The Runtime Complexity (innermost) 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(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

Rewrite Strategy: INNERMOST

(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(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

Types:
f :: nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:. → nil:.
g :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

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

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

(6) Obligation:

Innermost TRS:
Rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

Types:
f :: nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:. → nil:.
g :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(nil, gen_nil:.2_0(x))

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

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

Proved the following rewrite lemma:
f(gen_nil:.2_0(n4_0)) → gen_nil:.2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
f(gen_nil:.2_0(0)) →_R^Ω(1)
nil

Induction Step:
f(gen_nil:.2_0(+(n4_0, 1))) →_R^Ω(1)
.(nil, f(gen_nil:.2_0(n4_0))) →_IH
.(nil, gen_nil:.2_0(c5_0))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

Types:
f :: nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:. → nil:.
g :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Lemmas:
f(gen_nil:.2_0(n4_0)) → gen_nil:.2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(nil, gen_nil:.2_0(x))

The following defined symbols remain to be analysed:
g

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

Types:
f :: nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:. → nil:.
g :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Lemmas:
f(gen_nil:.2_0(n4_0)) → gen_nil:.2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(nil, gen_nil:.2_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_nil:.2_0(n4_0)) → gen_nil:.2_0(n4_0), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
f(nil) → nil
f(.(nil, y)) → .(nil, f(y))
f(.(.(x, y), z)) → f(.(x, .(y, z)))
g(nil) → nil
g(.(x, nil)) → .(g(x), nil)
g(.(x, .(y, z))) → g(.(.(x, y), z))

Types:
f :: nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:. → nil:.
g :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Lemmas:
f(gen_nil:.2_0(n4_0)) → gen_nil:.2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(nil, gen_nil:.2_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_nil:.2_0(n4_0)) → gen_nil:.2_0(n4_0), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)