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(terms(X)) |
→ |
terms(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(11) |
active(recip(X)) |
→ |
recip(active(X)) |
(12) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(13) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(14) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(15) |
active(dbl(X)) |
→ |
dbl(active(X)) |
(16) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(17) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(18) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(19) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(21) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(22) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(23) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(24) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(25) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
proper(terms(X)) |
→ |
terms(proper(X)) |
(28) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(29) |
proper(recip(X)) |
→ |
recip(proper(X)) |
(30) |
proper(sqr(X)) |
→ |
sqr(proper(X)) |
(31) |
proper(s(X)) |
→ |
s(proper(X)) |
(32) |
proper(0) |
→ |
ok(0) |
(33) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(34) |
proper(dbl(X)) |
→ |
dbl(proper(X)) |
(35) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(36) |
proper(nil) |
→ |
ok(nil) |
(37) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(38) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(40) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(41) |
s(ok(X)) |
→ |
ok(s(X)) |
(42) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(43) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(44) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
top(mark(X)) |
→ |
top(proper(X)) |
(46) |
top(ok(X)) |
→ |
top(active(X)) |
(47) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(48) |
proper#(terms(X)) |
→ |
terms#(proper(X)) |
(49) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(50) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(51) |
active#(terms(N)) |
→ |
cons#(recip(sqr(N)),terms(s(N))) |
(52) |
proper#(add(X1,X2)) |
→ |
add#(proper(X1),proper(X2)) |
(53) |
dbl#(ok(X)) |
→ |
dbl#(X) |
(54) |
active#(dbl(s(X))) |
→ |
dbl#(X) |
(55) |
active#(terms(X)) |
→ |
terms#(active(X)) |
(56) |
active#(first(X1,X2)) |
→ |
first#(X1,active(X2)) |
(57) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(58) |
proper#(s(X)) |
→ |
proper#(X) |
(59) |
active#(add(X1,X2)) |
→ |
add#(X1,active(X2)) |
(60) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(61) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(62) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(63) |
active#(dbl(X)) |
→ |
active#(X) |
(64) |
active#(terms(X)) |
→ |
active#(X) |
(65) |
proper#(dbl(X)) |
→ |
proper#(X) |
(66) |
recip#(mark(X)) |
→ |
recip#(X) |
(67) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(68) |
active#(terms(N)) |
→ |
recip#(sqr(N)) |
(69) |
active#(dbl(s(X))) |
→ |
s#(dbl(X)) |
(70) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(71) |
terms#(ok(X)) |
→ |
terms#(X) |
(72) |
s#(ok(X)) |
→ |
s#(X) |
(73) |
top#(ok(X)) |
→ |
top#(active(X)) |
(74) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(75) |
active#(recip(X)) |
→ |
recip#(active(X)) |
(76) |
active#(terms(N)) |
→ |
s#(N) |
(77) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(78) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(79) |
active#(terms(N)) |
→ |
terms#(s(N)) |
(80) |
top#(ok(X)) |
→ |
active#(X) |
(81) |
proper#(recip(X)) |
→ |
proper#(X) |
(82) |
proper#(first(X1,X2)) |
→ |
first#(proper(X1),proper(X2)) |
(83) |
recip#(ok(X)) |
→ |
recip#(X) |
(84) |
active#(sqr(s(X))) |
→ |
s#(add(sqr(X),dbl(X))) |
(85) |
proper#(recip(X)) |
→ |
recip#(proper(X)) |
(86) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(87) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(88) |
active#(recip(X)) |
→ |
active#(X) |
(89) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(90) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(91) |
active#(dbl(s(X))) |
→ |
s#(s(dbl(X))) |
(92) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(93) |
active#(sqr(s(X))) |
→ |
sqr#(X) |
(94) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(95) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(96) |
proper#(dbl(X)) |
→ |
dbl#(proper(X)) |
(97) |
proper#(terms(X)) |
→ |
proper#(X) |
(98) |
active#(sqr(s(X))) |
→ |
dbl#(X) |
(99) |
active#(add(X1,X2)) |
→ |
add#(active(X1),X2) |
(100) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(101) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(102) |
proper#(sqr(X)) |
→ |
proper#(X) |
(103) |
active#(dbl(X)) |
→ |
dbl#(active(X)) |
(104) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(105) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(106) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(107) |
active#(sqr(s(X))) |
→ |
add#(sqr(X),dbl(X)) |
(108) |
proper#(sqr(X)) |
→ |
sqr#(proper(X)) |
(109) |
active#(terms(N)) |
→ |
sqr#(N) |
(110) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(111) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(112) |
active#(sqr(X)) |
→ |
active#(X) |
(113) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(114) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(115) |
terms#(mark(X)) |
→ |
terms#(X) |
(116) |
top#(mark(X)) |
→ |
proper#(X) |
(117) |
active#(sqr(X)) |
→ |
sqr#(active(X)) |
(118) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(119) |
active#(first(X1,X2)) |
→ |
first#(active(X1),X2) |
(120) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(121) |
The dependency pairs are split into 11
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(74) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(91) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
16674 |
[recip(x1)] |
=
|
x1 + 38765 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 15943 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 0 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
240 |
[mark(x1)] |
=
|
x1 + 1 |
[first(x1, x2)] |
=
|
x1 + x2 + 5900 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + 39195 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 28872 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 77962 |
together with the usable
rules
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(18) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(15) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
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(dbl(X)) |
→ |
dbl(active(X)) |
(16) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(21) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(36) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(19) |
proper(s(X)) |
→ |
s(proper(X)) |
(32) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(17) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(34) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(22) |
proper(terms(X)) |
→ |
terms(proper(X)) |
(28) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(44) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
proper(0) |
→ |
ok(0) |
(33) |
active(terms(X)) |
→ |
terms(active(X)) |
(10) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(25) |
proper(recip(X)) |
→ |
recip(proper(X)) |
(30) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(14) |
proper(sqr(X)) |
→ |
sqr(proper(X)) |
(31) |
active(recip(X)) |
→ |
recip(active(X)) |
(12) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(23) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(24) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(11) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(13) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(40) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(38) |
proper(nil) |
→ |
ok(nil) |
(37) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(41) |
s(ok(X)) |
→ |
ok(s(X)) |
(42) |
proper(dbl(X)) |
→ |
dbl(proper(X)) |
(35) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(29) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(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)) |
(91) |
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#(cons(X1,X2)) |
→ |
proper#(X2) |
(119) |
proper#(recip(X)) |
→ |
proper#(X) |
(82) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(114) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(dbl(X)) |
→ |
proper#(X) |
(66) |
proper#(sqr(X)) |
→ |
proper#(X) |
(103) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(63) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(61) |
proper#(s(X)) |
→ |
proper#(X) |
(59) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(58) |
proper#(terms(X)) |
→ |
proper#(X) |
(98) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 23641 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
30137 |
[mark(x1)] |
=
|
16147 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 16143 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 3 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(119) |
proper#(recip(X)) |
→ |
proper#(X) |
(82) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(114) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(75) |
proper#(dbl(X)) |
→ |
proper#(X) |
(66) |
proper#(sqr(X)) |
→ |
proper#(X) |
(103) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(63) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(61) |
proper#(s(X)) |
→ |
proper#(X) |
(59) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(58) |
proper#(terms(X)) |
→ |
proper#(X) |
(98) |
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#(recip(X)) |
→ |
active#(X) |
(89) |
active#(sqr(X)) |
→ |
active#(X) |
(113) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(71) |
active#(dbl(X)) |
→ |
active#(X) |
(64) |
active#(terms(X)) |
→ |
active#(X) |
(65) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(62) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(96) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(50) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(90) |
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 + 1 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
2 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
30137 |
[mark(x1)] |
=
|
5 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 3 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(recip(X)) |
→ |
active#(X) |
(89) |
active#(sqr(X)) |
→ |
active#(X) |
(113) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(71) |
active#(dbl(X)) |
→ |
active#(X) |
(64) |
active#(terms(X)) |
→ |
active#(X) |
(65) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(62) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(96) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(50) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(90) |
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#(mark(X)) |
→ |
dbl#(X) |
(106) |
dbl#(ok(X)) |
→ |
dbl#(X) |
(54) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 1 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
x1 + 0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 49 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
6389 |
[mark(x1)] |
=
|
x1 + 4 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
dbl#(mark(X)) |
→ |
dbl#(X) |
(106) |
dbl#(ok(X)) |
→ |
dbl#(X) |
(54) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(111) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(95) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
x1 + 10392 |
[recip(x1)] |
=
|
x1 + 5744 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 9133 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
38312 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
6389 |
[mark(x1)] |
=
|
x1 + 80049 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 41735 |
[sqr(x1)] |
=
|
x1 + 18524 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(111) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(95) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
recip#(ok(X)) |
→ |
recip#(X) |
(84) |
recip#(mark(X)) |
→ |
recip#(X) |
(67) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 1 |
[recip#(x1)] |
=
|
x1 + 0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 18526 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 4592 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
recip#(ok(X)) |
→ |
recip#(X) |
(84) |
recip#(mark(X)) |
→ |
recip#(X) |
(67) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
7th
component contains the
pair
sqr#(mark(X)) |
→ |
sqr#(X) |
(79) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(93) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 12490 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 4 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 62970 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
sqr#(mark(X)) |
→ |
sqr#(X) |
(79) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(93) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
8th
component contains the
pair
terms#(mark(X)) |
→ |
terms#(X) |
(116) |
terms#(ok(X)) |
→ |
terms#(X) |
(72) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 1 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
26962 |
[mark(x1)] |
=
|
x1 + 4 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
terms#(mark(X)) |
→ |
terms#(X) |
(116) |
terms#(ok(X)) |
→ |
terms#(X) |
(72) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
9th
component contains the
pair
1.1.9 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 15530 |
[recip(x1)] |
=
|
x1 + 1 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 2089 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
x1 + 0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 55447 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 55444 |
[sqr(x1)] |
=
|
x1 + 19454 |
[terms(x1)] |
=
|
x1 + 54920 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.9.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
10th
component contains the
pair
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(115) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(112) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(101) |
1.1.10 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 1 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
0 |
[nil] |
=
|
26064 |
[mark(x1)] |
=
|
x1 + 47950 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 47947 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
x1 + 0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(115) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(101) |
could be deleted.
1.1.10.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
11th
component contains the
pair
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(88) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(78) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(68) |
1.1.11 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
x1 + 1 |
[recip(x1)] |
=
|
x1 + 1 |
[recip#(x1)] |
=
|
0 |
[dbl(x1)] |
=
|
x1 + 1 |
[top(x1)] |
=
|
0 |
[dbl#(x1)] |
=
|
0 |
[terms#(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[sqr#(x1)] |
=
|
0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
13644 |
[s#(x1)] |
=
|
0 |
[first#(x1, x2)] |
=
|
x1 + 0 |
[nil] |
=
|
1 |
[mark(x1)] |
=
|
x1 + 13647 |
[first(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
x1 + x2 + 1 |
[active#(x1)] |
=
|
0 |
[add#(x1, x2)] |
=
|
0 |
[add(x1, x2)] |
=
|
x1 + x2 + 1 |
[sqr(x1)] |
=
|
x1 + 1 |
[terms(x1)] |
=
|
x1 + 1 |
together with the usable
rules
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(26) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(27) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(39) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(20) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(78) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(68) |
could be deleted.
1.1.11.1 Dependency Graph Processor
The dependency pairs are split into 1
component.