(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(s(X)))
length(n__nil) → 0
length(n__cons(X, Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from(X) → n__from(X)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, 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:
from(X) → cons(X, n__from(s(X)))
length(n__nil) → 0
length(n__cons(X, Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from(X) → n__from(X)
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(X)
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
from/0
cons/0
n__from/0
n__cons/0
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
from → cons(n__from)
length(n__nil) → 0
length(n__cons(Y)) → s(length1(activate(Y)))
length1(X) → length(activate(X))
from → n__from
nil → n__nil
cons(X2) → n__cons(X2)
activate(n__from) → from
activate(n__nil) → nil
activate(n__cons(X2)) → cons(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(n__cons(n__from)) →+ s(length(n__cons(n__from)))
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)