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:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
bits(0) → 0
bits(s(x)) → s(bits(half(s(x))))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
bits(0') → 0'
bits(s(x)) → s(bits(half(s(x))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
bits(0') → 0'
bits(s(x)) → s(bits(half(s(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
bits :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
half, bits

They will be analysed ascendingly in the following order:
half < bits

(8) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
bits(0') → 0'
bits(s(x)) → s(bits(half(s(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
bits :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
half, bits

They will be analysed ascendingly in the following order:
half < bits

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

Proved the following rewrite lemma:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
half(gen_0':s2_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
half(gen_0':s2_0(*(2, +(n4_0, 1)))) →_R^Ω(1)
s(half(gen_0':s2_0(*(2, n4_0)))) →_IH
s(gen_0':s2_0(c5_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:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
bits(0') → 0'
bits(s(x)) → s(bits(half(s(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
bits :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
bits

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
bits(0') → 0'
bits(s(x)) → s(bits(half(s(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
bits :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
bits(0') → 0'
bits(s(x)) → s(bits(half(s(x))))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
bits :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
half(gen_0':s2_0(*(2, n4_0))) → gen_0':s2_0(n4_0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)