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

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(f(a, h(h(y))), x)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
The pumping substitution is [y / f(x, y)].
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(f(a, h(h(y))), x)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
h :: f:a → f:a
f :: f:a → f:a → f:a
a :: f:a
hole_f:a1_0 :: f:a
gen_f:a2_0 :: Nat → f:a

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

Heuristically decided to analyse the following defined symbols:
h

(8) Obligation:

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

Types:
h :: f:a → f:a
f :: f:a → f:a → f:a
a :: f:a
hole_f:a1_0 :: f:a
gen_f:a2_0 :: Nat → f:a

Generator Equations:
gen_f:a2_0(0) ⇔ a
gen_f:a2_0(+(x, 1)) ⇔ f(a, gen_f:a2_0(x))

The following defined symbols remain to be analysed:
h

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

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

(10) Obligation:

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

Types:
h :: f:a → f:a
f :: f:a → f:a → f:a
a :: f:a
hole_f:a1_0 :: f:a
gen_f:a2_0 :: Nat → f:a

Generator Equations:
gen_f:a2_0(0) ⇔ a
gen_f:a2_0(+(x, 1)) ⇔ f(a, gen_f:a2_0(x))

No more defined symbols left to analyse.