zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
U211(tt, IL, M, N) → ACTIVATE(M)
U121(tt, L) → ACTIVATE(L)
U221(tt, IL, M, N) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ACTIVATE(L)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
U221(tt, IL, M, N) → ACTIVATE(N)
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → U111(tt, activate(L))
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U211(tt, IL, M, N) → ACTIVATE(IL)
U111(tt, L) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
ACTIVATE(n__zeros) → ZEROS
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
U211(tt, IL, M, N) → ACTIVATE(M)
U121(tt, L) → ACTIVATE(L)
U221(tt, IL, M, N) → ACTIVATE(IL)
LENGTH(cons(N, L)) → ACTIVATE(L)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
U221(tt, IL, M, N) → ACTIVATE(N)
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → U111(tt, activate(L))
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U211(tt, IL, M, N) → ACTIVATE(IL)
U111(tt, L) → ACTIVATE(L)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
ACTIVATE(n__zeros) → ZEROS
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
U221(tt, IL, M, N) → ACTIVATE(N)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U211(tt, IL, M, N) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U211(tt, activate(IL), M, N)
Used ordering: Polynomial interpretation [25,35]:
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
U221(tt, IL, M, N) → ACTIVATE(N)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(IL)
U231(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(N)
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U211(tt, IL, M, N) → ACTIVATE(IL)
The value of delta used in the strict ordering is 4.
POL(U23(x1, x2, x3, x4)) = (4)x_2 + (2)x_3 + (4)x_4
POL(U22(x1, x2, x3, x4)) = (4)x_2 + (2)x_3 + (4)x_4
POL(n__zeros) = 0
POL(U211(x1, x2, x3, x4)) = (4)x_2 + (4)x_3 + (4)x_4
POL(activate(x1)) = x_1
POL(U21(x1, x2, x3, x4)) = (4)x_2 + (2)x_3 + (4)x_4
POL(take(x1, x2)) = x_1 + (4)x_2
POL(0) = 0
POL(TAKE(x1, x2)) = x_1 + (4)x_2
POL(cons(x1, x2)) = (4)x_1 + x_2
POL(tt) = 0
POL(U221(x1, x2, x3, x4)) = (4)x_2 + (4)x_3 + (4)x_4
POL(zeros) = 0
POL(U231(x1, x2, x3, x4)) = (4)x_2 + (4)x_3 + (4)x_4
POL(n__take(x1, x2)) = x_1 + (4)x_2
POL(s(x1)) = 4 + (4)x_1
POL(ACTIVATE(x1)) = x_1
POL(nil) = 0
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(X) → X
zeros → cons(0, n__zeros)
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
take(0, IL) → nil
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
U231(tt, IL, M, N) → ACTIVATE(N)
U221(tt, IL, M, N) → ACTIVATE(M)
U211(tt, IL, M, N) → ACTIVATE(M)
U221(tt, IL, M, N) → ACTIVATE(IL)
U221(tt, IL, M, N) → U231(tt, activate(IL), activate(M), activate(N))
U221(tt, IL, M, N) → ACTIVATE(N)
U211(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U211(tt, IL, M, N) → U221(tt, activate(IL), activate(M), activate(N))
U231(tt, IL, M, N) → ACTIVATE(M)
U231(tt, IL, M, N) → ACTIVATE(IL)
U211(tt, IL, M, N) → ACTIVATE(N)
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
U121(tt, L) → LENGTH(activate(L))
U111(tt, L) → U121(tt, activate(L))
LENGTH(cons(N, L)) → U111(tt, activate(L))
zeros → cons(0, n__zeros)
U11(tt, L) → U12(tt, activate(L))
U12(tt, L) → s(length(activate(L)))
U21(tt, IL, M, N) → U22(tt, activate(IL), activate(M), activate(N))
U22(tt, IL, M, N) → U23(tt, activate(IL), activate(M), activate(N))
U23(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
length(nil) → 0
length(cons(N, L)) → U11(tt, activate(L))
take(0, IL) → nil
take(s(M), cons(N, IL)) → U21(tt, activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X