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(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)

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(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
f/0
h/1

(4) Obligation:

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

g(f(y), z) → f(g(y, z))
g(h(x), z) → g(x, f(z))
g(x, h(y)) → h(g(x, y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
g(f(y), z) → f(g(y, z))
g(h(x), z) → g(x, f(z))
g(x, h(y)) → h(g(x, y))

Types:
g :: f:h → f:h → f:h
f :: f:h → f:h
h :: f:h → f:h
hole_f:h1_0 :: f:h
gen_f:h2_0 :: Nat → f:h

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

Heuristically decided to analyse the following defined symbols:
g

(8) Obligation:

Innermost TRS:
Rules:
g(f(y), z) → f(g(y, z))
g(h(x), z) → g(x, f(z))
g(x, h(y)) → h(g(x, y))

Types:
g :: f:h → f:h → f:h
f :: f:h → f:h
h :: f:h → f:h
hole_f:h1_0 :: f:h
gen_f:h2_0 :: Nat → f:h

Generator Equations:
gen_f:h2_0(0) ⇔ hole_f:h1_0
gen_f:h2_0(+(x, 1)) ⇔ f(gen_f:h2_0(x))

The following defined symbols remain to be analysed:
g

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

Proved the following rewrite lemma:
g(gen_f:h2_0(+(1, n4_0)), gen_f:h2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
g(gen_f:h2_0(+(1, 0)), gen_f:h2_0(b))

Induction Step:
g(gen_f:h2_0(+(1, +(n4_0, 1))), gen_f:h2_0(b)) →_R^Ω(1)
f(g(gen_f:h2_0(+(1, n4_0)), gen_f:h2_0(b))) →_IH
f(*3_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
g(f(y), z) → f(g(y, z))
g(h(x), z) → g(x, f(z))
g(x, h(y)) → h(g(x, y))

Types:
g :: f:h → f:h → f:h
f :: f:h → f:h
h :: f:h → f:h
hole_f:h1_0 :: f:h
gen_f:h2_0 :: Nat → f:h

Lemmas:
g(gen_f:h2_0(+(1, n4_0)), gen_f:h2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_f:h2_0(0) ⇔ hole_f:h1_0
gen_f:h2_0(+(x, 1)) ⇔ f(gen_f:h2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_f:h2_0(+(1, n4_0)), gen_f:h2_0(b)) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
g(f(y), z) → f(g(y, z))
g(h(x), z) → g(x, f(z))
g(x, h(y)) → h(g(x, y))

Types:
g :: f:h → f:h → f:h
f :: f:h → f:h
h :: f:h → f:h
hole_f:h1_0 :: f:h
gen_f:h2_0 :: Nat → f:h

Lemmas:
g(gen_f:h2_0(+(1, n4_0)), gen_f:h2_0(b)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_f:h2_0(0) ⇔ hole_f:h1_0
gen_f:h2_0(+(x, 1)) ⇔ f(gen_f:h2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_f:h2_0(+(1, n4_0)), gen_f:h2_0(b)) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)