The Runtime Complexity (innermost) 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:

g(x, 0) → 0
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0), y) → y
id(x) → f(x, s(0))

Rewrite Strategy: INNERMOST

(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:

g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
g, f

(6) Obligation:

Innermost TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
g, f

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Induction Base:
g(h, gen_0':s3_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
g(h, gen_0':s3_0(*(2, +(n5_0, 1)))) →_R^Ω(1)
s(g(h, gen_0':s3_0(*(2, n5_0)))) →_IH
s(gen_0':s3_0(c6_0))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
f

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
g(x, 0') → 0'
g(d, s(x)) → s(s(g(d, x)))
g(h, s(0')) → 0'
g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)
half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))
f(s(0'), y) → y
id(x) → f(x, s(0'))

Types:
g :: d:h → 0':s → 0':s
0' :: 0':s
d :: d:h
s :: 0':s → 0':s
h :: d:h
double :: 0':s → 0':s
half :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
id :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat → 0':s

Lemmas:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(h, gen_0':s3_0(*(2, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)