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:

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

Rewrite Strategy: FULL

(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) → f(h(x, y))
h(x, y) → g(x, f(y))

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
g/1
h/1

(4) Obligation:

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

g(f(x)) → f(h(x))
h(x) → g(x)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

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

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

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

(8) Obligation:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

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

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

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

Proved the following rewrite lemma:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
h(gen_f2_0(0))

Induction Step:
h(gen_f2_0(+(n4_0, 1))) →_R^Ω(1)
g(gen_f2_0(+(n4_0, 1))) →_R^Ω(1)
f(h(gen_f2_0(n4_0))) →_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:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

The following defined symbols remain to be analysed:
g

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

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

Proved the following rewrite lemma:
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n112₀)

Induction Base:
g(gen_f2_0(+(1, 0)))

Induction Step:
g(gen_f2_0(+(1, +(n112_0, 1)))) →_R^Ω(1)
f(h(gen_f2_0(+(1, n112_0)))) →_R^Ω(1)
f(g(gen_f2_0(+(1, n112_0)))) →_IH
f(*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:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n112₀)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

The following defined symbols remain to be analysed:
h

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

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

Proved the following rewrite lemma:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)

Induction Base:
h(gen_f2_0(0))

Induction Step:
h(gen_f2_0(+(n335_0, 1))) →_R^Ω(1)
g(gen_f2_0(+(n335_0, 1))) →_R^Ω(1)
f(h(gen_f2_0(n335_0))) →_IH
f(*3_0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n112₀)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n112₀)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n112₀)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

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

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(28) BOUNDS(n^1, INF)