### (0) Obligation:

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

w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
w(r(x)) →+ r(w(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / r(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:

w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Types:
w :: r → r
r :: r → r
b :: r → r
hole_r1_0 :: r
gen_r2_0 :: Nat → r

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

Heuristically decided to analyse the following defined symbols:
w, b

They will be analysed ascendingly in the following order:
w < b

### (8) Obligation:

TRS:
Rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Types:
w :: r → r
r :: r → r
b :: r → r
hole_r1_0 :: r
gen_r2_0 :: Nat → r

Generator Equations:
gen_r2_0(0) ⇔ hole_r1_0
gen_r2_0(+(x, 1)) ⇔ r(gen_r2_0(x))

The following defined symbols remain to be analysed:
w, b

They will be analysed ascendingly in the following order:
w < b

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

Proved the following rewrite lemma:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Induction Base:
w(gen_r2_0(+(1, 0)))

Induction Step:
w(gen_r2_0(+(1, +(n4_0, 1)))) →RΩ(1)
r(w(gen_r2_0(+(1, n4_0)))) →IH
r(*3_0)

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

### (11) Obligation:

TRS:
Rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Types:
w :: r → r
r :: r → r
b :: r → r
hole_r1_0 :: r
gen_r2_0 :: Nat → r

Lemmas:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_r2_0(0) ⇔ hole_r1_0
gen_r2_0(+(x, 1)) ⇔ r(gen_r2_0(x))

The following defined symbols remain to be analysed:
b

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

Proved the following rewrite lemma:
b(gen_r2_0(+(1, n130_0))) → *3_0, rt ∈ Ω(n1300)

Induction Base:
b(gen_r2_0(+(1, 0)))

Induction Step:
b(gen_r2_0(+(1, +(n130_0, 1)))) →RΩ(1)
r(b(gen_r2_0(+(1, n130_0)))) →IH
r(*3_0)

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

### (14) Obligation:

TRS:
Rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Types:
w :: r → r
r :: r → r
b :: r → r
hole_r1_0 :: r
gen_r2_0 :: Nat → r

Lemmas:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
b(gen_r2_0(+(1, n130_0))) → *3_0, rt ∈ Ω(n1300)

Generator Equations:
gen_r2_0(0) ⇔ hole_r1_0
gen_r2_0(+(x, 1)) ⇔ r(gen_r2_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

### (17) Obligation:

TRS:
Rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Types:
w :: r → r
r :: r → r
b :: r → r
hole_r1_0 :: r
gen_r2_0 :: Nat → r

Lemmas:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
b(gen_r2_0(+(1, n130_0))) → *3_0, rt ∈ Ω(n1300)

Generator Equations:
gen_r2_0(0) ⇔ hole_r1_0
gen_r2_0(+(x, 1)) ⇔ r(gen_r2_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

### (20) Obligation:

TRS:
Rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))

Types:
w :: r → r
r :: r → r
b :: r → r
hole_r1_0 :: r
gen_r2_0 :: Nat → r

Lemmas:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_r2_0(0) ⇔ hole_r1_0
gen_r2_0(+(x, 1)) ⇔ r(gen_r2_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
w(gen_r2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)