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:

append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
append

(8) Obligation:

Innermost TRS:
Rules:
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
append

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

Proved the following rewrite lemma:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
append(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) →_R^Ω(1)
gen_Cons:Nil3_0(b)

Induction Step:
append(gen_Cons:Nil3_0(+(n5_0, 1)), gen_Cons:Nil3_0(b)) →_R^Ω(1)
Cons(hole_a2_0, append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b))) →_IH
Cons(hole_a2_0, gen_Cons:Nil3_0(+(b, c6_0)))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_a2_0 :: a
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) → gen_Cons:Nil3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)