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:

++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

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:

++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: a → nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_a2_0 :: a
gen_nil:.3_0 :: Nat → nil:.

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

Heuristically decided to analyse the following defined symbols:
++

(6) Obligation:

TRS:
Rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: a → nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_a2_0 :: a
gen_nil:.3_0 :: Nat → nil:.

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.3_0(x))

The following defined symbols remain to be analysed:
++

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

Proved the following rewrite lemma:
++(gen_nil:.3_0(n5_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
++(gen_nil:.3_0(0), gen_nil:.3_0(b)) →_R^Ω(1)
gen_nil:.3_0(b)

Induction Step:
++(gen_nil:.3_0(+(n5_0, 1)), gen_nil:.3_0(b)) →_R^Ω(1)
.(hole_a2_0, ++(gen_nil:.3_0(n5_0), gen_nil:.3_0(b))) →_IH
.(hole_a2_0, gen_nil:.3_0(+(b, c6_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:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: a → nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_a2_0 :: a
gen_nil:.3_0 :: Nat → nil:.

Lemmas:
++(gen_nil:.3_0(n5_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.3_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
++(gen_nil:.3_0(n5_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

TRS:
Rules:
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: a → nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_a2_0 :: a
gen_nil:.3_0 :: Nat → nil:.

Lemmas:
++(gen_nil:.3_0(n5_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(hole_a2_0, gen_nil:.3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
++(gen_nil:.3_0(n5_0), gen_nil:.3_0(b)) → gen_nil:.3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(14) BOUNDS(n^1, INF)