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, x) → a
f(g(x), y) → f(x, y)

Rewrite Strategy: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(x, x) → a
f(g(x), y) → f(x, y)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

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

Types:
f :: g → g → a
a :: a
g :: g → g
hole_a1_0 :: a
hole_g2_0 :: g
gen_g3_0 :: Nat → g

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f

(6) Obligation:

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

Types:
f :: g → g → a
a :: a
g :: g → g
hole_a1_0 :: a
hole_g2_0 :: g
gen_g3_0 :: Nat → g

Generator Equations:
gen_g3_0(0) ⇔ hole_g2_0
gen_g3_0(+(x, 1)) ⇔ g(gen_g3_0(x))

The following defined symbols remain to be analysed:
f

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_g3_0(+(1, n5_0)), gen_g3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
f(gen_g3_0(+(1, 0)), gen_g3_0(b))

Induction Step:
f(gen_g3_0(+(1, +(n5_0, 1))), gen_g3_0(b)) →_R^Ω(1)
f(gen_g3_0(+(1, n5_0)), gen_g3_0(b)) →_IH
*4_0

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

(8) Complex Obligation (BEST)

(9) Obligation:

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

Types:
f :: g → g → a
a :: a
g :: g → g
hole_a1_0 :: a
hole_g2_0 :: g
gen_g3_0 :: Nat → g

Lemmas:
f(gen_g3_0(+(1, n5_0)), gen_g3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_g3_0(0) ⇔ hole_g2_0
gen_g3_0(+(x, 1)) ⇔ g(gen_g3_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_g3_0(+(1, n5_0)), gen_g3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

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

Types:
f :: g → g → a
a :: a
g :: g → g
hole_a1_0 :: a
hole_g2_0 :: g
gen_g3_0 :: Nat → g

Lemmas:
f(gen_g3_0(+(1, n5_0)), gen_g3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_g3_0(0) ⇔ hole_g2_0
gen_g3_0(+(x, 1)) ⇔ g(gen_g3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_g3_0(+(1, n5_0)), gen_g3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(14) BOUNDS(n^1, INF)