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

Infered types.

(6) Obligation:

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

Types:
h :: f → a → f
f :: f → f
g :: f → a → f
hole_f1_0 :: f
hole_a2_0 :: a
gen_f3_0 :: Nat → f

(7) 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

(8) Obligation:

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

Types:
h :: f → a → f
f :: f → f
g :: f → a → f
hole_f1_0 :: f
hole_a2_0 :: a
gen_f3_0 :: Nat → f

Generator Equations:
gen_f3_0(0) ⇔ hole_f1_0
gen_f3_0(+(x, 1)) ⇔ f(gen_f3_0(x))

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

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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

Types:
h :: f → a → f
f :: f → f
g :: f → a → f
hole_f1_0 :: f
hole_a2_0 :: a
gen_f3_0 :: Nat → f

Generator Equations:
gen_f3_0(0) ⇔ hole_f1_0
gen_f3_0(+(x, 1)) ⇔ f(gen_f3_0(x))

The following defined symbols remain to be analysed:
h

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

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

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

(12) Obligation:

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

Types:
h :: f → a → f
f :: f → f
g :: f → a → f
hole_f1_0 :: f
hole_a2_0 :: a
gen_f3_0 :: Nat → f

Generator Equations:
gen_f3_0(0) ⇔ hole_f1_0
gen_f3_0(+(x, 1)) ⇔ f(gen_f3_0(x))

No more defined symbols left to analyse.