The rewrite relation of the following TRS is considered.
active(terms(N)) |
→ |
mark(cons(recip(sqr(N)),terms(s(N)))) |
(1) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
active(sqr(s(X))) |
→ |
mark(s(add(sqr(X),dbl(X)))) |
(3) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(half(0)) |
→ |
mark(0) |
(10) |
active(half(s(0))) |
→ |
mark(0) |
(11) |
active(half(s(s(X)))) |
→ |
mark(s(half(X))) |
(12) |
active(half(dbl(X))) |
→ |
mark(X) |
(13) |
active(terms(X)) |
→ |
terms(active(X)) |
(14) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(15) |
active(recip(X)) |
→ |
recip(active(X)) |
(16) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(17) |
active(s(X)) |
→ |
s(active(X)) |
(18) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(19) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(20) |
active(dbl(X)) |
→ |
dbl(active(X)) |
(21) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(22) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(23) |
active(half(X)) |
→ |
half(active(X)) |
(24) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(25) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(26) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(27) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(28) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(30) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(31) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(32) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(33) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(34) |
half(mark(X)) |
→ |
mark(half(X)) |
(35) |
proper(terms(X)) |
→ |
terms(proper(X)) |
(36) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(37) |
proper(recip(X)) |
→ |
recip(proper(X)) |
(38) |
proper(sqr(X)) |
→ |
sqr(proper(X)) |
(39) |
proper(s(X)) |
→ |
s(proper(X)) |
(40) |
proper(0) |
→ |
ok(0) |
(41) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(42) |
proper(dbl(X)) |
→ |
dbl(proper(X)) |
(43) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(44) |
proper(nil) |
→ |
ok(nil) |
(45) |
proper(half(X)) |
→ |
half(proper(X)) |
(46) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(48) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(49) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(50) |
s(ok(X)) |
→ |
ok(s(X)) |
(51) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(52) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(53) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(54) |
half(ok(X)) |
→ |
ok(half(X)) |
(55) |
top(mark(X)) |
→ |
top(proper(X)) |
(56) |
top(ok(X)) |
→ |
top(active(X)) |
(57) |
proper#(dbl(X)) |
→ |
proper#(X) |
(58) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(59) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(60) |
active#(terms(X)) |
→ |
terms#(active(X)) |
(61) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(62) |
proper#(terms(X)) |
→ |
proper#(X) |
(63) |
proper#(sqr(X)) |
→ |
proper#(X) |
(64) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(65) |
active#(add(X1,X2)) |
→ |
add#(X1,active(X2)) |
(66) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(67) |
active#(sqr(s(X))) |
→ |
sqr#(X) |
(68) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(69) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(70) |
active#(sqr(s(X))) |
→ |
s#(add(sqr(X),dbl(X))) |
(71) |
active#(dbl(s(X))) |
→ |
s#(s(dbl(X))) |
(72) |
active#(terms(N)) |
→ |
cons#(recip(sqr(N)),terms(s(N))) |
(73) |
active#(sqr(s(X))) |
→ |
dbl#(X) |
(74) |
recip#(mark(X)) |
→ |
recip#(X) |
(75) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(76) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(77) |
proper#(half(X)) |
→ |
proper#(X) |
(78) |
active#(sqr(X)) |
→ |
sqr#(active(X)) |
(79) |
active#(first(X1,X2)) |
→ |
first#(active(X1),X2) |
(80) |
active#(dbl(X)) |
→ |
dbl#(active(X)) |
(81) |
terms#(mark(X)) |
→ |
terms#(X) |
(82) |
proper#(first(X1,X2)) |
→ |
first#(proper(X1),proper(X2)) |
(83) |
terms#(ok(X)) |
→ |
terms#(X) |
(84) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(85) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(86) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(87) |
active#(terms(N)) |
→ |
recip#(sqr(N)) |
(88) |
top#(ok(X)) |
→ |
top#(active(X)) |
(89) |
active#(recip(X)) |
→ |
active#(X) |
(90) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(91) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(92) |
active#(terms(N)) |
→ |
terms#(s(N)) |
(93) |
proper#(sqr(X)) |
→ |
sqr#(proper(X)) |
(94) |
active#(recip(X)) |
→ |
recip#(active(X)) |
(95) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(96) |
active#(half(s(s(X)))) |
→ |
half#(X) |
(97) |
active#(terms(N)) |
→ |
s#(N) |
(98) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(99) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(100) |
dbl#(ok(X)) |
→ |
dbl#(X) |
(101) |
active#(add(X1,X2)) |
→ |
add#(active(X1),X2) |
(102) |
active#(half(s(s(X)))) |
→ |
s#(half(X)) |
(103) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(104) |
active#(dbl(s(X))) |
→ |
s#(dbl(X)) |
(105) |
top#(ok(X)) |
→ |
active#(X) |
(106) |
proper#(s(X)) |
→ |
proper#(X) |
(107) |
proper#(add(X1,X2)) |
→ |
add#(proper(X1),proper(X2)) |
(108) |
recip#(ok(X)) |
→ |
recip#(X) |
(109) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(110) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(111) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(112) |
active#(sqr(s(X))) |
→ |
add#(sqr(X),dbl(X)) |
(113) |
proper#(half(X)) |
→ |
half#(proper(X)) |
(114) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(115) |
proper#(terms(X)) |
→ |
terms#(proper(X)) |
(116) |
active#(first(X1,X2)) |
→ |
first#(X1,active(X2)) |
(117) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(118) |
top#(mark(X)) |
→ |
proper#(X) |
(119) |
active#(s(X)) |
→ |
s#(active(X)) |
(120) |
active#(half(X)) |
→ |
active#(X) |
(121) |
active#(sqr(X)) |
→ |
active#(X) |
(122) |
proper#(recip(X)) |
→ |
proper#(X) |
(123) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(124) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(125) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(126) |
active#(half(X)) |
→ |
half#(active(X)) |
(127) |
active#(s(X)) |
→ |
active#(X) |
(128) |
active#(dbl(X)) |
→ |
active#(X) |
(129) |
active#(terms(X)) |
→ |
active#(X) |
(130) |
s#(ok(X)) |
→ |
s#(X) |
(131) |
proper#(recip(X)) |
→ |
recip#(proper(X)) |
(132) |
s#(mark(X)) |
→ |
s#(X) |
(133) |
active#(terms(N)) |
→ |
sqr#(N) |
(134) |
proper#(dbl(X)) |
→ |
dbl#(proper(X)) |
(135) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(136) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(137) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(138) |
half#(mark(X)) |
→ |
half#(X) |
(139) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(140) |
active#(dbl(s(X))) |
→ |
dbl#(X) |
(141) |
half#(ok(X)) |
→ |
half#(X) |
(142) |
The dependency pairs are split into 12
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(89) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(112) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(cons#) |
= |
1 |
π(dbl#) |
= |
1 |
π(top#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(active) |
= |
1 |
π(cons) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(s) |
= |
3 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(recip) |
= |
2 |
|
status(recip) |
= |
[1] |
|
list-extension(recip) |
= |
Lex |
prec(recip#) |
= |
0 |
|
status(recip#) |
= |
[] |
|
list-extension(recip#) |
= |
Lex |
prec(dbl) |
= |
5 |
|
status(dbl) |
= |
[1] |
|
list-extension(dbl) |
= |
Lex |
prec(top) |
= |
0 |
|
status(top) |
= |
[] |
|
list-extension(top) |
= |
Lex |
prec(terms#) |
= |
0 |
|
status(terms#) |
= |
[] |
|
list-extension(terms#) |
= |
Lex |
prec(half#) |
= |
0 |
|
status(half#) |
= |
[] |
|
list-extension(half#) |
= |
Lex |
prec(half) |
= |
3 |
|
status(half) |
= |
[1] |
|
list-extension(half) |
= |
Lex |
prec(sqr#) |
= |
0 |
|
status(sqr#) |
= |
[] |
|
list-extension(sqr#) |
= |
Lex |
prec(0) |
= |
3 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(first#) |
= |
0 |
|
status(first#) |
= |
[] |
|
list-extension(first#) |
= |
Lex |
prec(nil) |
= |
4 |
|
status(nil) |
= |
[] |
|
list-extension(nil) |
= |
Lex |
prec(mark) |
= |
2 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(first) |
= |
4 |
|
status(first) |
= |
[2, 1] |
|
list-extension(first) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
prec(add#) |
= |
0 |
|
status(add#) |
= |
[1, 2] |
|
list-extension(add#) |
= |
Lex |
prec(add) |
= |
4 |
|
status(add) |
= |
[1, 2] |
|
list-extension(add) |
= |
Lex |
prec(sqr) |
= |
5 |
|
status(sqr) |
= |
[1] |
|
list-extension(sqr) |
= |
Lex |
prec(terms) |
= |
6 |
|
status(terms) |
= |
[1] |
|
list-extension(terms) |
= |
Lex |
and the following
Max-polynomial interpretation
[s(x1)] |
=
|
x1 + 0 |
[recip(x1)] |
=
|
x1 + 0 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
x1 + 0 |
[sqr#(x1)] |
=
|
0 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
max(0) |
[nil] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[first(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[proper#(x1)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[add(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[sqr(x1)] |
=
|
x1 + 0 |
[terms(x1)] |
=
|
x1 + 0 |
together with the usable
rules
active(s(X)) |
→ |
s(active(X)) |
(18) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(50) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(15) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(54) |
active(terms(N)) |
→ |
mark(cons(recip(sqr(N)),terms(s(N)))) |
(1) |
active(sqr(s(X))) |
→ |
mark(s(add(sqr(X),dbl(X)))) |
(3) |
active(recip(X)) |
→ |
recip(active(X)) |
(16) |
active(dbl(X)) |
→ |
dbl(active(X)) |
(21) |
proper(terms(X)) |
→ |
terms(proper(X)) |
(36) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(26) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(19) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(32) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(17) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(27) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(34) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(22) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(28) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(44) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(33) |
active(half(0)) |
→ |
mark(0) |
(10) |
proper(sqr(X)) |
→ |
sqr(proper(X)) |
(39) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(20) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(25) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(49) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(52) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(30) |
active(terms(X)) |
→ |
terms(active(X)) |
(14) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(31) |
active(half(s(s(X)))) |
→ |
mark(s(half(X))) |
(12) |
proper(nil) |
→ |
ok(nil) |
(45) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(23) |
active(half(X)) |
→ |
half(active(X)) |
(24) |
active(half(s(0))) |
→ |
mark(0) |
(11) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(half(dbl(X))) |
→ |
mark(X) |
(13) |
s(ok(X)) |
→ |
ok(s(X)) |
(51) |
proper(s(X)) |
→ |
s(proper(X)) |
(40) |
half(ok(X)) |
→ |
ok(half(X)) |
(55) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
proper(recip(X)) |
→ |
recip(proper(X)) |
(38) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(48) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(53) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(47) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(37) |
proper(0) |
→ |
ok(0) |
(41) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(42) |
proper(half(X)) |
→ |
half(proper(X)) |
(46) |
half(mark(X)) |
→ |
mark(half(X)) |
(35) |
s(mark(X)) |
→ |
mark(s(X)) |
(29) |
proper(dbl(X)) |
→ |
dbl(proper(X)) |
(43) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(112) |
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#(cons(X1,X2)) |
→ |
active#(X1) |
(100) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(99) |
active#(recip(X)) |
→ |
active#(X) |
(90) |
active#(terms(X)) |
→ |
active#(X) |
(130) |
active#(dbl(X)) |
→ |
active#(X) |
(129) |
active#(s(X)) |
→ |
active#(X) |
(128) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(126) |
active#(sqr(X)) |
→ |
active#(X) |
(122) |
active#(half(X)) |
→ |
active#(X) |
(121) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(65) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(60) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 0 |
[recip(x1)] |
=
|
x1 + 0 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
x1 + 0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 0 |
[first(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 0 |
[sqr(x1)] |
=
|
x1 + 0 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(54) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(32) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(34) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(33) |
active(half(0)) |
→ |
mark(0) |
(10) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(52) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(30) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(31) |
active(half(s(0))) |
→ |
mark(0) |
(11) |
active(half(dbl(X))) |
→ |
mark(X) |
(13) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(53) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(terms(X)) |
→ |
active#(X) |
(130) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(104) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(96) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(137) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(86) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(85) |
proper#(recip(X)) |
→ |
proper#(X) |
(123) |
proper#(half(X)) |
→ |
proper#(X) |
(78) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(70) |
proper#(sqr(X)) |
→ |
proper#(X) |
(64) |
proper#(terms(X)) |
→ |
proper#(X) |
(63) |
proper#(dbl(X)) |
→ |
proper#(X) |
(58) |
proper#(s(X)) |
→ |
proper#(X) |
(107) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 13745 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 13457 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
x1 + 9123 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
3 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
15795 |
[first(x1, x2)] |
=
|
x1 + x2 + 22746 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
17406 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 6860 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(54) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(34) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(33) |
active(half(0)) |
→ |
mark(0) |
(10) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(52) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(31) |
active(half(s(0))) |
→ |
mark(0) |
(11) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(53) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(104) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(96) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(137) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(86) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(85) |
proper#(recip(X)) |
→ |
proper#(X) |
(123) |
proper#(half(X)) |
→ |
proper#(X) |
(78) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(70) |
proper#(sqr(X)) |
→ |
proper#(X) |
(64) |
proper#(terms(X)) |
→ |
proper#(X) |
(63) |
proper#(dbl(X)) |
→ |
proper#(X) |
(58) |
proper#(s(X)) |
→ |
proper#(X) |
(107) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
dbl#(ok(X)) |
→ |
dbl#(X) |
(101) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(76) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 9589 |
[recip(x1)] |
=
|
28967 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 49976 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
x1 + 0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
x1 + 51752 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 21639 |
[0] |
=
|
30728 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 28117 |
[first(x1, x2)] |
=
|
x1 + x2 + 8127 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 29518 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x2 + 27234 |
[sqr(x1)] |
=
|
26325 |
[terms(x1)] |
=
|
26534 |
together with the usable
rules
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(54) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(34) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
dbl#(ok(X)) |
→ |
dbl#(X) |
(101) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(76) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
half#(ok(X)) |
→ |
half#(X) |
(142) |
half#(mark(X)) |
→ |
half#(X) |
(139) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
8137 |
[recip(x1)] |
=
|
1428 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
x1 + 0 |
[half(x1)] |
=
|
x1 + 1 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 2874 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
23191 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
25338 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
x2 + 22612 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + 7803 |
[sqr(x1)] |
=
|
x1 + 21331 |
[terms(x1)] |
=
|
37936 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
half#(ok(X)) |
→ |
half#(X) |
(142) |
half#(mark(X)) |
→ |
half#(X) |
(139) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(124) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(77) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(67) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
33286 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
12341 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
47761 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
16689 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
31853 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
x1 + 0 |
[add(x1, x2)] |
=
|
7803 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
2 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(77) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(67) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
7th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(138) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(111) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
42037 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
12341 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
5609 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
21952 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
4686 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
26405 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
3343 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(138) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(111) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
8th
component contains the
pair
recip#(mark(X)) |
→ |
recip#(X) |
(75) |
recip#(ok(X)) |
→ |
recip#(X) |
(109) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
73860 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
x1 + 0 |
[dbl(x1)] |
=
|
12341 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 4000 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1610 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
21952 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
19315 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
26405 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
2 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
recip#(mark(X)) |
→ |
recip#(X) |
(75) |
recip#(ok(X)) |
→ |
recip#(X) |
(109) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
9th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(133) |
s#(ok(X)) |
→ |
s#(X) |
(131) |
1.1.9 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
89514 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
12341 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 408 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
x1 + 0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1203 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
21952 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
13069 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
23102 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
2 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
s#(mark(X)) |
→ |
s#(X) |
(133) |
s#(ok(X)) |
→ |
s#(X) |
(131) |
could be deleted.
1.1.9.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
10th
component contains the
pair
sqr#(mark(X)) |
→ |
sqr#(X) |
(136) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(62) |
1.1.10 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
117474 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
12341 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
9481 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
21952 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
31595 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
12761 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
2 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
sqr#(mark(X)) |
→ |
sqr#(X) |
(136) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(62) |
could be deleted.
1.1.10.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
11th
component contains the
pair
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(140) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(87) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(69) |
1.1.11 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
121223 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
2 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
x1 + 0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
31104 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
14862 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
12064 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
2 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(140) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(69) |
could be deleted.
1.1.11.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
12th
component contains the
pair
terms#(ok(X)) |
→ |
terms#(X) |
(84) |
terms#(mark(X)) |
→ |
terms#(X) |
(82) |
1.1.12 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
154603 |
[recip(x1)] |
=
|
2 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
29899 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
0 |
[half#(x1)] |
=
|
0 |
[half(x1)] |
=
|
2201 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 2 |
[first(x1, x2)] |
=
|
4037 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[cons(x1, x2)] |
=
|
26835 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
31781 |
[sqr(x1)] |
=
|
12115 |
[terms(x1)] |
=
|
20846 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
terms#(ok(X)) |
→ |
terms#(X) |
(84) |
terms#(mark(X)) |
→ |
terms#(X) |
(82) |
could be deleted.
1.1.12.1 Dependency Graph Processor
The dependency pairs are split into 0
components.