(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
and(tt, X) → activate(X)
length(nil) → 0
length(cons(N, L)) → s(length(activate(L)))
take(0, IL) → nil
take(s(M), cons(N, IL)) → cons(N, n__take(M, activate(IL)))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
Rewrite Strategy: FULL
(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)
Transformed TRS to relative TRS where S is empty.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
and(tt, X) → activate(X)
length(nil) → 0
length(cons(N, L)) → s(length(activate(L)))
take(0, IL) → nil
take(s(M), cons(N, IL)) → cons(N, n__take(M, activate(IL)))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
cons/0
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
zeros → cons(n__zeros)
and(tt, X) → activate(X)
length(nil) → 0
length(cons(L)) → s(length(activate(L)))
take(0, IL) → nil
take(s(M), cons(IL)) → cons(n__take(M, activate(IL)))
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) InfiniteLowerBoundProof (EQUIVALENT transformation)
The loop following loop proves infinite runtime complexity:
The rewrite sequence
length(cons(n__zeros)) →+ s(length(cons(n__zeros)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [ ].
The result substitution is [ ].
(6) BOUNDS(INF, INF)