The Runtime Complexity (innermost) 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(h(x), y) → h(f(y, f(x, h(f(a, a)))))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(h(x), y) →⁺ h(f(y, f(x, h(f(a, a)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [x / h(x)].
The result substitution is [y / h(f(a, a))].

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
f(h(x), y) → h(f(y, f(x, h(f(a, a)))))

Types:
f :: h:a → h:a → h:a
h :: h:a → h:a
a :: h:a
hole_h:a1_0 :: h:a
gen_h:a2_0 :: Nat → h:a

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

Heuristically decided to analyse the following defined symbols:
f

(8) Obligation:

Innermost TRS:
Rules:
f(h(x), y) → h(f(y, f(x, h(f(a, a)))))

Types:
f :: h:a → h:a → h:a
h :: h:a → h:a
a :: h:a
hole_h:a1_0 :: h:a
gen_h:a2_0 :: Nat → h:a

Generator Equations:
gen_h:a2_0(0) ⇔ a
gen_h:a2_0(+(x, 1)) ⇔ h(gen_h:a2_0(x))

The following defined symbols remain to be analysed:
f

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

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

(10) Obligation:

Innermost TRS:
Rules:
f(h(x), y) → h(f(y, f(x, h(f(a, a)))))

Types:
f :: h:a → h:a → h:a
h :: h:a → h:a
a :: h:a
hole_h:a1_0 :: h:a
gen_h:a2_0 :: Nat → h:a

Generator Equations:
gen_h:a2_0(0) ⇔ a
gen_h:a2_0(+(x, 1)) ⇔ h(gen_h:a2_0(x))

No more defined symbols left to analyse.