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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
h(gen_f:a2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
h(gen_f:a2_0(+(1, 0)))

Induction Step:
h(gen_f:a2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
f(f(a, h(h(gen_f:a2_0(+(1, n4_0))))), a) →_IH
f(f(a, h(*3_0)), a)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) 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

Lemmas:
h(gen_f:a2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

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.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_f:a2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(13) BOUNDS(n^1, INF)

(14) 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

Lemmas:
h(gen_f:a2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_f:a2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)