### (0) Obligation:

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

r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
r(nil, nil, zs, cons(w, cons(w63_0, ws64_0))) →+ r(nil, nil, cons(succ(zero), cons(succ(zero), zs)), ws64_0)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [ws64_0 / cons(w, cons(w63_0, ws64_0))].
The result substitution is [zs / cons(succ(zero), cons(succ(zero), zs))].

### (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:

r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(w, ws)) → r(xs, xs, cons(succ(zero), zs), ws)
r(xs, cons(y, ys), nil, cons(w, ws)) → r(xs, xs, cons(succ(zero), nil), ws)
r(xs, cons(y, ys), cons(z, zs), cons(w, ws)) → r(ys, cons(y, ys), zs, cons(succ(zero), cons(w, ws)))

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
cons/0
succ/0

### (6) Obligation:

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

r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(ws)) → r(xs, xs, cons(zs), ws)
r(xs, cons(ys), nil, cons(ws)) → r(xs, xs, cons(nil), ws)
r(xs, cons(ys), cons(zs), cons(ws)) → r(ys, cons(ys), zs, cons(cons(ws)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(ws)) → r(xs, xs, cons(zs), ws)
r(xs, cons(ys), nil, cons(ws)) → r(xs, xs, cons(nil), ws)
r(xs, cons(ys), cons(zs), cons(ws)) → r(ys, cons(ys), zs, cons(cons(ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: nil:cons → nil:cons
hole_nil:cons1_0 :: nil:cons
gen_nil:cons2_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
r

### (10) Obligation:

TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(ws)) → r(xs, xs, cons(zs), ws)
r(xs, cons(ys), nil, cons(ws)) → r(xs, xs, cons(nil), ws)
r(xs, cons(ys), cons(zs), cons(ws)) → r(ys, cons(ys), zs, cons(cons(ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: nil:cons → nil:cons
hole_nil:cons1_0 :: nil:cons
gen_nil:cons2_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons2_0(0) ⇔ nil
gen_nil:cons2_0(+(x, 1)) ⇔ cons(gen_nil:cons2_0(x))

The following defined symbols remain to be analysed:
r

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

Proved the following rewrite lemma:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(n4_0)) → gen_nil:cons2_0(0), rt ∈ Ω(1 + n40)

Induction Base:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(0)) →RΩ(1)
gen_nil:cons2_0(0)

Induction Step:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(+(n4_0, 1))) →RΩ(1)
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), cons(gen_nil:cons2_0(c)), gen_nil:cons2_0(n4_0)) →IH
gen_nil:cons2_0(0)

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

### (13) Obligation:

TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(ws)) → r(xs, xs, cons(zs), ws)
r(xs, cons(ys), nil, cons(ws)) → r(xs, xs, cons(nil), ws)
r(xs, cons(ys), cons(zs), cons(ws)) → r(ys, cons(ys), zs, cons(cons(ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: nil:cons → nil:cons
hole_nil:cons1_0 :: nil:cons
gen_nil:cons2_0 :: Nat → nil:cons

Lemmas:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(n4_0)) → gen_nil:cons2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_nil:cons2_0(0) ⇔ nil
gen_nil:cons2_0(+(x, 1)) ⇔ cons(gen_nil:cons2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(n4_0)) → gen_nil:cons2_0(0), rt ∈ Ω(1 + n40)

### (16) Obligation:

TRS:
Rules:
r(xs, ys, zs, nil) → xs
r(xs, nil, zs, cons(ws)) → r(xs, xs, cons(zs), ws)
r(xs, cons(ys), nil, cons(ws)) → r(xs, xs, cons(nil), ws)
r(xs, cons(ys), cons(zs), cons(ws)) → r(ys, cons(ys), zs, cons(cons(ws)))

Types:
r :: nil:cons → nil:cons → nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: nil:cons → nil:cons
hole_nil:cons1_0 :: nil:cons
gen_nil:cons2_0 :: Nat → nil:cons

Lemmas:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(n4_0)) → gen_nil:cons2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_nil:cons2_0(0) ⇔ nil
gen_nil:cons2_0(+(x, 1)) ⇔ cons(gen_nil:cons2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
r(gen_nil:cons2_0(0), gen_nil:cons2_0(0), gen_nil:cons2_0(c), gen_nil:cons2_0(n4_0)) → gen_nil:cons2_0(0), rt ∈ Ω(1 + n40)