0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 QDP
↳7 MRRProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 QDP
↳11 MRRProof (⇔)
↳12 QDP
↳13 DependencyGraphProof (⇔)
↳14 AND
↳15 QDP
↳16 UsableRulesProof (⇔)
↳17 QDP
↳18 QDPSizeChangeProof (⇔)
↳19 YES
↳20 QDP
↳21 Narrowing (⇔)
↳22 QDP
↳23 DependencyGraphProof (⇔)
↳24 QDP
↳25 Narrowing (⇔)
↳26 QDP
↳27 DependencyGraphProof (⇔)
↳28 QDP
↳29 QDPOrderProof (⇔)
↳30 QDP
↳31 QDPOrderProof (⇔)
↳32 QDP
↳33 QDPOrderProof (⇔)
↳34 QDP
↳35 QDPOrderProof (⇔)
↳36 QDP
↳37 NonTerminationProof (⇔)
↳38 NO
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(tt, L)
a__take(0, IL) → nil
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = x1 + x2
POL(U21(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + x4
POL(U22(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + x4
POL(U23(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + x4
POL(a__U11(x1, x2)) = x1 + x2
POL(a__U12(x1, x2)) = x1 + x2
POL(a__U21(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + x4
POL(a__U22(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + x4
POL(a__U23(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + x4
POL(a__length(x1)) = x1
POL(a__take(x1, x2)) = 2 + x1 + x2
POL(a__zeros) = 0
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 1
POL(s(x1)) = x1
POL(take(x1, x2)) = 2 + x1 + x2
POL(tt) = 0
POL(zeros) = 0
a__length(nil) → 0
a__take(0, IL) → nil
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__U12(tt, L) → MARK(L)
A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(zeros) → A__ZEROS
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → MARK(L)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
MARK(U21(X1, X2, X3, X4)) → A__U21(mark(X1), X2, X3, X4)
A__U22(tt, IL, M, N) → A__U23(tt, IL, M, N)
MARK(U21(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2, X3, X4)) → A__U22(mark(X1), X2, X3, X4)
MARK(U22(X1, X2, X3, X4)) → MARK(X1)
MARK(U23(X1, X2, X3, X4)) → A__U23(mark(X1), X2, X3, X4)
MARK(U23(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
POL(0) = 0
POL(A__LENGTH(x1)) = 2·x1
POL(A__TAKE(x1, x2)) = 1 + 2·x1 + 2·x2
POL(A__U11(x1, x2)) = 2·x1 + 2·x2
POL(A__U12(x1, x2)) = 2·x1 + 2·x2
POL(A__U21(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + 2·x4
POL(A__U22(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + 2·x4
POL(A__U23(x1, x2, x3, x4)) = 2·x1 + x2 + x3 + 2·x4
POL(MARK(x1)) = 2·x1
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = 2·x1 + x2
POL(U21(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + 2·x4
POL(U22(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + 2·x4
POL(U23(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + 2·x4
POL(a__U11(x1, x2)) = x1 + x2
POL(a__U12(x1, x2)) = 2·x1 + x2
POL(a__U21(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + 2·x4
POL(a__U22(x1, x2, x3, x4)) = 1 + 2·x1 + x2 + 2·x3 + 2·x4
POL(a__U23(x1, x2, x3, x4)) = 1 + x1 + x2 + 2·x3 + 2·x4
POL(a__length(x1)) = x1
POL(a__take(x1, x2)) = 1 + 2·x1 + x2
POL(a__zeros) = 0
POL(cons(x1, x2)) = 2·x1 + x2
POL(length(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + 2·x1 + x2
POL(tt) = 0
POL(zeros) = 0
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → MARK(L)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
A__U21(tt, IL, M, N) → A__U22(tt, IL, M, N)
A__U23(tt, IL, M, N) → MARK(N)
A__TAKE(s(M), cons(N, IL)) → A__U21(tt, IL, M, N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__U12(tt, L) → MARK(L)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
POL(0) = 0
POL(A__LENGTH(x1)) = x1
POL(A__U11(x1, x2)) = 2·x1 + x2
POL(A__U12(x1, x2)) = 2·x1 + x2
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = 1 + 2·x1 + 2·x2
POL(U12(x1, x2)) = 1 + 2·x1 + 2·x2
POL(U21(x1, x2, x3, x4)) = x1 + x2 + x3 + 2·x4
POL(U22(x1, x2, x3, x4)) = x1 + x2 + x3 + 2·x4
POL(U23(x1, x2, x3, x4)) = x1 + x2 + x3 + 2·x4
POL(a__U11(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a__U12(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a__U21(x1, x2, x3, x4)) = x1 + x2 + x3 + 2·x4
POL(a__U22(x1, x2, x3, x4)) = x1 + x2 + x3 + 2·x4
POL(a__U23(x1, x2, x3, x4)) = x1 + x2 + x3 + 2·x4
POL(a__length(x1)) = 1 + 2·x1
POL(a__take(x1, x2)) = x1 + x2
POL(a__zeros) = 0
POL(cons(x1, x2)) = 2·x1 + x2
POL(length(x1)) = 1 + 2·x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(tt) = 0
POL(zeros) = 0
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__U12(tt, L) → MARK(L)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
From the DPs we obtained the following set of size-change graphs:
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, 0) → A__LENGTH(0)
A__U12(tt, tt) → A__LENGTH(tt)
A__U12(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U12(tt, nil) → A__LENGTH(nil)
A__U11(tt, L) → A__U12(tt, L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, 0) → A__LENGTH(0)
A__U12(tt, tt) → A__LENGTH(tt)
A__U12(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U12(tt, nil) → A__LENGTH(nil)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U12(tt, zeros) → A__LENGTH(zeros)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U12(tt, zeros) → A__LENGTH(zeros)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U12(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U12(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U12(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
POL( A__LENGTH(x1) ) = max{0, 2x1 - 2}
POL( a__U11(x1, x2) ) = max{0, -1}
POL( a__U12(x1, x2) ) = max{0, -2}
POL( a__U21(x1, ..., x4) ) = 2
POL( a__U22(x1, ..., x4) ) = 2
POL( a__U23(x1, ..., x4) ) = 2
POL( a__length(x1) ) = 0
POL( a__take(x1, x2) ) = 2
POL( cons(x1, x2) ) = 2
POL( s(x1) ) = max{0, -2}
POL( mark(x1) ) = 2
POL( zeros ) = 0
POL( a__zeros ) = 0
POL( U11(x1, x2) ) = max{0, -1}
POL( U12(x1, x2) ) = max{0, -1}
POL( length(x1) ) = max{0, -1}
POL( U21(x1, ..., x4) ) = 2
POL( U22(x1, ..., x4) ) = 2
POL( U23(x1, ..., x4) ) = 2
POL( take(x1, x2) ) = 2
POL( 0 ) = 0
POL( tt ) = 0
POL( nil ) = 1
POL( A__U11(x1, x2) ) = 2
POL( A__U12(x1, x2) ) = 2
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(tt, L) → a__U12(tt, L)
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt, L) → s(a__length(mark(L)))
a__U12(X1, X2) → U12(X1, X2)
a__length(cons(N, L)) → a__U11(tt, L)
a__length(X) → length(X)
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__take(X1, X2) → take(X1, X2)
A__U11(tt, L) → A__U12(tt, L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U12(tt, U22(x0, x1, x2, x3)) → A__LENGTH(a__U22(mark(x0), x1, x2, x3))
A__U12(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
POL( A__LENGTH(x1) ) = x1 + 1
POL( a__U11(x1, x2) ) = 2
POL( a__U12(x1, x2) ) = 1
POL( a__U21(x1, ..., x4) ) = 1
POL( a__U22(x1, ..., x4) ) = 1
POL( a__U23(x1, ..., x4) ) = 1
POL( a__length(x1) ) = 2
POL( a__take(x1, x2) ) = 1
POL( cons(x1, x2) ) = 2x2 + 1
POL( s(x1) ) = 1
POL( mark(x1) ) = x1 + 1
POL( zeros ) = 2
POL( a__zeros ) = 0
POL( U11(x1, x2) ) = 1
POL( U12(x1, x2) ) = 1
POL( length(x1) ) = 1
POL( U21(x1, ..., x4) ) = max{0, -1}
POL( U22(x1, ..., x4) ) = 1
POL( U23(x1, ..., x4) ) = max{0, -2}
POL( take(x1, x2) ) = max{0, -2}
POL( 0 ) = 1
POL( tt ) = 0
POL( nil ) = 0
POL( A__U11(x1, x2) ) = 2x2 + 2
POL( A__U12(x1, x2) ) = 2x2 + 2
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__take(X1, X2) → take(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__length(cons(N, L)) → a__U11(tt, L)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
A__U11(tt, L) → A__U12(tt, L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U12(tt, U21(x0, x1, x2, x3)) → A__LENGTH(a__U21(mark(x0), x1, x2, x3))
POL( A__LENGTH(x1) ) = max{0, 2x1 - 2}
POL( a__U11(x1, x2) ) = max{0, -2}
POL( a__U12(x1, x2) ) = max{0, -2}
POL( a__U21(x1, ..., x4) ) = 2
POL( a__U22(x1, ..., x4) ) = x1
POL( a__U23(x1, ..., x4) ) = 2
POL( a__length(x1) ) = 2x1 + 2
POL( a__take(x1, x2) ) = 2
POL( cons(x1, x2) ) = 2x2 + 2
POL( s(x1) ) = max{0, -2}
POL( mark(x1) ) = 2x1 + 2
POL( zeros ) = 0
POL( a__zeros ) = 2
POL( U11(x1, x2) ) = 0
POL( U12(x1, x2) ) = 0
POL( length(x1) ) = 2x1 + 2
POL( U21(x1, ..., x4) ) = 2
POL( U22(x1, ..., x4) ) = x1
POL( U23(x1, ..., x4) ) = max{0, -2}
POL( take(x1, x2) ) = max{0, -2}
POL( 0 ) = 1
POL( tt ) = 2
POL( nil ) = 2
POL( A__U11(x1, x2) ) = 2x2 + 2
POL( A__U12(x1, x2) ) = max{0, 2x1 + x2 - 2}
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__take(X1, X2) → take(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__length(cons(N, L)) → a__U11(tt, L)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__zeros → cons(0, zeros)
a__zeros → zeros
A__U11(tt, L) → A__U12(tt, L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U12(tt, U23(x0, x1, x2, x3)) → A__LENGTH(a__U23(mark(x0), x1, x2, x3))
POL( A__LENGTH(x1) ) = max{0, 2x1 - 2}
POL( a__U11(x1, x2) ) = 2
POL( a__U12(x1, x2) ) = 2
POL( a__U21(x1, ..., x4) ) = 2
POL( a__U22(x1, ..., x4) ) = 2
POL( a__U23(x1, ..., x4) ) = 2
POL( a__length(x1) ) = 2
POL( a__take(x1, x2) ) = 2
POL( cons(x1, x2) ) = x2 + 1
POL( s(x1) ) = 2
POL( mark(x1) ) = 2x1
POL( zeros ) = 0
POL( a__zeros ) = 0
POL( U11(x1, x2) ) = 2
POL( U12(x1, x2) ) = 2
POL( length(x1) ) = 1
POL( U21(x1, ..., x4) ) = 2
POL( U22(x1, ..., x4) ) = 1
POL( U23(x1, ..., x4) ) = 2
POL( take(x1, x2) ) = 1
POL( 0 ) = 1
POL( tt ) = 0
POL( nil ) = 0
POL( A__U11(x1, x2) ) = 2x2
POL( A__U12(x1, x2) ) = 2x2
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
a__take(X1, X2) → take(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__length(cons(N, L)) → a__U11(tt, L)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
A__U11(tt, L) → A__U12(tt, L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U12(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__U21(tt, IL, M, N) → a__U22(tt, IL, M, N)
a__U22(tt, IL, M, N) → a__U23(tt, IL, M, N)
a__U23(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__U11(tt, L)
a__take(s(M), cons(N, IL)) → a__U21(tt, IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U21(X1, X2, X3, X4)) → a__U21(mark(X1), X2, X3, X4)
mark(U22(X1, X2, X3, X4)) → a__U22(mark(X1), X2, X3, X4)
mark(U23(X1, X2, X3, X4)) → a__U23(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
a__U21(X1, X2, X3, X4) → U21(X1, X2, X3, X4)
a__U22(X1, X2, X3, X4) → U22(X1, X2, X3, X4)
a__U23(X1, X2, X3, X4) → U23(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)