### (0) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0) → 0
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(from(X)) →+ cons(mark(mark(X)), from(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X / from(X)].
The result substitution is [ ].

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

a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

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

Heuristically decided to analyse the following defined symbols:
a__from, mark, a__length, a__length1

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

### (8) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
a__length1, a__from, mark, a__length

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

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

Could not prove a rewrite lemma for the defined symbol a__length1.

### (10) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
a__length, a__from, mark

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

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

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

### (12) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
mark, a__from

They will be analysed ascendingly in the following order:
a__from = mark

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

Proved the following rewrite lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n580)

Induction Base:
mark(gen_s:from:cons:nil:0':length:length12_0(0)) →RΩ(1)
nil

Induction Step:
mark(gen_s:from:cons:nil:0':length:length12_0(+(n58_0, 1))) →RΩ(1)
cons(mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)), nil) →IH
cons(gen_s:from:cons:nil:0':length:length12_0(c59_0), nil)

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

### (15) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n580)

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

The following defined symbols remain to be analysed:
a__from

They will be analysed ascendingly in the following order:
a__from = mark

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

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

### (17) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n580)

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n580)

### (20) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__length(nil) → 0'
a__length(cons(X, Y)) → s(a__length1(Y))
a__length1(X) → a__length(X)
mark(from(X)) → a__from(mark(X))
mark(length(X)) → a__length(X)
mark(length1(X)) → a__length1(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(nil) → nil
mark(0') → 0'
a__from(X) → from(X)
a__length(X) → length(X)
a__length1(X) → length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 → s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat → s:from:cons:nil:0':length:length1

Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n580)

Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) ⇔ nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) ⇔ cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) → gen_s:from:cons:nil:0':length:length12_0(n58_0), rt ∈ Ω(1 + n580)