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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
g(f(x), y) →⁺ f(g(x, f(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 [y / f(y)].

(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:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

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

Types:
g :: f → f → f
f :: f → f
h :: f → 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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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

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

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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

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

Types:
g :: f → f → f
f :: f → f
h :: f → 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))

No more defined symbols left to analyse.