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

Infered types.

(4) 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
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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

(5) 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

(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
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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

Generator Equations:
gen_:::nil4_0(0) ⇔ nil
gen_:::nil4_0(+(x, 1)) ⇔ ::(leaf, gen_:::nil4_0(x))
gen_leaf:node5_0(0) ⇔ leaf
gen_leaf:node5_0(+(x, 1)) ⇔ node(hole_a3_0, leaf, gen_leaf:node5_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

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol subtrees#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
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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

Generator Equations:
gen_:::nil4_0(0) ⇔ nil
gen_:::nil4_0(+(x, 1)) ⇔ ::(leaf, gen_:::nil4_0(x))
gen_leaf:node5_0(0) ⇔ leaf
gen_leaf:node5_0(+(x, 1)) ⇔ node(hole_a3_0, leaf, gen_leaf:node5_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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) 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
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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

Generator Equations:
gen_:::nil4_0(0) ⇔ nil
gen_:::nil4_0(+(x, 1)) ⇔ ::(leaf, gen_:::nil4_0(x))
gen_leaf:node5_0(0) ⇔ leaf
gen_leaf:node5_0(+(x, 1)) ⇔ node(hole_a3_0, leaf, gen_leaf:node5_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

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

Proved the following rewrite lemma:
append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b)) → gen_:::nil4_0(+(n6995_0, b)), rt ∈ Ω(1 + n6995₀)

Induction Base:
append#1(gen_:::nil4_0(0), gen_:::nil4_0(b)) →_R^Ω(1)
gen_:::nil4_0(b)

Induction Step:
append#1(gen_:::nil4_0(+(n6995_0, 1)), gen_:::nil4_0(b)) →_R^Ω(1)
::(leaf, append(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b))) →_R^Ω(1)
::(leaf, append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b))) →_IH
::(leaf, gen_:::nil4_0(+(b, c6996_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(@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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

Lemmas:
append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b)) → gen_:::nil4_0(+(n6995_0, b)), rt ∈ Ω(1 + n6995₀)

Generator Equations:
gen_:::nil4_0(0) ⇔ nil
gen_:::nil4_0(+(x, 1)) ⇔ ::(leaf, gen_:::nil4_0(x))
gen_leaf:node5_0(0) ⇔ leaf
gen_leaf:node5_0(+(x, 1)) ⇔ node(hole_a3_0, leaf, gen_leaf:node5_0(x))

The following defined symbols remain to be analysed:
append

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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) 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
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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

Lemmas:
append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b)) → gen_:::nil4_0(+(n6995_0, b)), rt ∈ Ω(1 + n6995₀)

Generator Equations:
gen_:::nil4_0(0) ⇔ nil
gen_:::nil4_0(+(x, 1)) ⇔ ::(leaf, gen_:::nil4_0(x))
gen_leaf:node5_0(0) ⇔ leaf
gen_leaf:node5_0(+(x, 1)) ⇔ node(hole_a3_0, leaf, gen_leaf:node5_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b)) → gen_:::nil4_0(+(n6995_0, b)), rt ∈ Ω(1 + n6995₀)

(17) BOUNDS(n^1, INF)

(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
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))

Types:
append :: :::nil → :::nil → :::nil
append#1 :: :::nil → :::nil → :::nil
:: :: leaf:node → :::nil → :::nil
nil :: :::nil
subtrees :: leaf:node → :::nil
subtrees#1 :: leaf:node → :::nil
leaf :: leaf:node
node :: a → leaf:node → leaf:node → leaf:node
subtrees#2 :: :::nil → leaf:node → leaf:node → a → :::nil
subtrees#3 :: :::nil → :::nil → leaf:node → leaf:node → a → :::nil
hole_:::nil1_0 :: :::nil
hole_leaf:node2_0 :: leaf:node
hole_a3_0 :: a
gen_:::nil4_0 :: Nat → :::nil
gen_leaf:node5_0 :: Nat → leaf:node

Lemmas:
append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b)) → gen_:::nil4_0(+(n6995_0, b)), rt ∈ Ω(1 + n6995₀)

Generator Equations:
gen_:::nil4_0(0) ⇔ nil
gen_:::nil4_0(+(x, 1)) ⇔ ::(leaf, gen_:::nil4_0(x))
gen_leaf:node5_0(0) ⇔ leaf
gen_leaf:node5_0(+(x, 1)) ⇔ node(hole_a3_0, leaf, gen_leaf:node5_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
append#1(gen_:::nil4_0(n6995_0), gen_:::nil4_0(b)) → gen_:::nil4_0(+(n6995_0, b)), rt ∈ Ω(1 + n6995₀)

(20) BOUNDS(n^1, INF)