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
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@x, @t1, @t2)) → subtrees#2(subtrees(@t1), @t1, @t2, @x)
subtrees#2(@l1, @t1, @t2, @x) → subtrees#3(subtrees(@t2), @l1, @t1, @t2, @x)
subtrees#3(@l2, @l1, @t1, @t2, @x) → ::(node(@x, @t1, @t2), append(@l1, @l2))

Rewrite Strategy: INNERMOST

(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:

append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@x, @xs), @l2) → ::(@x, append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@x, @t1, @t2)) → subtrees#2(subtrees(@t1), @t1, @t2, @x)
subtrees#2(@l1, @t1, @t2, @x) → subtrees#3(subtrees(@t2), @l1, @t1, @t2, @x)
subtrees#3(@l2, @l1, @t1, @t2, @x) → ::(node(@x, @t1, @t2), append(@l1, @l2))

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
::/0
node/0
subtrees#2/1
subtrees#2/3
subtrees#3/2
subtrees#3/3
subtrees#3/4

(4) Obligation:

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

append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

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(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

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

Heuristically decided to analyse the following defined symbols:
append, append#1, subtrees, subtrees#1

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

(8) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

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

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

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

Proved the following rewrite lemma:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

Induction Base:
subtrees#1(gen_leaf:node4_0(0)) →_R^Ω(1)
nil

Induction Step:
subtrees#1(gen_leaf:node4_0(+(n6_0, 1))) →_R^Ω(1)
subtrees#2(subtrees(leaf), gen_leaf:node4_0(n6_0)) →_R^Ω(1)
subtrees#2(subtrees#1(leaf), gen_leaf:node4_0(n6_0)) →_R^Ω(1)
subtrees#2(nil, gen_leaf:node4_0(n6_0)) →_R^Ω(1)
subtrees#3(subtrees(gen_leaf:node4_0(n6_0)), nil) →_R^Ω(1)
subtrees#3(subtrees#1(gen_leaf:node4_0(n6_0)), nil) →_IH
subtrees#3(gen_:::nil3_0(c7_0), nil) →_R^Ω(1)
::(append(nil, gen_:::nil3_0(n6_0))) →_R^Ω(1)
::(append#1(nil, gen_:::nil3_0(n6_0))) →_R^Ω(1)
::(gen_:::nil3_0(n6_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(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Lemmas:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

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

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

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

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

(13) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Lemmas:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

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

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

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

Proved the following rewrite lemma:
append#1(gen_:::nil3_0(n383_0), gen_:::nil3_0(b)) → gen_:::nil3_0(+(n383_0, b)), rt ∈ Ω(1 + n383₀)

Induction Base:
append#1(gen_:::nil3_0(0), gen_:::nil3_0(b)) →_R^Ω(1)
gen_:::nil3_0(b)

Induction Step:
append#1(gen_:::nil3_0(+(n383_0, 1)), gen_:::nil3_0(b)) →_R^Ω(1)
::(append(gen_:::nil3_0(n383_0), gen_:::nil3_0(b))) →_R^Ω(1)
::(append#1(gen_:::nil3_0(n383_0), gen_:::nil3_0(b))) →_IH
::(gen_:::nil3_0(+(b, c384_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(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Lemmas:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)
append#1(gen_:::nil3_0(n383_0), gen_:::nil3_0(b)) → gen_:::nil3_0(+(n383_0, b)), rt ∈ Ω(1 + n383₀)

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

The following defined symbols remain to be analysed:
append

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

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

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

(18) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Lemmas:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)
append#1(gen_:::nil3_0(n383_0), gen_:::nil3_0(b)) → gen_:::nil3_0(+(n383_0, b)), rt ∈ Ω(1 + n383₀)

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Lemmas:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)
append#1(gen_:::nil3_0(n383_0), gen_:::nil3_0(b)) → gen_:::nil3_0(+(n383_0, b)), rt ∈ Ω(1 + n383₀)

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
append(@l1, @l2) → append#1(@l1, @l2)
append#1(::(@xs), @l2) → ::(append(@xs, @l2))
append#1(nil, @l2) → @l2
subtrees(@t) → subtrees#1(@t)
subtrees#1(leaf) → nil
subtrees#1(node(@t1, @t2)) → subtrees#2(subtrees(@t1), @t2)
subtrees#2(@l1, @t2) → subtrees#3(subtrees(@t2), @l1)
subtrees#3(@l2, @l1) → ::(append(@l1, @l2))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → :::nil
subtrees#3 :: :::nil → :::nil → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
gen_:::nil3_0 :: Nat → :::nil
gen_leaf:node4_0 :: Nat → leaf:node

Lemmas:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_:::nil3_0(0) ⇔ nil
gen_:::nil3_0(+(x, 1)) ⇔ ::(gen_:::nil3_0(x))
gen_leaf:node4_0(0) ⇔ leaf
gen_leaf:node4_0(+(x, 1)) ⇔ node(leaf, gen_leaf:node4_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
subtrees#1(gen_leaf:node4_0(n6_0)) → gen_:::nil3_0(n6_0), rt ∈ Ω(1 + n6₀)

(26) BOUNDS(n^1, INF)