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:

f(x, y) → g(x, y)
g(h(x), y) → h(f(x, y))
g(h(x), y) → h(g(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
f(h(x3_0), y) →⁺ h(f(x3_0, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x3_0 / h(x3_0)].
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:

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: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: h → a → h
g :: h → a → h
h :: h → h
hole_h1_0 :: h
hole_a2_0 :: a
gen_h3_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:

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

Types:
f :: h → a → h
g :: h → a → h
h :: h → h
hole_h1_0 :: h
hole_a2_0 :: a
gen_h3_0 :: Nat → h

Generator Equations:
gen_h3_0(0) ⇔ hole_h1_0
gen_h3_0(+(x, 1)) ⇔ h(gen_h3_0(x))

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

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

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

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

(10) Obligation:

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

Types:
f :: h → a → h
g :: h → a → h
h :: h → h
hole_h1_0 :: h
hole_a2_0 :: a
gen_h3_0 :: Nat → h

Generator Equations:
gen_h3_0(0) ⇔ hole_h1_0
gen_h3_0(+(x, 1)) ⇔ h(gen_h3_0(x))

The following defined symbols remain to be analysed:
f

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

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

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

(12) Obligation:

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

Types:
f :: h → a → h
g :: h → a → h
h :: h → h
hole_h1_0 :: h
hole_a2_0 :: a
gen_h3_0 :: Nat → h

Generator Equations:
gen_h3_0(0) ⇔ hole_h1_0
gen_h3_0(+(x, 1)) ⇔ h(gen_h3_0(x))

No more defined symbols left to analyse.