### (0) Obligation:

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

from(X) → cons(X, n__from(s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
sel(s(X), cons(Y, Z)) →+ sel(X, Z)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X / s(X), Z / cons(Y, Z)].
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:

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
from :: s:0' → n__from:cons:nil:n__first
cons :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
n__from :: s:0' → n__from:cons:nil:n__first
s :: s:0' → s:0'
first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
0' :: s:0'
nil :: n__from:cons:nil:n__first
n__first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
activate :: n__from:cons:nil:n__first → n__from:cons:nil:n__first
sel :: s:0' → n__from:cons:nil:n__first → s:0'
hole_n__from:cons:nil:n__first1_0 :: n__from:cons:nil:n__first
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__first3_0 :: Nat → n__from:cons:nil:n__first
gen_s:0'4_0 :: Nat → s:0'

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

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

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

### (8) Obligation:

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

Types:
from :: s:0' → n__from:cons:nil:n__first
cons :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
n__from :: s:0' → n__from:cons:nil:n__first
s :: s:0' → s:0'
first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
0' :: s:0'
nil :: n__from:cons:nil:n__first
n__first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
activate :: n__from:cons:nil:n__first → n__from:cons:nil:n__first
sel :: s:0' → n__from:cons:nil:n__first → s:0'
hole_n__from:cons:nil:n__first1_0 :: n__from:cons:nil:n__first
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__first3_0 :: Nat → n__from:cons:nil:n__first
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__first3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__first3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__first3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

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

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

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

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

### (10) Obligation:

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

Types:
from :: s:0' → n__from:cons:nil:n__first
cons :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
n__from :: s:0' → n__from:cons:nil:n__first
s :: s:0' → s:0'
first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
0' :: s:0'
nil :: n__from:cons:nil:n__first
n__first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
activate :: n__from:cons:nil:n__first → n__from:cons:nil:n__first
sel :: s:0' → n__from:cons:nil:n__first → s:0'
hole_n__from:cons:nil:n__first1_0 :: n__from:cons:nil:n__first
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__first3_0 :: Nat → n__from:cons:nil:n__first
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__first3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__first3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__first3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
sel

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

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

### (12) Obligation:

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

Types:
from :: s:0' → n__from:cons:nil:n__first
cons :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
n__from :: s:0' → n__from:cons:nil:n__first
s :: s:0' → s:0'
first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
0' :: s:0'
nil :: n__from:cons:nil:n__first
n__first :: s:0' → n__from:cons:nil:n__first → n__from:cons:nil:n__first
activate :: n__from:cons:nil:n__first → n__from:cons:nil:n__first
sel :: s:0' → n__from:cons:nil:n__first → s:0'
hole_n__from:cons:nil:n__first1_0 :: n__from:cons:nil:n__first
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__first3_0 :: Nat → n__from:cons:nil:n__first
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__first3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__first3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__first3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.