |
[pair(x1, x2)] |
=
|
19072 |
[top#(x1)] |
=
|
0 |
[zip#(x1, x2)] |
=
|
0 |
[tail(x1)] |
=
|
18331 |
[proper(x1)] |
=
|
12892 |
[ok(x1)] |
=
|
x1 + 16209 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[nil] |
=
|
23332 |
[tail#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 2766 |
[incr#(x1)] |
=
|
0 |
[pairNs] |
=
|
32662 |
[oddNs] |
=
|
2284 |
[proper#(x1)] |
=
|
0 |
[repItems#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + 46918 |
[active#(x1)] |
=
|
0 |
[pair#(x1, x2)] |
=
|
0 |
[zip(x1, x2)] |
=
|
21654 |
together with the usable
rules
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(129) |
take#(ok(X1),ok(X2)) |
→ |
take#(X1,X2) |
(79) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(116) |
could be deleted.
1.1.11.1 Dependency Graph Processor
The dependency pairs are split into 0
components.