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(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
append(::(@x3_0, @xs4_0), @l2) →⁺ ::(@x3_0, append(@xs4_0, @l2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [@xs4_0 / ::(@x3_0, @xs4_0)].
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(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

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

Heuristically decided to analyse the following defined symbols:
append, append#1, appendAll, appendAll#1, appendAll2, appendAll2#1, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
append = append#1
append < appendAll#1
append < appendAll2#1
append < appendAll3#1
appendAll = appendAll#1
appendAll < appendAll2#1
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

(8) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
append#1, append, appendAll, appendAll#1, appendAll2, appendAll2#1, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
append = append#1
append < appendAll#1
append < appendAll2#1
append < appendAll3#1
appendAll = appendAll#1
appendAll < appendAll2#1
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

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

Proved the following rewrite lemma:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
append#1(gen_:::nil2_0(0), gen_:::nil2_0(b)) →_R^Ω(1)
gen_:::nil2_0(b)

Induction Step:
append#1(gen_:::nil2_0(+(n4_0, 1)), gen_:::nil2_0(b)) →_R^Ω(1)
::(nil, append(gen_:::nil2_0(n4_0), gen_:::nil2_0(b))) →_R^Ω(1)
::(nil, append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b))) →_IH
::(nil, gen_:::nil2_0(+(b, c5_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(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
append, appendAll, appendAll#1, appendAll2, appendAll2#1, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
append = append#1
append < appendAll#1
append < appendAll2#1
append < appendAll3#1
appendAll = appendAll#1
appendAll < appendAll2#1
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
appendAll#1, appendAll, appendAll2, appendAll2#1, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
appendAll = appendAll#1
appendAll < appendAll2#1
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)

Induction Base:
appendAll#1(gen_:::nil2_0(0)) →_R^Ω(1)
nil

Induction Step:
appendAll#1(gen_:::nil2_0(+(n641_0, 1))) →_R^Ω(1)
append(nil, appendAll(gen_:::nil2_0(n641_0))) →_R^Ω(1)
append(nil, appendAll#1(gen_:::nil2_0(n641_0))) →_IH
append(nil, gen_:::nil2_0(0)) →_R^Ω(1)
append#1(nil, gen_:::nil2_0(0)) →_L^Ω(1)
gen_:::nil2_0(+(0, 0))

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
appendAll, appendAll2, appendAll2#1, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
appendAll = appendAll#1
appendAll < appendAll2#1
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
appendAll2#1, appendAll2, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)

Induction Base:
appendAll2#1(gen_:::nil2_0(0)) →_R^Ω(1)
nil

Induction Step:
appendAll2#1(gen_:::nil2_0(+(n1022_0, 1))) →_R^Ω(1)
append(appendAll(nil), appendAll2(gen_:::nil2_0(n1022_0))) →_R^Ω(1)
append(appendAll#1(nil), appendAll2(gen_:::nil2_0(n1022_0))) →_L^Ω(1)
append(gen_:::nil2_0(0), appendAll2(gen_:::nil2_0(n1022_0))) →_R^Ω(1)
append(gen_:::nil2_0(0), appendAll2#1(gen_:::nil2_0(n1022_0))) →_IH
append(gen_:::nil2_0(0), gen_:::nil2_0(0)) →_R^Ω(1)
append#1(gen_:::nil2_0(0), gen_:::nil2_0(0)) →_L^Ω(1)
gen_:::nil2_0(+(0, 0))

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

(20) Complex Obligation (BEST)

(21) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
appendAll2, appendAll3, appendAll3#1

They will be analysed ascendingly in the following order:
appendAll2 = appendAll2#1
appendAll2 < appendAll3#1
appendAll3 = appendAll3#1

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol appendAll2.

(23) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
appendAll3#1, appendAll3

They will be analysed ascendingly in the following order:
appendAll3 = appendAll3#1

(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
appendAll3#1(gen_:::nil2_0(n1764_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1764₀)

Induction Base:
appendAll3#1(gen_:::nil2_0(0)) →_R^Ω(1)
nil

Induction Step:
appendAll3#1(gen_:::nil2_0(+(n1764_0, 1))) →_R^Ω(1)
append(appendAll2(nil), appendAll3(gen_:::nil2_0(n1764_0))) →_R^Ω(1)
append(appendAll2#1(nil), appendAll3(gen_:::nil2_0(n1764_0))) →_L^Ω(1)
append(gen_:::nil2_0(0), appendAll3(gen_:::nil2_0(n1764_0))) →_R^Ω(1)
append(gen_:::nil2_0(0), appendAll3#1(gen_:::nil2_0(n1764_0))) →_IH
append(gen_:::nil2_0(0), gen_:::nil2_0(0)) →_R^Ω(1)
append#1(gen_:::nil2_0(0), gen_:::nil2_0(0)) →_L^Ω(1)
gen_:::nil2_0(+(0, 0))

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

(25) Complex Obligation (BEST)

(26) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)
appendAll3#1(gen_:::nil2_0(n1764_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1764₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

The following defined symbols remain to be analysed:
appendAll3

They will be analysed ascendingly in the following order:
appendAll3 = appendAll3#1

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol appendAll3.

(28) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)
appendAll3#1(gen_:::nil2_0(n1764_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1764₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(30) BOUNDS(n^1, INF)

(31) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)
appendAll3#1(gen_:::nil2_0(n1764_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1764₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)
appendAll2#1(gen_:::nil2_0(n1022_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n1022₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(36) BOUNDS(n^1, INF)

(37) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
appendAll#1(gen_:::nil2_0(n641_0)) → gen_:::nil2_0(0), rt ∈ Ω(1 + n641₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

No more defined symbols left to analyse.

(38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(39) BOUNDS(n^1, INF)

(40) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
appendAll(@l) → appendAll#1(@l)
appendAll#1(::(@l1, @ls)) → append(@l1, appendAll(@ls))
appendAll#1(nil) → nil
appendAll2(@l) → appendAll2#1(@l)
appendAll2#1(::(@l1, @ls)) → append(appendAll(@l1), appendAll2(@ls))
appendAll2#1(nil) → nil
appendAll3(@l) → appendAll3#1(@l)
appendAll3#1(::(@l1, @ls)) → append(appendAll2(@l1), appendAll3(@ls))
appendAll3#1(nil) → nil

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil → :::nil
nil :: :::nil
appendAll :: :::nil → :::nil
appendAll#1 :: :::nil → :::nil
appendAll2 :: :::nil → :::nil
appendAll2#1 :: :::nil → :::nil
appendAll3 :: :::nil → :::nil
appendAll3#1 :: :::nil → :::nil
hole_:::nil1_0 :: :::nil
gen_:::nil2_0 :: Nat → :::nil

Lemmas:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_:::nil2_0(0) ⇔ nil
gen_:::nil2_0(+(x, 1)) ⇔ ::(nil, gen_:::nil2_0(x))

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil2_0(n4_0), gen_:::nil2_0(b)) → gen_:::nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(42) BOUNDS(n^1, INF)