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

(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, a) → x
f(x, g(y)) → f(g(x), y)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
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

(8) Obligation:

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

Types:
f :: a:g → a:g → a:g
a :: a:g
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:
f

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

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

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

Induction Step:
f(gen_a:g2_0(a), gen_a:g2_0(+(n4_0, 1))) →_R^Ω(1)
f(g(gen_a:g2_0(a)), gen_a:g2_0(n4_0)) →_IH
gen_a:g2_0(+(+(a, 1), c5_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(x, a) → x
f(x, g(y)) → f(g(x), y)

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

Lemmas:
f(gen_a:g2_0(a), gen_a:g2_0(n4_0)) → gen_a:g2_0(+(n4_0, a)), 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.

(12) LowerBoundsProof (EQUIVALENT transformation)

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

(13) BOUNDS(n^1, INF)

(14) Obligation:

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

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

Lemmas:
f(gen_a:g2_0(a), gen_a:g2_0(n4_0)) → gen_a:g2_0(+(n4_0, a)), 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.

(15) LowerBoundsProof (EQUIVALENT transformation)

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

(16) BOUNDS(n^1, INF)