The rewrite relation of the following TRS is considered.
active(and(tt,T)) |
→ |
mark(T) |
(1) |
active(isNatIList(IL)) |
→ |
mark(isNatList(IL)) |
(2) |
active(isNat(0)) |
→ |
mark(tt) |
(3) |
active(isNat(s(N))) |
→ |
mark(isNat(N)) |
(4) |
active(isNat(length(L))) |
→ |
mark(isNatList(L)) |
(5) |
active(isNatIList(zeros)) |
→ |
mark(tt) |
(6) |
active(isNatIList(cons(N,IL))) |
→ |
mark(and(isNat(N),isNatIList(IL))) |
(7) |
active(isNatList(nil)) |
→ |
mark(tt) |
(8) |
active(isNatList(cons(N,L))) |
→ |
mark(and(isNat(N),isNatList(L))) |
(9) |
active(isNatList(take(N,IL))) |
→ |
mark(and(isNat(N),isNatIList(IL))) |
(10) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(11) |
active(take(0,IL)) |
→ |
mark(uTake1(isNatIList(IL))) |
(12) |
active(uTake1(tt)) |
→ |
mark(nil) |
(13) |
active(take(s(M),cons(N,IL))) |
→ |
mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(14) |
active(uTake2(tt,M,N,IL)) |
→ |
mark(cons(N,take(M,IL))) |
(15) |
active(length(cons(N,L))) |
→ |
mark(uLength(and(isNat(N),isNatList(L)),L)) |
(16) |
active(uLength(tt,L)) |
→ |
mark(s(length(L))) |
(17) |
mark(and(X1,X2)) |
→ |
active(and(mark(X1),mark(X2))) |
(18) |
mark(tt) |
→ |
active(tt) |
(19) |
mark(isNatIList(X)) |
→ |
active(isNatIList(X)) |
(20) |
mark(isNatList(X)) |
→ |
active(isNatList(X)) |
(21) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(22) |
mark(0) |
→ |
active(0) |
(23) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(24) |
mark(length(X)) |
→ |
active(length(mark(X))) |
(25) |
mark(zeros) |
→ |
active(zeros) |
(26) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(27) |
mark(nil) |
→ |
active(nil) |
(28) |
mark(take(X1,X2)) |
→ |
active(take(mark(X1),mark(X2))) |
(29) |
mark(uTake1(X)) |
→ |
active(uTake1(mark(X))) |
(30) |
mark(uTake2(X1,X2,X3,X4)) |
→ |
active(uTake2(mark(X1),X2,X3,X4)) |
(31) |
mark(uLength(X1,X2)) |
→ |
active(uLength(mark(X1),X2)) |
(32) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(34) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(35) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(36) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(37) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(38) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(39) |
isNatList(active(X)) |
→ |
isNatList(X) |
(40) |
isNat(mark(X)) |
→ |
isNat(X) |
(41) |
isNat(active(X)) |
→ |
isNat(X) |
(42) |
s(mark(X)) |
→ |
s(X) |
(43) |
s(active(X)) |
→ |
s(X) |
(44) |
length(mark(X)) |
→ |
length(X) |
(45) |
length(active(X)) |
→ |
length(X) |
(46) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(47) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(48) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(49) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(50) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(51) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(52) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(53) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(54) |
uTake1(mark(X)) |
→ |
uTake1(X) |
(55) |
uTake1(active(X)) |
→ |
uTake1(X) |
(56) |
uTake2(mark(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(57) |
uTake2(X1,mark(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(58) |
uTake2(X1,X2,mark(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(59) |
uTake2(X1,X2,X3,mark(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(60) |
uTake2(active(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(61) |
uTake2(X1,active(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(62) |
uTake2(X1,X2,active(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(63) |
uTake2(X1,X2,X3,active(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(64) |
uLength(mark(X1),X2) |
→ |
uLength(X1,X2) |
(65) |
uLength(X1,mark(X2)) |
→ |
uLength(X1,X2) |
(66) |
uLength(active(X1),X2) |
→ |
uLength(X1,X2) |
(67) |
uLength(X1,active(X2)) |
→ |
uLength(X1,X2) |
(68) |
The evaluation strategy is innermost.There are 115 ruless (increase limit for explicit display).
The dependency pairs are split into 12
components.
-
The
1st
component contains the
pair
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),mark(X2))) |
(115) |
active#(and(tt,T)) |
→ |
mark#(T) |
(69) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(117) |
mark#(and(X1,X2)) |
→ |
mark#(X2) |
(118) |
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(120) |
active#(isNatIList(IL)) |
→ |
mark#(isNatList(IL)) |
(70) |
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(121) |
active#(isNatList(cons(N,L))) |
→ |
mark#(and(isNat(N),isNatList(L))) |
(83) |
active#(isNatList(take(N,IL))) |
→ |
mark#(and(isNat(N),isNatIList(IL))) |
(87) |
active#(isNatIList(cons(N,IL))) |
→ |
mark#(and(isNat(N),isNatIList(IL))) |
(78) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(122) |
active#(isNat(s(N))) |
→ |
mark#(isNat(N)) |
(73) |
active#(isNat(length(L))) |
→ |
mark#(isNatList(L)) |
(75) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(124) |
active#(take(0,IL)) |
→ |
mark#(uTake1(isNatIList(IL))) |
(93) |
mark#(uTake1(X)) |
→ |
active#(uTake1(mark(X))) |
(139) |
active#(take(s(M),cons(N,IL))) |
→ |
mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(97) |
mark#(uTake2(X1,X2,X3,X4)) |
→ |
active#(uTake2(mark(X1),X2,X3,X4)) |
(142) |
active#(uTake2(tt,M,N,IL)) |
→ |
mark#(cons(N,take(M,IL))) |
(104) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(131) |
active#(length(cons(N,L))) |
→ |
mark#(uLength(and(isNat(N),isNatList(L)),L)) |
(107) |
mark#(uLength(X1,X2)) |
→ |
active#(uLength(mark(X1),X2)) |
(145) |
active#(uLength(tt,L)) |
→ |
mark#(s(length(L))) |
(112) |
mark#(s(X)) |
→ |
mark#(X) |
(126) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(127) |
mark#(length(X)) |
→ |
mark#(X) |
(129) |
mark#(zeros) |
→ |
active#(zeros) |
(130) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(91) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(133) |
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(135) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(137) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(138) |
mark#(uTake1(X)) |
→ |
mark#(X) |
(141) |
mark#(uTake2(X1,X2,X3,X4)) |
→ |
mark#(X1) |
(144) |
mark#(uLength(X1,X2)) |
→ |
mark#(X1) |
(147) |
1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
mark(and(X1,X2)) |
→ |
active(and(mark(X1),mark(X2))) |
(18) |
active(and(tt,T)) |
→ |
mark(T) |
(1) |
mark(isNatIList(X)) |
→ |
active(isNatIList(X)) |
(20) |
active(isNatIList(IL)) |
→ |
mark(isNatList(IL)) |
(2) |
mark(isNatList(X)) |
→ |
active(isNatList(X)) |
(21) |
active(isNatList(cons(N,L))) |
→ |
mark(and(isNat(N),isNatList(L))) |
(9) |
active(isNatList(take(N,IL))) |
→ |
mark(and(isNat(N),isNatIList(IL))) |
(10) |
active(isNatIList(cons(N,IL))) |
→ |
mark(and(isNat(N),isNatIList(IL))) |
(7) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(22) |
active(isNat(s(N))) |
→ |
mark(isNat(N)) |
(4) |
active(isNat(length(L))) |
→ |
mark(isNatList(L)) |
(5) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(24) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(11) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(27) |
active(take(0,IL)) |
→ |
mark(uTake1(isNatIList(IL))) |
(12) |
mark(uTake1(X)) |
→ |
active(uTake1(mark(X))) |
(30) |
active(take(s(M),cons(N,IL))) |
→ |
mark(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(14) |
mark(uTake2(X1,X2,X3,X4)) |
→ |
active(uTake2(mark(X1),X2,X3,X4)) |
(31) |
active(uTake2(tt,M,N,IL)) |
→ |
mark(cons(N,take(M,IL))) |
(15) |
active(length(cons(N,L))) |
→ |
mark(uLength(and(isNat(N),isNatList(L)),L)) |
(16) |
mark(uLength(X1,X2)) |
→ |
active(uLength(mark(X1),X2)) |
(32) |
active(uLength(tt,L)) |
→ |
mark(s(length(L))) |
(17) |
mark(length(X)) |
→ |
active(length(mark(X))) |
(25) |
mark(zeros) |
→ |
active(zeros) |
(26) |
mark(take(X1,X2)) |
→ |
active(take(mark(X1),mark(X2))) |
(29) |
mark(tt) |
→ |
active(tt) |
(19) |
mark(0) |
→ |
active(0) |
(23) |
mark(nil) |
→ |
active(nil) |
(28) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(52) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(51) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(53) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(54) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(34) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(33) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(35) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(36) |
active(isNat(0)) |
→ |
mark(tt) |
(3) |
active(isNatIList(zeros)) |
→ |
mark(tt) |
(6) |
active(isNatList(nil)) |
→ |
mark(tt) |
(8) |
active(uTake1(tt)) |
→ |
mark(nil) |
(13) |
s(active(X)) |
→ |
s(X) |
(44) |
s(mark(X)) |
→ |
s(X) |
(43) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(48) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(47) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(49) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(50) |
uTake1(active(X)) |
→ |
uTake1(X) |
(56) |
uTake1(mark(X)) |
→ |
uTake1(X) |
(55) |
uTake2(X1,mark(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(58) |
uTake2(mark(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(57) |
uTake2(X1,X2,mark(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(59) |
uTake2(X1,X2,X3,mark(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(60) |
uTake2(active(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(61) |
uTake2(X1,active(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(62) |
uTake2(X1,X2,active(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(63) |
uTake2(X1,X2,X3,active(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(64) |
uLength(X1,mark(X2)) |
→ |
uLength(X1,X2) |
(66) |
uLength(mark(X1),X2) |
→ |
uLength(X1,X2) |
(65) |
uLength(active(X1),X2) |
→ |
uLength(X1,X2) |
(67) |
uLength(X1,active(X2)) |
→ |
uLength(X1,X2) |
(68) |
length(active(X)) |
→ |
length(X) |
(46) |
length(mark(X)) |
→ |
length(X) |
(45) |
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[and(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 |
[length(x1)] |
= |
2 |
[s(x1)] |
= |
-2 |
[take(x1, x2)] |
= |
2 |
[uLength(x1, x2)] |
= |
2 |
[uTake1(x1)] |
= |
-2 |
[uTake2(x1,...,x4)] |
= |
2 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
-2 + x1
|
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
2 |
[isNatList(x1)] |
= |
2 |
[isNat(x1)] |
= |
2 |
[zeros] |
= |
2 |
[0] |
= |
0 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(34) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(33) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(35) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(36) |
s(active(X)) |
→ |
s(X) |
(44) |
s(mark(X)) |
→ |
s(X) |
(43) |
uTake1(active(X)) |
→ |
uTake1(X) |
(56) |
uTake1(mark(X)) |
→ |
uTake1(X) |
(55) |
uTake2(X1,mark(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(58) |
uTake2(mark(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(57) |
uTake2(X1,X2,mark(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(59) |
uTake2(X1,X2,X3,mark(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(60) |
uTake2(active(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(61) |
uTake2(X1,active(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(62) |
uTake2(X1,X2,active(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(63) |
uTake2(X1,X2,X3,active(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(64) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(48) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(47) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(49) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(50) |
uLength(X1,mark(X2)) |
→ |
uLength(X1,X2) |
(66) |
uLength(mark(X1),X2) |
→ |
uLength(X1,X2) |
(65) |
uLength(active(X1),X2) |
→ |
uLength(X1,X2) |
(67) |
uLength(X1,active(X2)) |
→ |
uLength(X1,X2) |
(68) |
length(active(X)) |
→ |
length(X) |
(46) |
length(mark(X)) |
→ |
length(X) |
(45) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(52) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(51) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(53) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(54) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(124) |
mark#(uTake1(X)) |
→ |
active#(uTake1(mark(X))) |
(139) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(131) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 · x1
|
[and(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[isNat(x1)] |
= |
0 |
[take(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 + 1 · x1
|
[0] |
= |
0 |
[uTake1(x1)] |
= |
1 · x1
|
[uTake2(x1,...,x4)] |
= |
1 · x1 + 1 · x2 + 1 · x3 + 1 · x4
|
[uLength(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
[zeros] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
the
pairs
mark#(length(X)) |
→ |
mark#(X) |
(129) |
mark#(uLength(X1,X2)) |
→ |
mark#(X1) |
(147) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 + 2 · x1
|
[and(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[length(x1)] |
= |
0 |
[take(x1, x2)] |
= |
x1 + x2
|
[uLength(x1, x2)] |
= |
-2 |
[uTake2(x1,...,x4)] |
= |
x1 + x2 + 2 · x3
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
2 · x1
|
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
x1 |
[zeros] |
= |
2 |
[0] |
= |
1 |
[uTake1(x1)] |
= |
x1 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 + 2 · x1
|
the
pair
active#(take(0,IL)) |
→ |
mark#(uTake1(isNatIList(IL))) |
(93) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[and(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[length(x1)] |
= |
-2 |
[take(x1, x2)] |
= |
2 + x1 + x2
|
[uLength(x1, x2)] |
= |
-2 |
[uTake2(x1,...,x4)] |
= |
2 · x1 + x2 + 2 · x3
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
2 · x1
|
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
x1 |
[zeros] |
= |
0 |
[0] |
= |
0 |
[uTake1(x1)] |
= |
x1 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
x1 |
the
pairs
active#(take(s(M),cons(N,IL))) |
→ |
mark#(uTake2(and(isNat(M),and(isNat(N),isNatIList(IL))),M,N,IL)) |
(97) |
mark#(take(X1,X2)) |
→ |
mark#(X1) |
(137) |
mark#(take(X1,X2)) |
→ |
mark#(X2) |
(138) |
could be deleted.
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
|
[and(x1, x2)] |
= |
1 |
[length(x1)] |
= |
1 |
[take(x1, x2)] |
= |
-2 |
[uLength(x1, x2)] |
= |
1 |
[uTake2(x1,...,x4)] |
= |
1 |
[mark(x1)] |
= |
2 |
[active(x1)] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
1 |
[isNatList(x1)] |
= |
1 |
[cons(x1, x2)] |
= |
-2 |
[isNat(x1)] |
= |
1 |
[s(x1)] |
= |
-2 + x1
|
[zeros] |
= |
1 |
[0] |
= |
0 |
[uTake1(x1)] |
= |
0 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(34) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(33) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(35) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(36) |
uTake2(X1,mark(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(58) |
uTake2(mark(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(57) |
uTake2(X1,X2,mark(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(59) |
uTake2(X1,X2,X3,mark(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(60) |
uTake2(active(X1),X2,X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(61) |
uTake2(X1,active(X2),X3,X4) |
→ |
uTake2(X1,X2,X3,X4) |
(62) |
uTake2(X1,X2,active(X3),X4) |
→ |
uTake2(X1,X2,X3,X4) |
(63) |
uTake2(X1,X2,X3,active(X4)) |
→ |
uTake2(X1,X2,X3,X4) |
(64) |
uLength(X1,mark(X2)) |
→ |
uLength(X1,X2) |
(66) |
uLength(mark(X1),X2) |
→ |
uLength(X1,X2) |
(65) |
uLength(active(X1),X2) |
→ |
uLength(X1,X2) |
(67) |
uLength(X1,active(X2)) |
→ |
uLength(X1,X2) |
(68) |
length(active(X)) |
→ |
length(X) |
(46) |
length(mark(X)) |
→ |
length(X) |
(45) |
take(X1,mark(X2)) |
→ |
take(X1,X2) |
(52) |
take(mark(X1),X2) |
→ |
take(X1,X2) |
(51) |
take(active(X1),X2) |
→ |
take(X1,X2) |
(53) |
take(X1,active(X2)) |
→ |
take(X1,X2) |
(54) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(take(X1,X2)) |
→ |
active#(take(mark(X1),mark(X2))) |
(135) |
could be deleted.
1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
2 + x1
|
[and(x1, x2)] |
= |
2 · x1 + x2
|
[length(x1)] |
= |
2 |
[uLength(x1, x2)] |
= |
2 + x1
|
[uTake2(x1,...,x4)] |
= |
2 + 2 · x1 + x3
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
2 + x1
|
[isNat(x1)] |
= |
0 |
[take(x1, x2)] |
= |
2 + x2
|
[s(x1)] |
= |
x1 |
[zeros] |
= |
2 |
[0] |
= |
0 |
[uTake1(x1)] |
= |
1 + x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 + x1
|
the
pairs
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(133) |
mark#(uTake1(X)) |
→ |
mark#(X) |
(141) |
mark#(uTake2(X1,X2,X3,X4)) |
→ |
mark#(X1) |
(144) |
could be deleted.
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
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(149) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(148) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(150) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(151) |
1.1.2 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.2.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.2.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(149) |
|
1 |
≥ |
1 |
2 |
> |
2 |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(148) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(150) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(151) |
|
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
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(153) |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(152) |
1.1.3 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.3.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.3.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(153) |
|
1 |
> |
1 |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(152) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
isNatList#(active(X)) |
→ |
isNatList#(X) |
(155) |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(154) |
1.1.4 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.4.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.4.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNatList#(active(X)) |
→ |
isNatList#(X) |
(155) |
|
1 |
> |
1 |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(154) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
isNat#(active(X)) |
→ |
isNat#(X) |
(157) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(156) |
1.1.5 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.5.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.5.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNat#(active(X)) |
→ |
isNat#(X) |
(157) |
|
1 |
> |
1 |
isNat#(mark(X)) |
→ |
isNat#(X) |
(156) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(159) |
s#(mark(X)) |
→ |
s#(X) |
(158) |
1.1.6 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.6.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.6.1.1 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) |
(159) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(158) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(161) |
length#(mark(X)) |
→ |
length#(X) |
(160) |
1.1.7 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.7.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.7.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length#(active(X)) |
→ |
length#(X) |
(161) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(160) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(163) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(162) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(164) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(165) |
1.1.8 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.8.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.8.1.1 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) |
(163) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(162) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(164) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(165) |
|
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
take#(X1,mark(X2)) |
→ |
take#(X1,X2) |
(167) |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(166) |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(168) |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(169) |
1.1.9 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.9.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.9.1.1 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) |
(167) |
|
1 |
≥ |
1 |
2 |
> |
2 |
take#(mark(X1),X2) |
→ |
take#(X1,X2) |
(166) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(active(X1),X2) |
→ |
take#(X1,X2) |
(168) |
|
1 |
> |
1 |
2 |
≥ |
2 |
take#(X1,active(X2)) |
→ |
take#(X1,X2) |
(169) |
|
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
uTake1#(active(X)) |
→ |
uTake1#(X) |
(171) |
uTake1#(mark(X)) |
→ |
uTake1#(X) |
(170) |
1.1.10 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.10.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.10.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
uTake1#(active(X)) |
→ |
uTake1#(X) |
(171) |
|
1 |
> |
1 |
uTake1#(mark(X)) |
→ |
uTake1#(X) |
(170) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
uTake2#(X1,mark(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(173) |
uTake2#(mark(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(172) |
uTake2#(X1,X2,mark(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(174) |
uTake2#(X1,X2,X3,mark(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(175) |
uTake2#(active(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(176) |
uTake2#(X1,active(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(177) |
uTake2#(X1,X2,active(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(178) |
uTake2#(X1,X2,X3,active(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(179) |
1.1.11 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.11.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.11.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
uTake2#(X1,mark(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(173) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(mark(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(172) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,mark(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(174) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,X3,mark(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(175) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
> |
4 |
uTake2#(active(X1),X2,X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(176) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(X1,active(X2),X3,X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(177) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,active(X3),X4) |
→ |
uTake2#(X1,X2,X3,X4) |
(178) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
4 |
≥ |
4 |
uTake2#(X1,X2,X3,active(X4)) |
→ |
uTake2#(X1,X2,X3,X4) |
(179) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
4 |
> |
4 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
12th
component contains the
pair
uLength#(X1,mark(X2)) |
→ |
uLength#(X1,X2) |
(181) |
uLength#(mark(X1),X2) |
→ |
uLength#(X1,X2) |
(180) |
uLength#(active(X1),X2) |
→ |
uLength#(X1,X2) |
(182) |
uLength#(X1,active(X2)) |
→ |
uLength#(X1,X2) |
(183) |
1.1.12 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.12.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(and(tt,x0)) |
active(isNatIList(x0)) |
active(isNat(0)) |
active(isNat(s(x0))) |
active(isNat(length(x0))) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(isNatList(take(x0,x1))) |
active(zeros) |
active(take(0,x0)) |
active(uTake1(tt)) |
active(take(s(x0),cons(x1,x2))) |
active(uTake2(tt,x0,x1,x2)) |
active(length(cons(x0,x1))) |
active(uLength(tt,x0)) |
mark(and(x0,x1)) |
mark(tt) |
mark(isNatIList(x0)) |
mark(isNatList(x0)) |
mark(isNat(x0)) |
mark(0) |
mark(s(x0)) |
mark(length(x0)) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(nil) |
mark(take(x0,x1)) |
mark(uTake1(x0)) |
mark(uTake2(x0,x1,x2,x3)) |
mark(uLength(x0,x1)) |
1.1.12.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
uLength#(X1,mark(X2)) |
→ |
uLength#(X1,X2) |
(181) |
|
1 |
≥ |
1 |
2 |
> |
2 |
uLength#(mark(X1),X2) |
→ |
uLength#(X1,X2) |
(180) |
|
1 |
> |
1 |
2 |
≥ |
2 |
uLength#(active(X1),X2) |
→ |
uLength#(X1,X2) |
(182) |
|
1 |
> |
1 |
2 |
≥ |
2 |
uLength#(X1,active(X2)) |
→ |
uLength#(X1,X2) |
(183) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.