The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) 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

(5) 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

(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

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

(7) 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)))) →_R^Ω(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 irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) 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

(10) 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)))) →_R^Ω(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 irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) 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.

(13) 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)

(14) BOUNDS(n^1, INF)

(15) 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.

(16) 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) BOUNDS(n^1, INF)

(18) 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.

(19) 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) BOUNDS(n^1, INF)