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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(6) Obligation:

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

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

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

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
append

(10) Obligation:

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

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
append

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

Proved the following rewrite lemma:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
append(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) →_R^Ω(1)
gen_Cons:Nil2_0(b)

Induction Step:
append(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) →_R^Ω(1)
Cons(append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b))) →_IH
Cons(gen_Cons:Nil2_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:

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

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
append :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)