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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
g :: a:g → a:g
h :: a:g → a:g → a:g → a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat → a:g

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

Heuristically decided to analyse the following defined symbols:
f, h

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

(8) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
g :: a:g → a:g
h :: a:g → a:g → a:g → a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat → a:g

Generator Equations:
gen_a:g2_0(0) ⇔ a
gen_a:g2_0(+(x, 1)) ⇔ g(gen_a:g2_0(x))

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

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) → gen_a:g2_0(+(1, n4_0)), rt ∈ Ω(1 + n4₀)

Induction Base:
h(gen_a:g2_0(0), gen_a:g2_0(0), gen_a:g2_0(1)) →_R^Ω(1)
gen_a:g2_0(1)

Induction Step:
h(gen_a:g2_0(+(n4_0, 1)), gen_a:g2_0(0), gen_a:g2_0(1)) →_R^Ω(1)
f(gen_a:g2_0(0), h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1))) →_IH
f(gen_a:g2_0(0), gen_a:g2_0(+(1, c5_0))) →_R^Ω(1)
g(g(gen_a:g2_0(n4_0)))

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

(10) Complex Obligation (BEST)

(11) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
g :: a:g → a:g
h :: a:g → a:g → a:g → a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat → a:g

Lemmas:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) → gen_a:g2_0(+(1, n4_0)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_a:g2_0(0) ⇔ a
gen_a:g2_0(+(x, 1)) ⇔ g(gen_a:g2_0(x))

The following defined symbols remain to be analysed:
f

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
g :: a:g → a:g
h :: a:g → a:g → a:g → a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat → a:g

Lemmas:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) → gen_a:g2_0(+(1, n4_0)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_a:g2_0(0) ⇔ a
gen_a:g2_0(+(x, 1)) ⇔ g(gen_a:g2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) → gen_a:g2_0(+(1, n4_0)), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
g :: a:g → a:g
h :: a:g → a:g → a:g → a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat → a:g

Lemmas:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) → gen_a:g2_0(+(1, n4_0)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_a:g2_0(0) ⇔ a
gen_a:g2_0(+(x, 1)) ⇔ g(gen_a:g2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) → gen_a:g2_0(+(1, n4_0)), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)