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) 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 [ ].

(2) BOUNDS(n^1, INF)

(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

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

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)))) →_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).

(10) Complex Obligation (BEST)

(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)))) →_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).

(13) Complex Obligation (BEST)

(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)

(16) BOUNDS(n^1, INF)

(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)

(19) BOUNDS(n^1, INF)

(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)

(22) BOUNDS(n^1, INF)