(0) Obligation:

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

2nd(cons1(X, cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(X, activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(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__from(X)) →+ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].

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

2nd(cons1(X, cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(X, activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
cons1/0

(6) Obligation:

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

2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

Infered types.

(8) Obligation:

TRS:
Rules:
2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from

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

Heuristically decided to analyse the following defined symbols:
2nd, activate

They will be analysed ascendingly in the following order:
activate < 2nd

(10) Obligation:

TRS:
Rules:
2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from

Generator Equations:
gen_cons:cons1:n__s:n__from2_0(0) ⇔ hole_cons:cons1:n__s:n__from1_0
gen_cons:cons1:n__s:n__from2_0(+(x, 1)) ⇔ cons(hole_cons:cons1:n__s:n__from1_0, gen_cons:cons1:n__s:n__from2_0(x))

The following defined symbols remain to be analysed:
activate, 2nd

They will be analysed ascendingly in the following order:
activate < 2nd

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

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

(12) Obligation:

TRS:
Rules:
2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from

Generator Equations:
gen_cons:cons1:n__s:n__from2_0(0) ⇔ hole_cons:cons1:n__s:n__from1_0
gen_cons:cons1:n__s:n__from2_0(+(x, 1)) ⇔ cons(hole_cons:cons1:n__s:n__from1_0, gen_cons:cons1:n__s:n__from2_0(x))

The following defined symbols remain to be analysed:
2nd

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

Could not prove a rewrite lemma for the defined symbol 2nd.

(14) Obligation:

TRS:
Rules:
2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from

Generator Equations:
gen_cons:cons1:n__s:n__from2_0(0) ⇔ hole_cons:cons1:n__s:n__from1_0
gen_cons:cons1:n__s:n__from2_0(+(x, 1)) ⇔ cons(hole_cons:cons1:n__s:n__from1_0, gen_cons:cons1:n__s:n__from2_0(x))

No more defined symbols left to analyse.