(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

fst(0, Z) → nil
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(n__s(X)))
len(nil) → 0
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
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__fst(n__from(X7745_3), X2)) →+ fst(cons(activate(X7745_3), n__from(n__s(activate(X7745_3)))), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X7745_3 / n__fst(n__from(X7745_3), X2)].
The result substitution is [ ].

The rewrite sequence
activate(n__fst(n__from(X7745_3), X2)) →+ fst(cons(activate(X7745_3), n__from(n__s(activate(X7745_3)))), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X7745_3 / n__fst(n__from(X7745_3), X2)].
The result substitution is [ ].

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

fst(0', Z) → nil
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(n__s(X)))
len(nil) → 0'
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/0

(6) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

fst(0', Z) → nil
fst(s(X), cons(Z)) → cons(n__fst(activate(X), activate(Z)))
from(X) → cons(n__from(n__s(X)))
len(nil) → 0'
len(cons(Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

Infered types.

(8) Obligation:

TRS:
Rules:
fst(0', Z) → nil
fst(s(X), cons(Z)) → cons(n__fst(activate(X), activate(Z)))
from(X) → cons(n__from(n__s(X)))
len(nil) → 0'
len(cons(Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
activate(X) → X

Types:

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
activate, len

They will be analysed ascendingly in the following order:
activate = len

(10) Obligation:

TRS:
Rules:
fst(0', Z) → nil
fst(s(X), cons(Z)) → cons(n__fst(activate(X), activate(Z)))
from(X) → cons(n__from(n__s(X)))
len(nil) → 0'
len(cons(Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
activate(X) → X

Types:

Generator Equations:

The following defined symbols remain to be analysed:
len, activate

They will be analysed ascendingly in the following order:
activate = len

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol len.

(12) Obligation:

TRS:
Rules:
fst(0', Z) → nil
fst(s(X), cons(Z)) → cons(n__fst(activate(X), activate(Z)))
from(X) → cons(n__from(n__s(X)))
len(nil) → 0'
len(cons(Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
activate(X) → X

Types:

Generator Equations:

The following defined symbols remain to be analysed:
activate

They will be analysed ascendingly in the following order:
activate = len

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol activate.

(14) Obligation:

TRS:
Rules:
fst(0', Z) → nil
fst(s(X), cons(Z)) → cons(n__fst(activate(X), activate(Z)))
from(X) → cons(n__from(n__s(X)))
len(nil) → 0'
len(cons(Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(X)
activate(n__len(X)) → len(activate(X))
activate(X) → X

Types: