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:

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

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

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
h/1
g/1

(6) Obligation:

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

h(f(x)) → f(g(x))
g(x) → h(x)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

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

(9) OrderProof (LOWER BOUND(ID) transformation)

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

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

(10) Obligation:

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

Types:
h :: f → f
f :: f → f
g :: 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:
g, h

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

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
g(gen_f2_0(0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

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

Lemmas:
g(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:
h

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

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

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

(15) Complex Obligation (BEST)

(16) Obligation:

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

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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
g(gen_f2_0(0))

Induction Step:
g(gen_f2_0(+(n335_0, 1))) →_R^Ω(1)
h(gen_f2_0(+(n335_0, 1))) →_R^Ω(1)
f(g(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).

(18) Complex Obligation (BEST)

(19) Obligation:

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

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

Lemmas:
g(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)
h(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.

(20) LowerBoundsProof (EQUIVALENT transformation)

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

(21) BOUNDS(n^1, INF)

(22) Obligation:

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

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

Lemmas:
g(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n335₀)
h(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.

(23) LowerBoundsProof (EQUIVALENT transformation)

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

(24) BOUNDS(n^1, INF)

(25) Obligation:

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

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

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

(26) LowerBoundsProof (EQUIVALENT transformation)

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

(27) BOUNDS(n^1, INF)

(28) Obligation:

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

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

Lemmas:
g(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.

(29) LowerBoundsProof (EQUIVALENT transformation)

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

(30) BOUNDS(n^1, INF)