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:

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

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:

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

S is empty.
Rewrite Strategy: INNERMOST

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

Sliced the following arguments:
f/1
g/1

(4) Obligation:

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

f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

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

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

They will be analysed ascendingly in the following order:
f = g

(8) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

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

They will be analysed ascendingly in the following order:
f = g

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

Proved the following rewrite lemma:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
g(gen_h2_0(+(1, 0)))

Induction Step:
g(gen_h2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
h(g(gen_h2_0(+(1, n4_0)))) →_IH
h(*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:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Lemmas:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
f = g

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_h2_0(n215_0)) → *3_0, rt ∈ Ω(n215₀)

Induction Base:
f(gen_h2_0(0))

Induction Step:
f(gen_h2_0(+(n215_0, 1))) →_R^Ω(1)
g(gen_h2_0(+(n215_0, 1))) →_R^Ω(1)
h(f(gen_h2_0(n215_0))) →_IH
h(*3_0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Lemmas:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
f(gen_h2_0(n215_0)) → *3_0, rt ∈ Ω(n215₀)

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

The following defined symbols remain to be analysed:
g

They will be analysed ascendingly in the following order:
f = g

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_h2_0(+(1, n477_0))) → *3_0, rt ∈ Ω(n477₀)

Induction Base:
g(gen_h2_0(+(1, 0)))

Induction Step:
g(gen_h2_0(+(1, +(n477_0, 1)))) →_R^Ω(1)
h(g(gen_h2_0(+(1, n477_0)))) →_IH
h(*3_0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Lemmas:
g(gen_h2_0(+(1, n477_0))) → *3_0, rt ∈ Ω(n477₀)
f(gen_h2_0(n215_0)) → *3_0, rt ∈ Ω(n215₀)

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_h2_0(+(1, n477_0))) → *3_0, rt ∈ Ω(n477₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Lemmas:
g(gen_h2_0(+(1, n477_0))) → *3_0, rt ∈ Ω(n477₀)
f(gen_h2_0(n215_0)) → *3_0, rt ∈ Ω(n215₀)

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_h2_0(+(1, n477_0))) → *3_0, rt ∈ Ω(n477₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Lemmas:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
f(gen_h2_0(n215_0)) → *3_0, rt ∈ Ω(n215₀)

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
f(x) → g(x)
g(h(x)) → h(f(x))
g(h(x)) → h(g(x))

Types:
f :: h → h
g :: h → h
h :: h → h
hole_h1_0 :: h
gen_h2_0 :: Nat → h

Lemmas:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_h2_0(0) ⇔ hole_h1_0
gen_h2_0(+(x, 1)) ⇔ h(gen_h2_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_h2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(28) BOUNDS(n^1, INF)