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) |
mark(pairNs) |
→ |
active(pairNs) |
(12) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(13) |
mark(0) |
→ |
active(0) |
(14) |
mark(incr(X)) |
→ |
active(incr(mark(X))) |
(15) |
mark(oddNs) |
→ |
active(oddNs) |
(16) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(17) |
mark(take(X1,X2)) |
→ |
active(take(mark(X1),mark(X2))) |
(18) |
mark(nil) |
→ |
active(nil) |
(19) |
mark(zip(X1,X2)) |
→ |
active(zip(mark(X1),mark(X2))) |
(20) |
mark(pair(X1,X2)) |
→ |
active(pair(mark(X1),mark(X2))) |
(21) |
mark(tail(X)) |
→ |
active(tail(mark(X))) |
(22) |
mark(repItems(X)) |
→ |
active(repItems(mark(X))) |
(23) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(24) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(25) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(26) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(27) |
incr(mark(X)) |
→ |
incr(X) |
(28) |
incr(active(X)) |
→ |
incr(X) |
(29) |
s(mark(X)) |
→ |
s(X) |
(30) |
s(active(X)) |
→ |
s(X) |
(31) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(32) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(33) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(34) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(35) |
zip(mark(X1),X2) |
→ |
zip(X1,X2) |
(36) |
zip(X1,mark(X2)) |
→ |
zip(X1,X2) |
(37) |
zip(active(X1),X2) |
→ |
zip(X1,X2) |
(38) |
zip(X1,active(X2)) |
→ |
zip(X1,X2) |
(39) |
pair(mark(X1),X2) |
→ |
pair(X1,X2) |
(40) |
pair(X1,mark(X2)) |
→ |
pair(X1,X2) |
(41) |
pair(active(X1),X2) |
→ |
pair(X1,X2) |
(42) |
pair(X1,active(X2)) |
→ |
pair(X1,X2) |
(43) |
tail(mark(X)) |
→ |
tail(X) |
(44) |
tail(active(X)) |
→ |
tail(X) |
(45) |
repItems(mark(X)) |
→ |
repItems(X) |
(46) |
repItems(active(X)) |
→ |
repItems(X) |
(47) |
active#(pairNs) |
→ |
mark#(cons(0,incr(oddNs))) |
(48) |
active#(pairNs) |
→ |
cons#(0,incr(oddNs)) |
(49) |
active#(pairNs) |
→ |
incr#(oddNs) |
(50) |
active#(oddNs) |
→ |
mark#(incr(pairNs)) |
(51) |
active#(oddNs) |
→ |
incr#(pairNs) |
(52) |
active#(incr(cons(X,XS))) |
→ |
mark#(cons(s(X),incr(XS))) |
(53) |
active#(incr(cons(X,XS))) |
→ |
cons#(s(X),incr(XS)) |
(54) |
active#(incr(cons(X,XS))) |
→ |
s#(X) |
(55) |
active#(incr(cons(X,XS))) |
→ |
incr#(XS) |
(56) |
active#(take(s(N),cons(X,XS))) |
→ |
mark#(cons(X,take(N,XS))) |
(57) |
active#(take(s(N),cons(X,XS))) |
→ |
cons#(X,take(N,XS)) |
(58) |
active#(take(s(N),cons(X,XS))) |
→ |
take#(N,XS) |
(59) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark#(cons(pair(X,Y),zip(XS,YS))) |
(60) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
cons#(pair(X,Y),zip(XS,YS)) |
(61) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
pair#(X,Y) |
(62) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
zip#(XS,YS) |
(63) |
active#(repItems(cons(X,XS))) |
→ |
mark#(cons(X,cons(X,repItems(XS)))) |
(64) |
active#(repItems(cons(X,XS))) |
→ |
cons#(X,cons(X,repItems(XS))) |
(65) |
active#(repItems(cons(X,XS))) |
→ |
cons#(X,repItems(XS)) |
(66) |
active#(repItems(cons(X,XS))) |
→ |
repItems#(XS) |
(67) |
mark#(pairNs) |
→ |
active#(pairNs) |
(68) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(69) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(70) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(71) |
mark#(0) |
→ |
active#(0) |
(72) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(73) |
mark#(incr(X)) |
→ |
incr#(mark(X)) |
(74) |
mark#(incr(X)) |
→ |
mark#(X) |
(75) |
mark#(oddNs) |
→ |
active#(oddNs) |
(76) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(77) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(78) |
mark#(s(X)) |
→ |
mark#(X) |
(79) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(80) |
mark#(take(X1,X2)) |
→ |
take#(mark(X1),mark(X2)) |
(81) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(82) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(83) |
mark#(nil) |
→ |
active#(nil) |
(84) |
mark#(zip(X1,X2)) |
→ |
active#(zip(mark(X1),mark(X2))) |
(85) |
mark#(zip(X1,X2)) |
→ |
zip#(mark(X1),mark(X2)) |
(86) |
mark#(zip(X1,X2)) |
→ |
mark#(X1) |
(87) |
mark#(zip(X1,X2)) |
→ |
mark#(X2) |
(88) |
mark#(pair(X1,X2)) |
→ |
active#(pair(mark(X1),mark(X2))) |
(89) |
mark#(pair(X1,X2)) |
→ |
pair#(mark(X1),mark(X2)) |
(90) |
mark#(pair(X1,X2)) |
→ |
mark#(X1) |
(91) |
mark#(pair(X1,X2)) |
→ |
mark#(X2) |
(92) |
mark#(tail(X)) |
→ |
active#(tail(mark(X))) |
(93) |
mark#(tail(X)) |
→ |
tail#(mark(X)) |
(94) |
mark#(tail(X)) |
→ |
mark#(X) |
(95) |
mark#(repItems(X)) |
→ |
active#(repItems(mark(X))) |
(96) |
mark#(repItems(X)) |
→ |
repItems#(mark(X)) |
(97) |
mark#(repItems(X)) |
→ |
mark#(X) |
(98) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(99) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(100) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(101) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(102) |
incr#(mark(X)) |
→ |
incr#(X) |
(103) |
incr#(active(X)) |
→ |
incr#(X) |
(104) |
s#(mark(X)) |
→ |
s#(X) |
(105) |
s#(active(X)) |
→ |
s#(X) |
(106) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(107) |
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(108) |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(109) |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(110) |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(111) |
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(112) |
zip#(active(X1),X2) |
→ |
zip#(X1,X2) |
(113) |
zip#(X1,active(X2)) |
→ |
zip#(X1,X2) |
(114) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(115) |
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(116) |
pair#(active(X1),X2) |
→ |
pair#(X1,X2) |
(117) |
pair#(X1,active(X2)) |
→ |
pair#(X1,X2) |
(118) |
tail#(mark(X)) |
→ |
tail#(X) |
(119) |
tail#(active(X)) |
→ |
tail#(X) |
(120) |
repItems#(mark(X)) |
→ |
repItems#(X) |
(121) |
repItems#(active(X)) |
→ |
repItems#(X) |
(122) |
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(69) |
active#(incr(cons(X,XS))) |
→ |
mark#(cons(s(X),incr(XS))) |
(53) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(71) |
mark#(pairNs) |
→ |
active#(pairNs) |
(68) |
active#(pairNs) |
→ |
mark#(cons(0,incr(oddNs))) |
(48) |
mark#(incr(X)) |
→ |
active#(incr(mark(X))) |
(73) |
active#(take(s(N),cons(X,XS))) |
→ |
mark#(cons(X,take(N,XS))) |
(57) |
mark#(incr(X)) |
→ |
mark#(X) |
(75) |
mark#(oddNs) |
→ |
active#(oddNs) |
(76) |
active#(oddNs) |
→ |
mark#(incr(pairNs)) |
(51) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(77) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark#(cons(pair(X,Y),zip(XS,YS))) |
(60) |
mark#(s(X)) |
→ |
mark#(X) |
(79) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(80) |
active#(repItems(cons(X,XS))) |
→ |
mark#(cons(X,cons(X,repItems(XS)))) |
(64) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(82) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(83) |
mark#(zip(X1,X2)) |
→ |
active#(zip(mark(X1),mark(X2))) |
(85) |
mark#(zip(X1,X2)) |
→ |
mark#(X1) |
(87) |
mark#(zip(X1,X2)) |
→ |
mark#(X2) |
(88) |
mark#(pair(X1,X2)) |
→ |
active#(pair(mark(X1),mark(X2))) |
(89) |
mark#(pair(X1,X2)) |
→ |
mark#(X1) |
(91) |
mark#(pair(X1,X2)) |
→ |
mark#(X2) |
(92) |
mark#(tail(X)) |
→ |
active#(tail(mark(X))) |
(93) |
mark#(tail(X)) |
→ |
mark#(X) |
(95) |
mark#(repItems(X)) |
→ |
active#(repItems(mark(X))) |
(96) |
mark#(repItems(X)) |
→ |
mark#(X) |
(98) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor
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)] |
= |
1 · x1
|
[oddNs] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[take(x1, x2)] |
= |
2 + 1 · x1 + 1 · x2
|
[zip(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[pair(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[repItems(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[tail(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pairs
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(82) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(83) |
and
no rules
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
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)] |
= |
1 · x1
|
[take(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[zip(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[pair(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[repItems(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[tail(x1)] |
= |
2 + 1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
the
pair
mark#(tail(X)) |
→ |
mark#(X) |
(95) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[pairNs] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
1 · x1
|
[oddNs] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[take(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[zip(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
[pair(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[repItems(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[tail(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pairs
mark#(zip(X1,X2)) |
→ |
mark#(X1) |
(87) |
mark#(zip(X1,X2)) |
→ |
mark#(X2) |
(88) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[pairNs] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[0] |
= |
0 |
[incr(x1)] |
= |
2 · x1
|
[oddNs] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[take(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[zip(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[pair(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[repItems(x1)] |
= |
1 + 2 · x1
|
[nil] |
= |
0 |
[tail(x1)] |
= |
2 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
the
pair
mark#(repItems(X)) |
→ |
mark#(X) |
(98) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[cons(x1, x2)] |
= |
-2 |
[incr(x1)] |
= |
1 |
[pair(x1, x2)] |
= |
0 |
[repItems(x1)] |
= |
1 |
[s(x1)] |
= |
0 |
[tail(x1)] |
= |
-2 |
[take(x1, x2)] |
= |
1 |
[zip(x1, x2)] |
= |
1 |
[mark(x1)] |
= |
1 |
[active(x1)] |
= |
2 |
[pairNs] |
= |
1 |
[0] |
= |
0 |
[oddNs] |
= |
1 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(25) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(24) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(26) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(27) |
s(active(X)) |
→ |
s(X) |
(31) |
s(mark(X)) |
→ |
s(X) |
(30) |
incr(active(X)) |
→ |
incr(X) |
(29) |
incr(mark(X)) |
→ |
incr(X) |
(28) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(33) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(32) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(34) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(35) |
pair(X1,mark(X2)) |
→ |
pair(X1,X2) |
(41) |
pair(mark(X1),X2) |
→ |
pair(X1,X2) |
(40) |
pair(active(X1),X2) |
→ |
pair(X1,X2) |
(42) |
pair(X1,active(X2)) |
→ |
pair(X1,X2) |
(43) |
zip(X1,mark(X2)) |
→ |
zip(X1,X2) |
(37) |
zip(mark(X1),X2) |
→ |
zip(X1,X2) |
(36) |
zip(active(X1),X2) |
→ |
zip(X1,X2) |
(38) |
zip(X1,active(X2)) |
→ |
zip(X1,X2) |
(39) |
repItems(active(X)) |
→ |
repItems(X) |
(47) |
repItems(mark(X)) |
→ |
repItems(X) |
(46) |
tail(active(X)) |
→ |
tail(X) |
(45) |
tail(mark(X)) |
→ |
tail(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(69) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(77) |
mark#(pair(X1,X2)) |
→ |
active#(pair(mark(X1),mark(X2))) |
(89) |
mark#(tail(X)) |
→ |
active#(tail(mark(X))) |
(93) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(pairNs) |
= |
1 |
|
weight(pairNs) |
= |
1 |
|
|
|
prec(0) |
= |
0 |
|
weight(0) |
= |
1 |
|
|
|
prec(oddNs) |
= |
2 |
|
weight(oddNs) |
= |
1 |
|
|
|
prec(zip) |
= |
4 |
|
weight(zip) |
= |
2 |
|
|
|
prec(pair) |
= |
5 |
|
weight(pair) |
= |
1 |
|
|
|
prec(tail) |
= |
3 |
|
weight(tail) |
= |
2 |
|
|
|
prec(nil) |
= |
6 |
|
weight(nil) |
= |
2 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(incr) |
= |
1 |
π(cons) |
= |
1 |
π(mark#) |
= |
1 |
π(s) |
= |
1 |
π(pairNs) |
= |
[] |
π(0) |
= |
[] |
π(mark) |
= |
1 |
π(take) |
= |
2 |
π(oddNs) |
= |
[] |
π(zip) |
= |
[1,2] |
π(pair) |
= |
[1,2] |
π(repItems) |
= |
1 |
π(active) |
= |
1 |
π(tail) |
= |
[] |
π(nil) |
= |
[] |
the
pairs
active#(pairNs) |
→ |
mark#(cons(0,incr(oddNs))) |
(48) |
active#(oddNs) |
→ |
mark#(incr(pairNs)) |
(51) |
active#(zip(cons(X,XS),cons(Y,YS))) |
→ |
mark#(cons(pair(X,Y),zip(XS,YS))) |
(60) |
mark#(pair(X1,X2)) |
→ |
mark#(X1) |
(91) |
mark#(pair(X1,X2)) |
→ |
mark#(X2) |
(92) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(100) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(99) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(101) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(102) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(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.
1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(100) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(99) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(101) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(102) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
incr#(active(X)) |
→ |
incr#(X) |
(104) |
incr#(mark(X)) |
→ |
incr#(X) |
(103) |
1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(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.
1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
incr#(active(X)) |
→ |
incr#(X) |
(104) |
|
1 |
> |
1 |
incr#(mark(X)) |
→ |
incr#(X) |
(103) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(106) |
s#(mark(X)) |
→ |
s#(X) |
(105) |
1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(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.
1.1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(106) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(105) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(108) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(107) |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(109) |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(110) |
1.1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(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.
1.1.1.1.1.5.1 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) |
(108) |
|
1 |
≥ |
1 |
2 |
> |
2 |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(107) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(109) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(110) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
zip#(X1,mark(X2)) |
→ |
zip#(X1,X2) |
(112) |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(111) |
zip#(active(X1),X2) |
→ |
zip#(X1,X2) |
(113) |
zip#(X1,active(X2)) |
→ |
zip#(X1,X2) |
(114) |
1.1.1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(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.
1.1.1.1.1.6.1 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) |
(112) |
|
1 |
≥ |
1 |
2 |
> |
2 |
zip#(mark(X1),X2) |
→ |
zip#(X1,X2) |
(111) |
|
1 |
> |
1 |
2 |
≥ |
2 |
zip#(active(X1),X2) |
→ |
zip#(X1,X2) |
(113) |
|
1 |
> |
1 |
2 |
≥ |
2 |
zip#(X1,active(X2)) |
→ |
zip#(X1,X2) |
(114) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
pair#(X1,mark(X2)) |
→ |
pair#(X1,X2) |
(116) |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(115) |
pair#(active(X1),X2) |
→ |
pair#(X1,X2) |
(117) |
pair#(X1,active(X2)) |
→ |
pair#(X1,X2) |
(118) |
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
|
[active(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.
1.1.1.1.1.7.1 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) |
(116) |
|
1 |
≥ |
1 |
2 |
> |
2 |
pair#(mark(X1),X2) |
→ |
pair#(X1,X2) |
(115) |
|
1 |
> |
1 |
2 |
≥ |
2 |
pair#(active(X1),X2) |
→ |
pair#(X1,X2) |
(117) |
|
1 |
> |
1 |
2 |
≥ |
2 |
pair#(X1,active(X2)) |
→ |
pair#(X1,X2) |
(118) |
|
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
tail#(active(X)) |
→ |
tail#(X) |
(120) |
tail#(mark(X)) |
→ |
tail#(X) |
(119) |
1.1.1.1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(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.
1.1.1.1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
tail#(active(X)) |
→ |
tail#(X) |
(120) |
|
1 |
> |
1 |
tail#(mark(X)) |
→ |
tail#(X) |
(119) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
repItems#(active(X)) |
→ |
repItems#(X) |
(122) |
repItems#(mark(X)) |
→ |
repItems#(X) |
(121) |
1.1.1.1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(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.
1.1.1.1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
repItems#(active(X)) |
→ |
repItems#(X) |
(122) |
|
1 |
> |
1 |
repItems#(mark(X)) |
→ |
repItems#(X) |
(121) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.