### (0) Obligation:

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

from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0) → 0
minus(s(X), s(Y)) → minus(X, Y)
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(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(N), cons(X, XS)) →+ sel(N, XS)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [N / s(N), XS / cons(X, XS)].
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)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(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)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

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

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

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

### (8) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_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:
minus, sel, activate, quot

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

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

Proved the following rewrite lemma:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

Induction Base:
minus(gen_s:0'4_0(0), gen_s:0'4_0(0)) →RΩ(1)
0'

Induction Step:
minus(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) →RΩ(1)
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) →IH
gen_s:0'4_0(0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (11) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_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:
quot, sel, activate

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

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

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

### (13) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_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

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

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

### (15) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_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

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

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

### (17) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

### (20) Obligation:

TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
minus(X, 0') → 0'
minus(s(X), s(Y)) → minus(X, Y)
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) → nil
zWquot(nil, XS) → nil
zWquot(cons(X, XS), cons(Y, YS)) → cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) → n__from(X)
zWquot(X1, X2) → n__zWquot(X1, X2)
activate(n__from(X)) → from(X)
activate(n__zWquot(X1, X2)) → zWquot(X1, X2)
activate(X) → X

Types:
from :: s:0' → n__from:cons:nil:n__zWquot
cons :: s:0' → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
n__from :: s:0' → n__from:cons:nil:n__zWquot
s :: s:0' → s:0'
sel :: s:0' → n__from:cons:nil:n__zWquot → s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
minus :: s:0' → s:0' → s:0'
quot :: s:0' → s:0' → s:0'
zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot → n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat → n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) ⇔ n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons:nil:n__zWquot3_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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) → gen_s:0'4_0(0), rt ∈ Ω(1 + n60)