### (0) Obligation:

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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
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__first(n__from(X110_3), X2)) →+ first(cons(activate(X110_3), n__from(n__s(activate(X110_3)))), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X110_3 / n__first(n__from(X110_3), X2)].
The result substitution is [ ].

The rewrite sequence
activate(n__first(n__from(X110_3), X2)) →+ first(cons(activate(X110_3), n__from(n__s(activate(X110_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 [X110_3 / n__first(n__from(X110_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:

first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
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:
cons/0

### (6) Obligation:

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

first(0', X) → nil
first(s(X), cons(Z)) → cons(n__first(X, activate(Z)))
from(X) → cons(n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
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:
first(0', X) → nil
first(s(X), cons(Z)) → cons(n__first(X, activate(Z)))
from(X) → cons(n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
first :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
0' :: 0':nil:cons:n__first:n__s:n__from
nil :: 0':nil:cons:n__first:n__s:n__from
s :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
cons :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__first :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
activate :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
from :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__from :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__s :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
hole_0':nil:cons:n__first:n__s:n__from1_0 :: 0':nil:cons:n__first:n__s:n__from
gen_0':nil:cons:n__first:n__s:n__from2_0 :: Nat → 0':nil:cons:n__first:n__s:n__from

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

Heuristically decided to analyse the following defined symbols:
activate

### (10) Obligation:

TRS:
Rules:
first(0', X) → nil
first(s(X), cons(Z)) → cons(n__first(X, activate(Z)))
from(X) → cons(n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
first :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
0' :: 0':nil:cons:n__first:n__s:n__from
nil :: 0':nil:cons:n__first:n__s:n__from
s :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
cons :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__first :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
activate :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
from :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__from :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__s :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
hole_0':nil:cons:n__first:n__s:n__from1_0 :: 0':nil:cons:n__first:n__s:n__from
gen_0':nil:cons:n__first:n__s:n__from2_0 :: Nat → 0':nil:cons:n__first:n__s:n__from

Generator Equations:
gen_0':nil:cons:n__first:n__s:n__from2_0(0) ⇔ 0'
gen_0':nil:cons:n__first:n__s:n__from2_0(+(x, 1)) ⇔ cons(gen_0':nil:cons:n__first:n__s:n__from2_0(x))

The following defined symbols remain to be analysed:
activate

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

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

### (12) Obligation:

TRS:
Rules:
first(0', X) → nil
first(s(X), cons(Z)) → cons(n__first(X, activate(Z)))
from(X) → cons(n__from(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Types:
first :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
0' :: 0':nil:cons:n__first:n__s:n__from
nil :: 0':nil:cons:n__first:n__s:n__from
s :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
cons :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__first :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
activate :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
from :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__from :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
n__s :: 0':nil:cons:n__first:n__s:n__from → 0':nil:cons:n__first:n__s:n__from
hole_0':nil:cons:n__first:n__s:n__from1_0 :: 0':nil:cons:n__first:n__s:n__from
gen_0':nil:cons:n__first:n__s:n__from2_0 :: Nat → 0':nil:cons:n__first:n__s:n__from

Generator Equations:
gen_0':nil:cons:n__first:n__s:n__from2_0(0) ⇔ 0'
gen_0':nil:cons:n__first:n__s:n__from2_0(+(x, 1)) ⇔ cons(gen_0':nil:cons:n__first:n__s:n__from2_0(x))

No more defined symbols left to analyse.