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) |
active#(pairNs) |
→ |
cons#(0,incr(oddNs)) |
(56) |
active#(pairNs) |
→ |
incr#(oddNs) |
(57) |
active#(oddNs) |
→ |
incr#(pairNs) |
(58) |
active#(incr(cons(X,XS))) |
→ |
cons#(s(X),incr(XS)) |
(59) |
active#(incr(cons(X,XS))) |
→ |
s#(X) |
(60) |
active#(incr(cons(X,XS))) |
→ |
incr#(XS) |
(61) |
active#(take(s(N),cons(X,XS))) |
→ |
cons#(X,take(N,XS)) |
(62) |
active#(take(s(N),cons(X,XS))) |
→ |
take#(N,XS) |
(63) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
cons#(pair(X,Y),zip(XS,YS)) |
(64) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
pair#(X,Y) |
(65) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
zip#(XS,YS) |
(66) |
active#(repItems(cons(X,XS))) |
→ |
cons#(X,cons(X,repItems(XS))) |
(67) |
active#(repItems(cons(X,XS))) |
→ |
cons#(X,repItems(XS)) |
(68) |
active#(repItems(cons(X,XS))) |
→ |
repItems#(XS) |
(69) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(70) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(71) |
active#(incr(X)) |
→ |
incr#(active(X)) |
(72) |
active#(incr(X)) |
→ |
active#(X) |
(73) |
active#(s(X)) |
→ |
s#(active(X)) |
(74) |
active#(s(X)) |
→ |
active#(X) |
(75) |
active#(take(X1,X2)) |
→ |
take#(active(X1),X2) |
(76) |
active#(take(X1,X2)) |
→ |
active#(X1) |
(77) |
active#(take(X1,X2)) |
→ |
take#(X1,active(X2)) |
(78) |
active#(take(X1,X2)) |
→ |
active#(X2) |
(79) |
active#(zip(X1,X2)) |
→ |
zip#(active(X1),X2) |
(80) |
active#(zip(X1,X2)) |
→ |
active#(X1) |
(81) |
active#(zip(X1,X2)) |
→ |
zip#(X1,active(X2)) |
(82) |
active#(zip(X1,X2)) |
→ |
active#(X2) |
(83) |
active#(pair(X1,X2)) |
→ |
pair#(active(X1),X2) |
(84) |
active#(pair(X1,X2)) |
→ |
active#(X1) |
(85) |
active#(pair(X1,X2)) |
→ |
pair#(X1,active(X2)) |
(86) |
active#(pair(X1,X2)) |
→ |
active#(X2) |
(87) |
active#(tail(X)) |
→ |
tail#(active(X)) |
(88) |
active#(tail(X)) |
→ |
active#(X) |
(89) |
active#(repItems(X)) |
→ |
repItems#(active(X)) |
(90) |
active#(repItems(X)) |
→ |
active#(X) |
(91) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(92) |
incr#(mark(X)) |
→ |
incr#(X) |
(93) |
s#(mark(X)) |
→ |
s#(X) |
(94) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(95) |
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(96) |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(97) |
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(98) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(99) |
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(100) |
tail#(mark(X)) |
→ |
tail#(X) |
(101) |
repItems#(mark(X)) |
→ |
repItems#(X) |
(102) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(103) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(104) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(105) |
proper#(incr(X)) |
→ |
incr#(proper(X)) |
(106) |
proper#(incr(X)) |
→ |
proper#(X) |
(107) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(108) |
proper#(s(X)) |
→ |
proper#(X) |
(109) |
proper#(take(X1,X2)) |
→ |
take#(proper(X1),proper(X2)) |
(110) |
proper#(take(X1,X2)) |
→ |
proper#(X1) |
(111) |
proper#(take(X1,X2)) |
→ |
proper#(X2) |
(112) |
proper#(zip(X1,X2)) |
→ |
zip#(proper(X1),proper(X2)) |
(113) |
proper#(zip(X1,X2)) |
→ |
proper#(X1) |
(114) |
proper#(zip(X1,X2)) |
→ |
proper#(X2) |
(115) |
proper#(pair(X1,X2)) |
→ |
pair#(proper(X1),proper(X2)) |
(116) |
proper#(pair(X1,X2)) |
→ |
proper#(X1) |
(117) |
proper#(pair(X1,X2)) |
→ |
proper#(X2) |
(118) |
proper#(tail(X)) |
→ |
tail#(proper(X)) |
(119) |
proper#(tail(X)) |
→ |
proper#(X) |
(120) |
proper#(repItems(X)) |
→ |
repItems#(proper(X)) |
(121) |
proper#(repItems(X)) |
→ |
proper#(X) |
(122) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(123) |
incr#(ok(X)) |
→ |
incr#(X) |
(124) |
s#(ok(X)) |
→ |
s#(X) |
(125) |
take#(ok(X1),ok(X2)) |
→ |
take#(X1,X2) |
(126) |
zip#(ok(X1),ok(X2)) |
→ |
zip#(X1,X2) |
(127) |
pair#(ok(X1),ok(X2)) |
→ |
pair#(X1,X2) |
(128) |
tail#(ok(X)) |
→ |
tail#(X) |
(129) |
repItems#(ok(X)) |
→ |
repItems#(X) |
(130) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(131) |
top#(mark(X)) |
→ |
proper#(X) |
(132) |
top#(ok(X)) |
→ |
top#(active(X)) |
(133) |
top#(ok(X)) |
→ |
active#(X) |
(134) |
The dependency pairs are split into 11
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(133) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(131) |
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[pairNs] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
2 · x1
|
[oddNs] |
= |
0 |
[s(x1)] |
= |
2 · x1
|
[take(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[nil] |
= |
0 |
[zip(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[pair(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[tail(x1)] |
= |
1 · x1
|
[repItems(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
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) |
repItems(mark(X)) |
→ |
mark(repItems(X)) |
(33) |
repItems(ok(X)) |
→ |
ok(repItems(X)) |
(53) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(32) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(52) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
zip(mark(X1),X2) |
→ |
mark(zip(X1,X2)) |
(28) |
zip(X1,mark(X2)) |
→ |
mark(zip(X1,X2)) |
(29) |
zip(ok(X1),ok(X2)) |
→ |
ok(zip(X1,X2)) |
(50) |
take(mark(X1),X2) |
→ |
mark(take(X1,X2)) |
(26) |
take(X1,mark(X2)) |
→ |
mark(take(X1,X2)) |
(27) |
take(ok(X1),ok(X2)) |
→ |
ok(take(X1,X2)) |
(49) |
s(mark(X)) |
→ |
mark(s(X)) |
(25) |
s(ok(X)) |
→ |
ok(s(X)) |
(48) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
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(s(N),cons(X,XS))) |
→ |
mark(cons(X,take(N,XS))) |
(5) |
active(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(pair(X,Y),zip(XS,YS))) |
(8) |
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) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(mark) |
= |
1 |
|
stat(mark) |
= |
lex
|
prec(pairNs) |
= |
3 |
|
stat(pairNs) |
= |
lex
|
prec(0) |
= |
2 |
|
stat(0) |
= |
lex
|
prec(incr) |
= |
1 |
|
stat(incr) |
= |
lex
|
prec(oddNs) |
= |
4 |
|
stat(oddNs) |
= |
lex
|
prec(take) |
= |
5 |
|
stat(take) |
= |
lex
|
prec(zip) |
= |
7 |
|
stat(zip) |
= |
lex
|
prec(pair) |
= |
6 |
|
stat(pair) |
= |
lex
|
prec(repItems) |
= |
8 |
|
stat(repItems) |
= |
lex
|
prec(tail) |
= |
1 |
|
stat(tail) |
= |
lex
|
prec(nil) |
= |
9 |
|
stat(nil) |
= |
lex
|
π(top#) |
= |
[1] |
π(ok) |
= |
1 |
π(active) |
= |
1 |
π(mark) |
= |
[1] |
π(proper) |
= |
1 |
π(pairNs) |
= |
[] |
π(cons) |
= |
1 |
π(0) |
= |
[] |
π(incr) |
= |
[1] |
π(oddNs) |
= |
[] |
π(s) |
= |
1 |
π(take) |
= |
[1,2] |
π(zip) |
= |
[1,2] |
π(pair) |
= |
[2,1] |
π(repItems) |
= |
[1] |
π(tail) |
= |
[1] |
π(nil) |
= |
[] |
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(131) |
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[pairNs] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
2 · x1
|
[oddNs] |
= |
0 |
[s(x1)] |
= |
2 · x1
|
[take(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[zip(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[pair(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[repItems(x1)] |
= |
2 · x1
|
[tail(x1)] |
= |
2 · x1
|
[ok(x1)] |
= |
2 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
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(s(N),cons(X,XS))) |
→ |
mark(cons(X,take(N,XS))) |
(5) |
active(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(pair(X,Y),zip(XS,YS))) |
(8) |
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) |
repItems(mark(X)) |
→ |
mark(repItems(X)) |
(33) |
repItems(ok(X)) |
→ |
ok(repItems(X)) |
(53) |
tail(mark(X)) |
→ |
mark(tail(X)) |
(32) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(52) |
pair(mark(X1),X2) |
→ |
mark(pair(X1,X2)) |
(30) |
pair(X1,mark(X2)) |
→ |
mark(pair(X1,X2)) |
(31) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
zip(mark(X1),X2) |
→ |
mark(zip(X1,X2)) |
(28) |
zip(X1,mark(X2)) |
→ |
mark(zip(X1,X2)) |
(29) |
zip(ok(X1),ok(X2)) |
→ |
ok(zip(X1,X2)) |
(50) |
take(mark(X1),X2) |
→ |
mark(take(X1,X2)) |
(26) |
take(X1,mark(X2)) |
→ |
mark(take(X1,X2)) |
(27) |
take(ok(X1),ok(X2)) |
→ |
ok(take(X1,X2)) |
(49) |
s(mark(X)) |
→ |
mark(s(X)) |
(25) |
s(ok(X)) |
→ |
ok(s(X)) |
(48) |
incr(mark(X)) |
→ |
mark(incr(X)) |
(24) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(23) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
2 · x1
|
[pairNs] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
2 · x1
|
[oddNs] |
= |
0 |
[s(x1)] |
= |
2 · x1
|
[take(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[zip(x1, x2)] |
= |
1 + 1 · x1 + 2 · x2
|
[pair(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[repItems(x1)] |
= |
2 · x1
|
[tail(x1)] |
= |
2 · x1
|
[ok(x1)] |
= |
1 + 2 · x1
|
[top#(x1)] |
= |
2 · x1
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(133) |
and
the
rules
active(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark(cons(pair(X,Y),zip(XS,YS))) |
(8) |
active(zip(X1,X2)) |
→ |
zip(active(X1),X2) |
(17) |
active(zip(X1,X2)) |
→ |
zip(X1,active(X2)) |
(18) |
repItems(ok(X)) |
→ |
ok(repItems(X)) |
(53) |
tail(ok(X)) |
→ |
ok(tail(X)) |
(52) |
pair(ok(X1),ok(X2)) |
→ |
ok(pair(X1,X2)) |
(51) |
zip(ok(X1),ok(X2)) |
→ |
ok(zip(X1,X2)) |
(50) |
take(ok(X1),ok(X2)) |
→ |
ok(take(X1,X2)) |
(49) |
s(ok(X)) |
→ |
ok(s(X)) |
(48) |
incr(ok(X)) |
→ |
ok(incr(X)) |
(47) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(46) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(incr(X)) |
→ |
active#(X) |
(73) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(71) |
active#(s(X)) |
→ |
active#(X) |
(75) |
active#(take(X1,X2)) |
→ |
active#(X1) |
(77) |
active#(take(X1,X2)) |
→ |
active#(X2) |
(79) |
active#(zip(X1,X2)) |
→ |
active#(X1) |
(81) |
active#(zip(X1,X2)) |
→ |
active#(X2) |
(83) |
active#(pair(X1,X2)) |
→ |
active#(X1) |
(85) |
active#(pair(X1,X2)) |
→ |
active#(X2) |
(87) |
active#(tail(X)) |
→ |
active#(X) |
(89) |
active#(repItems(X)) |
→ |
active#(X) |
(91) |
1.1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[incr(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[take(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[zip(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[pair(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[tail(x1)] |
= |
1 · x1
|
[repItems(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(incr(X)) |
→ |
active#(X) |
(73) |
|
1 |
> |
1 |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(71) |
|
1 |
> |
1 |
active#(s(X)) |
→ |
active#(X) |
(75) |
|
1 |
> |
1 |
active#(take(X1,X2)) |
→ |
active#(X1) |
(77) |
|
1 |
> |
1 |
active#(take(X1,X2)) |
→ |
active#(X2) |
(79) |
|
1 |
> |
1 |
active#(zip(X1,X2)) |
→ |
active#(X1) |
(81) |
|
1 |
> |
1 |
active#(zip(X1,X2)) |
→ |
active#(X2) |
(83) |
|
1 |
> |
1 |
active#(pair(X1,X2)) |
→ |
active#(X1) |
(85) |
|
1 |
> |
1 |
active#(pair(X1,X2)) |
→ |
active#(X2) |
(87) |
|
1 |
> |
1 |
active#(tail(X)) |
→ |
active#(X) |
(89) |
|
1 |
> |
1 |
active#(repItems(X)) |
→ |
active#(X) |
(91) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(105) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(104) |
proper#(incr(X)) |
→ |
proper#(X) |
(107) |
proper#(s(X)) |
→ |
proper#(X) |
(109) |
proper#(take(X1,X2)) |
→ |
proper#(X1) |
(111) |
proper#(take(X1,X2)) |
→ |
proper#(X2) |
(112) |
proper#(zip(X1,X2)) |
→ |
proper#(X1) |
(114) |
proper#(zip(X1,X2)) |
→ |
proper#(X2) |
(115) |
proper#(pair(X1,X2)) |
→ |
proper#(X1) |
(117) |
proper#(pair(X1,X2)) |
→ |
proper#(X2) |
(118) |
proper#(tail(X)) |
→ |
proper#(X) |
(120) |
proper#(repItems(X)) |
→ |
proper#(X) |
(122) |
1.1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[incr(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[take(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[zip(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[pair(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[tail(x1)] |
= |
1 · x1
|
[repItems(x1)] |
= |
1 · x1
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(105) |
|
1 |
> |
1 |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(104) |
|
1 |
> |
1 |
proper#(incr(X)) |
→ |
proper#(X) |
(107) |
|
1 |
> |
1 |
proper#(s(X)) |
→ |
proper#(X) |
(109) |
|
1 |
> |
1 |
proper#(take(X1,X2)) |
→ |
proper#(X1) |
(111) |
|
1 |
> |
1 |
proper#(take(X1,X2)) |
→ |
proper#(X2) |
(112) |
|
1 |
> |
1 |
proper#(zip(X1,X2)) |
→ |
proper#(X1) |
(114) |
|
1 |
> |
1 |
proper#(zip(X1,X2)) |
→ |
proper#(X2) |
(115) |
|
1 |
> |
1 |
proper#(pair(X1,X2)) |
→ |
proper#(X1) |
(117) |
|
1 |
> |
1 |
proper#(pair(X1,X2)) |
→ |
proper#(X2) |
(118) |
|
1 |
> |
1 |
proper#(tail(X)) |
→ |
proper#(X) |
(120) |
|
1 |
> |
1 |
proper#(repItems(X)) |
→ |
proper#(X) |
(122) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(123) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(92) |
1.1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(123) |
|
1 |
> |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(92) |
|
1 |
> |
1 |
2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
incr#(ok(X)) |
→ |
incr#(X) |
(124) |
incr#(mark(X)) |
→ |
incr#(X) |
(93) |
1.1.1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[incr#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
incr#(ok(X)) |
→ |
incr#(X) |
(124) |
|
1 |
> |
1 |
incr#(mark(X)) |
→ |
incr#(X) |
(93) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(125) |
s#(mark(X)) |
→ |
s#(X) |
(94) |
1.1.1.1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(ok(X)) |
→ |
s#(X) |
(125) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(94) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(96) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(95) |
take#(ok(X1),ok(X2)) |
→ |
take#(X1,X2) |
(126) |
1.1.1.1.1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[take#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(96) |
|
1 |
≥ |
1 |
2 |
> |
2 |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(95) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(ok(X1),ok(X2)) |
→ |
take#(X1,X2) |
(126) |
|
1 |
> |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(98) |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(97) |
zip#(ok(X1),ok(X2)) |
→ |
zip#(X1,X2) |
(127) |
1.1.1.1.1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[zip#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(98) |
|
1 |
≥ |
1 |
2 |
> |
2 |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(97) |
|
1 |
> |
1 |
2 |
≥ |
2 |
zip#(ok(X1),ok(X2)) |
→ |
zip#(X1,X2) |
(127) |
|
1 |
> |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(100) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(99) |
pair#(ok(X1),ok(X2)) |
→ |
pair#(X1,X2) |
(128) |
1.1.1.1.1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 · x1
|
[pair#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(100) |
|
1 |
≥ |
1 |
2 |
> |
2 |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(99) |
|
1 |
> |
1 |
2 |
≥ |
2 |
pair#(ok(X1),ok(X2)) |
→ |
pair#(X1,X2) |
(128) |
|
1 |
> |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
tail#(ok(X)) |
→ |
tail#(X) |
(129) |
tail#(mark(X)) |
→ |
tail#(X) |
(101) |
1.1.1.1.1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[tail#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tail#(ok(X)) |
→ |
tail#(X) |
(129) |
|
1 |
> |
1 |
tail#(mark(X)) |
→ |
tail#(X) |
(101) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
repItems#(ok(X)) |
→ |
repItems#(X) |
(130) |
repItems#(mark(X)) |
→ |
repItems#(X) |
(102) |
1.1.1.1.1.1.11 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[repItems#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.11.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
repItems#(ok(X)) |
→ |
repItems#(X) |
(130) |
|
1 |
> |
1 |
repItems#(mark(X)) |
→ |
repItems#(X) |
(102) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.