0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
↳5 QDP
↳6 QDPOrderProof (⇔)
↳7 QDP
↳8 PisEmptyProof (⇔)
↳9 TRUE
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 PisEmptyProof (⇔)
↳14 TRUE
↳15 QDP
↳16 QDPOrderProof (⇔)
↳17 QDP
↳18 PisEmptyProof (⇔)
↳19 TRUE
↳20 QDP
↳21 QDPOrderProof (⇔)
↳22 QDP
↳23 QDPOrderProof (⇔)
↳24 QDP
↳25 PisEmptyProof (⇔)
↳26 TRUE
↳27 QDP
↳28 QDPOrderProof (⇔)
↳29 QDP
↳30 QDPOrderProof (⇔)
↳31 QDP
↳32 PisEmptyProof (⇔)
↳33 TRUE
↳34 QDP
↳35 QDPOrderProof (⇔)
↳36 QDP
↳37 QDPOrderProof (⇔)
↳38 QDP
↳39 PisEmptyProof (⇔)
↳40 TRUE
↳41 QDP
↳42 QDPOrderProof (⇔)
↳43 QDP
↳44 PisEmptyProof (⇔)
↳45 TRUE
↳46 QDP
↳47 QDPOrderProof (⇔)
↳48 QDP
↳49 PisEmptyProof (⇔)
↳50 TRUE
↳51 QDP
↳52 QDPOrderProof (⇔)
↳53 QDP
↳54 PisEmptyProof (⇔)
↳55 TRUE
↳56 QDP
↳57 QDPOrderProof (⇔)
↳58 QDP
↳59 QDPOrderProof (⇔)
↳60 QDP
↳61 PisEmptyProof (⇔)
↳62 TRUE
↳63 QDP
↳64 QDPOrderProof (⇔)
↳65 QDP
↳66 QDPOrderProof (⇔)
↳67 QDP
↳68 PisEmptyProof (⇔)
↳69 TRUE
↳70 QDP
↳71 QDPOrderProof (⇔)
↳72 QDP
↳73 QDPOrderProof (⇔)
↳74 QDP
↳75 QDPOrderProof (⇔)
↳76 QDP
↳77 PisEmptyProof (⇔)
↳78 TRUE
↳79 QDP
↳80 QDPOrderProof (⇔)
↳81 QDP
↳82 PisEmptyProof (⇔)
↳83 TRUE
↳84 QDP
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U11(tt, L)) → S(length(L))
ACTIVE(U11(tt, L)) → LENGTH(L)
ACTIVE(U31(tt, IL, M, N)) → CONS(N, take(M, IL))
ACTIVE(U31(tt, IL, M, N)) → TAKE(M, IL)
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(isNatList(take(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(isNatList(take(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(take(V1, V2))) → ISNATILIST(V2)
ACTIVE(length(cons(N, L))) → U111(and(isNatList(L), isNat(N)), L)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(take(0, IL)) → U211(isNatIList(IL))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → U311(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNatIList(IL), and(isNat(M), isNat(N)))
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), isNat(N))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U21(X)) → U211(active(X))
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3, X4)) → U311(active(X1), X2, X3, X4)
ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(active(X1), X2)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → TAKE(X1, active(X2))
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → AND(active(X1), X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
CONS(mark(X1), X2) → CONS(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
S(mark(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
U211(mark(X)) → U211(X)
U311(mark(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
PROPER(length(X)) → PROPER(X)
PROPER(U21(X)) → U211(proper(X))
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → U311(proper(X1), proper(X2), proper(X3), proper(X4))
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → TAKE(proper(X1), proper(X2))
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → AND(proper(X1), proper(X2))
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → ISNAT(proper(X))
PROPER(isNat(X)) → PROPER(X)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
PROPER(isNatIList(X)) → PROPER(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
U111(ok(X1), ok(X2)) → U111(X1, X2)
S(ok(X)) → S(X)
LENGTH(ok(X)) → LENGTH(X)
U211(ok(X)) → U211(X)
U311(ok(X1), ok(X2), ok(X3), ok(X4)) → U311(X1, X2, X3, X4)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
AND(ok(X1), ok(X2)) → AND(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
ISNATLIST(ok(X)) → ISNATLIST(X)
ISNATILIST(ok(X)) → ISNATILIST(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ISNATILIST(ok(X)) → ISNATILIST(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNATILIST(ok(X)) → ISNATILIST(X)
[active1, s1, U211, nil, U312, proper1] > zeros
[active1, s1, U211, nil, U312, proper1] > 0 > [and1, isNatIList1] > [ISNATILIST1, ok1]
[active1, s1, U211, nil, U312, proper1] > U111 > [ISNATILIST1, ok1]
[active1, s1, U211, nil, U312, proper1] > tt > take2 > [and1, isNatIList1] > [ISNATILIST1, ok1]
[active1, s1, U211, nil, U312, proper1] > isNatList1 > [and1, isNatIList1] > [ISNATILIST1, ok1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ISNATLIST(ok(X)) → ISNATLIST(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNATLIST(ok(X)) → ISNATLIST(X)
[active1, s1, U211, nil, U312, proper1] > zeros
[active1, s1, U211, nil, U312, proper1] > 0 > [and1, isNatIList1] > [ISNATLIST1, ok1]
[active1, s1, U211, nil, U312, proper1] > U111 > [ISNATLIST1, ok1]
[active1, s1, U211, nil, U312, proper1] > tt > take2 > [and1, isNatIList1] > [ISNATLIST1, ok1]
[active1, s1, U211, nil, U312, proper1] > isNatList1 > [and1, isNatIList1] > [ISNATLIST1, ok1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ISNAT(ok(X)) → ISNAT(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ISNAT(ok(X)) → ISNAT(X)
[active1, s1, U211, nil, U312, proper1] > zeros
[active1, s1, U211, nil, U312, proper1] > 0 > [and1, isNatIList1] > [ISNAT1, ok1]
[active1, s1, U211, nil, U312, proper1] > U111 > [ISNAT1, ok1]
[active1, s1, U211, nil, U312, proper1] > tt > take2 > [and1, isNatIList1] > [ISNAT1, ok1]
[active1, s1, U211, nil, U312, proper1] > isNatList1 > [and1, isNatIList1] > [ISNAT1, ok1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
AND(ok(X1), ok(X2)) → AND(X1, X2)
AND(mark(X1), X2) → AND(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
AND(ok(X1), ok(X2)) → AND(X1, X2)
AND1 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > zeros > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > 0 > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > nil > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > U311 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > isNat1 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
top > [ok1, mark, active1, cons1, U111, length1, isNatList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
AND(mark(X1), X2) → AND(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
AND(mark(X1), X2) → AND(X1, X2)
AND1 > [mark1, s1, length1, top]
active1 > zeros > cons2 > isNat > [mark1, s1, length1, top]
active1 > zeros > 0 > [mark1, s1, length1, top]
active1 > zeros > tt > [mark1, s1, length1, top]
active1 > U112 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > 0 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > tt > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > U314 > cons2 > isNat > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > and2 > [mark1, s1, length1, top]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(ok(X1), ok(X2)) → TAKE(X1, X2)
proper1 > [active1, 0] > [U112, s1] > [zeros, cons2, U314] > [mark1, tt] > TAKE1 > ok1
proper1 > [active1, 0] > [U112, s1] > [zeros, cons2, U314] > [mark1, tt] > top > ok1
proper1 > [active1, 0] > [U112, s1] > and2 > [mark1, tt] > TAKE1 > ok1
proper1 > [active1, 0] > [U112, s1] > and2 > [mark1, tt] > top > ok1
proper1 > [active1, 0] > take2 > [zeros, cons2, U314] > [mark1, tt] > TAKE1 > ok1
proper1 > [active1, 0] > take2 > [zeros, cons2, U314] > [mark1, tt] > top > ok1
proper1 > [active1, 0] > take2 > U211 > nil > [mark1, tt] > TAKE1 > ok1
proper1 > [active1, 0] > take2 > U211 > nil > [mark1, tt] > top > ok1
proper1 > [active1, 0] > take2 > and2 > [mark1, tt] > TAKE1 > ok1
proper1 > [active1, 0] > take2 > and2 > [mark1, tt] > top > ok1
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
TAKE(mark(X1), X2) → TAKE(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE1 > [mark1, s1, length1, top]
active1 > zeros > cons2 > isNat > [mark1, s1, length1, top]
active1 > zeros > 0 > [mark1, s1, length1, top]
active1 > zeros > tt > [mark1, s1, length1, top]
active1 > U112 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > 0 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > tt > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > U314 > cons2 > isNat > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > and2 > [mark1, s1, length1, top]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
U311(ok(X1), ok(X2), ok(X3), ok(X4)) → U311(X1, X2, X3, X4)
U311(mark(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U311(mark(X1), X2, X3, X4) → U311(X1, X2, X3, X4)
active1 > zeros > cons2 > U112 > [mark1, s1] > [isNatList1, isNatIList1]
active1 > zeros > cons2 > and2 > [mark1, s1] > [isNatList1, isNatIList1]
active1 > zeros > tt
active1 > 0 > [mark1, s1] > [isNatList1, isNatIList1]
active1 > 0 > tt
active1 > nil > [mark1, s1] > [isNatList1, isNatIList1]
active1 > nil > tt
active1 > U314 > cons2 > U112 > [mark1, s1] > [isNatList1, isNatIList1]
active1 > U314 > cons2 > and2 > [mark1, s1] > [isNatList1, isNatIList1]
active1 > U314 > take2 > and2 > [mark1, s1] > [isNatList1, isNatIList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
U311(ok(X1), ok(X2), ok(X3), ok(X4)) → U311(X1, X2, X3, X4)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U311(ok(X1), ok(X2), ok(X3), ok(X4)) → U311(X1, X2, X3, X4)
zeros > [ok1, active1, proper1] > 0 > [mark, tt, nil]
zeros > [ok1, active1, proper1] > top > [mark, tt, nil]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
U211(ok(X)) → U211(X)
U211(mark(X)) → U211(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U211(ok(X)) → U211(X)
U211(mark(X)) → U211(X)
U21^11 > [ok1, mark1, tt, isNatList1]
zeros > [active1, cons2, U112, length1, U314, take2] > [0, U211, nil] > [ok1, mark1, tt, isNatList1]
zeros > [active1, cons2, U112, length1, U314, take2] > [s1, and2] > [ok1, mark1, tt, isNatList1]
[proper1, top] > [active1, cons2, U112, length1, U314, take2] > [0, U211, nil] > [ok1, mark1, tt, isNatList1]
[proper1, top] > [active1, cons2, U112, length1, U314, take2] > [s1, and2] > [ok1, mark1, tt, isNatList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH1 > [ok1, mark1, tt, isNatList1]
zeros > [active1, cons2, U112, length1, U314, take2] > [0, U211, nil] > [ok1, mark1, tt, isNatList1]
zeros > [active1, cons2, U112, length1, U314, take2] > [s1, and2] > [ok1, mark1, tt, isNatList1]
[proper1, top] > [active1, cons2, U112, length1, U314, take2] > [0, U211, nil] > [ok1, mark1, tt, isNatList1]
[proper1, top] > [active1, cons2, U112, length1, U314, take2] > [s1, and2] > [ok1, mark1, tt, isNatList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S(ok(X)) → S(X)
S(mark(X)) → S(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S(ok(X)) → S(X)
S(mark(X)) → S(X)
S1 > [ok1, mark1, tt, isNatList1]
zeros > [active1, cons2, U112, length1, U314, take2] > [0, U211, nil] > [ok1, mark1, tt, isNatList1]
zeros > [active1, cons2, U112, length1, U314, take2] > [s1, and2] > [ok1, mark1, tt, isNatList1]
[proper1, top] > [active1, cons2, U112, length1, U314, take2] > [0, U211, nil] > [ok1, mark1, tt, isNatList1]
[proper1, top] > [active1, cons2, U112, length1, U314, take2] > [s1, and2] > [ok1, mark1, tt, isNatList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
U111(ok(X1), ok(X2)) → U111(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U111(ok(X1), ok(X2)) → U111(X1, X2)
U11^11 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > zeros > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > 0 > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > nil > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > U311 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > isNat1 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
top > [ok1, mark, active1, cons1, U111, length1, isNatList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
U111(mark(X1), X2) → U111(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U111(mark(X1), X2) → U111(X1, X2)
U11^11 > [mark1, s1, length1, top]
active1 > zeros > cons2 > isNat > [mark1, s1, length1, top]
active1 > zeros > 0 > [mark1, s1, length1, top]
active1 > zeros > tt > [mark1, s1, length1, top]
active1 > U112 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > 0 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > tt > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > U314 > cons2 > isNat > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > and2 > [mark1, s1, length1, top]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS1 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > zeros > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > 0 > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > nil > tt > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > U311 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
[isNatIList1, proper1] > isNat1 > [ok1, mark, active1, cons1, U111, length1, isNatList1]
top > [ok1, mark, active1, cons1, U111, length1, isNatList1]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
CONS(mark(X1), X2) → CONS(X1, X2)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(mark(X1), X2) → CONS(X1, X2)
CONS1 > [mark1, s1, length1, top]
active1 > zeros > cons2 > isNat > [mark1, s1, length1, top]
active1 > zeros > 0 > [mark1, s1, length1, top]
active1 > zeros > tt > [mark1, s1, length1, top]
active1 > U112 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > 0 > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > nil > tt > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > U314 > cons2 > isNat > [mark1, s1, length1, top]
active1 > [U211, take2, isNatList, isNatIList] > and2 > [mark1, s1, length1, top]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(length(X)) → PROPER(X)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X1)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X2)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X3)
PROPER(U31(X1, X2, X3, X4)) → PROPER(X4)
PROPER(take(X1, X2)) → PROPER(X1)
PROPER(take(X1, X2)) → PROPER(X2)
PROPER(and(X1, X2)) → PROPER(X1)
PROPER(and(X1, X2)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER1 > [cons2, and2, zeros, tt, top]
[U314, isNatList1, active1, proper1] > U112 > length1 > [cons2, and2, zeros, tt, top]
[U314, isNatList1, active1, proper1] > take2 > isNatIList1 > [cons2, and2, zeros, tt, top]
[U314, isNatList1, active1, proper1] > [0, nil] > isNatIList1 > [cons2, and2, zeros, tt, top]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
PROPER(s(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER(U21(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
isNat1 > isNatList > tt > [U211, active1, mark, cons]
zeros > 0 > [U211, active1, mark, cons]
zeros > tt > [U211, active1, mark, cons]
U111 > [U211, active1, mark, cons]
nil > 0 > [U211, active1, mark, cons]
nil > tt > [U211, active1, mark, cons]
top > [U211, active1, mark, cons]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
PROPER(s(X)) → PROPER(X)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PROPER(s(X)) → PROPER(X)
[U111, isNat, proper1] > [active1, U311, and1] > s1 > [length, U21, take2, ok, top] > [mark, nil]
[U111, isNat, proper1] > [active1, U311, and1] > cons1 > [length, U21, take2, ok, top] > [mark, nil]
[U111, isNat, proper1] > [active1, U311, and1] > 0 > [length, U21, take2, ok, top] > [mark, nil]
[U111, isNat, proper1] > zeros > 0 > [length, U21, take2, ok, top] > [mark, nil]
[U111, isNat, proper1] > zeros > tt > s1 > [length, U21, take2, ok, top] > [mark, nil]
[U111, isNat, proper1] > zeros > tt > cons1 > [length, U21, take2, ok, top] > [mark, nil]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U31(X1, X2, X3, X4)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X1)
ACTIVE(take(X1, X2)) → ACTIVE(X2)
ACTIVE(and(X1, X2)) → ACTIVE(X1)
zeros > [U112, cons2, U211, U314, active1] > take2 > and2 > [length1, 0, top]
zeros > [U112, cons2, U211, U314, active1] > [tt, nil] > s1 > [length1, 0, top]
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))
active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))