(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U101(tt, V1, V2) → U102(isNatKind(activate(V1)), activate(V1), activate(V2))
U102(tt, V1, V2) → U103(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U103(tt, V1, V2) → U104(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U104(tt, V1, V2) → U105(isNat(activate(V1)), activate(V2))
U105(tt, V2) → U106(isNatIList(activate(V2)))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U111(tt, L, N) → U112(isNatIListKind(activate(L)), activate(L), activate(N))
U112(tt, L, N) → U113(isNat(activate(N)), activate(L), activate(N))
U113(tt, L, N) → U114(isNatKind(activate(N)), activate(L))
U114(tt, L) → s(length(activate(L)))
U12(tt, V1) → U13(isNatList(activate(V1)))
U121(tt, IL) → U122(isNatIListKind(activate(IL)))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N))
U132(tt, IL, M, N) → U133(isNat(activate(M)), activate(IL), activate(M), activate(N))
U133(tt, IL, M, N) → U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N))
U134(tt, IL, M, N) → U135(isNat(activate(N)), activate(IL), activate(M), activate(N))
U135(tt, IL, M, N) → U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N))
U136(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(activate(V2)))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(activate(V1)), activate(V1), activate(V2))
U92(tt, V1, V2) → U93(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U93(tt, V1, V2) → U94(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U94(tt, V1, V2) → U95(isNat(activate(V1)), activate(V2))
U95(tt, V2) → U96(isNatList(activate(V2)))
U96(tt) → tt
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__take(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U71(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U81(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U91(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U101(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U111(isNatList(activate(L)), activate(L), N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__take(n__0, X2)) →+ U121(isNatIList(activate(X2)), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X2 / n__take(n__0, X2)].
The result substitution is [ ].
The rewrite sequence
activate(n__take(n__0, X2)) →+ U121(isNatIList(activate(X2)), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X2 / n__take(n__0, X2)].
The result substitution is [ ].
(2) BOUNDS(2^n, INF)