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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
++(.(x, y), z) →⁺ .(x, ++(y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [y / .(x, y)].
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:

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

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
./0

(6) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
++

(10) Obligation:

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

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: 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)) ⇔ .(gen_nil:.2_0(x))

The following defined symbols remain to be analysed:
++

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
++(gen_nil:.2_0(0), gen_nil:.2_0(b)) →_R^Ω(1)
gen_nil:.2_0(b)

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

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

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

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

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
++ :: nil:. → nil:. → nil:.
nil :: nil:.
. :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

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

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

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)