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:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

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:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Types:
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
lessleaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_leaf:cons1_0 :: leaf:cons
hole_false:true2_0 :: false:true
gen_leaf:cons3_0 :: Nat → leaf:cons

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
concat, lessleaves

They will be analysed ascendingly in the following order:
concat < lessleaves

(6) Obligation:

Innermost TRS:
Rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Types:
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
lessleaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_leaf:cons1_0 :: leaf:cons
hole_false:true2_0 :: false:true
gen_leaf:cons3_0 :: Nat → leaf:cons

Generator Equations:
gen_leaf:cons3_0(0) ⇔ leaf
gen_leaf:cons3_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons3_0(x))

The following defined symbols remain to be analysed:
concat, lessleaves

They will be analysed ascendingly in the following order:
concat < lessleaves

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
concat(gen_leaf:cons3_0(0), gen_leaf:cons3_0(b)) →_R^Ω(1)
gen_leaf:cons3_0(b)

Induction Step:
concat(gen_leaf:cons3_0(+(n5_0, 1)), gen_leaf:cons3_0(b)) →_R^Ω(1)
cons(leaf, concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b))) →_IH
cons(leaf, gen_leaf:cons3_0(+(b, c6_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Types:
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
lessleaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_leaf:cons1_0 :: leaf:cons
hole_false:true2_0 :: false:true
gen_leaf:cons3_0 :: Nat → leaf:cons

Lemmas:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_leaf:cons3_0(0) ⇔ leaf
gen_leaf:cons3_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons3_0(x))

The following defined symbols remain to be analysed:
lessleaves

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
lessleaves(gen_leaf:cons3_0(n460_0), gen_leaf:cons3_0(n460_0)) → false, rt ∈ Ω(1 + n460₀)

Induction Base:
lessleaves(gen_leaf:cons3_0(0), gen_leaf:cons3_0(0)) →_R^Ω(1)
false

Induction Step:
lessleaves(gen_leaf:cons3_0(+(n460_0, 1)), gen_leaf:cons3_0(+(n460_0, 1))) →_R^Ω(1)
lessleaves(concat(leaf, gen_leaf:cons3_0(n460_0)), concat(leaf, gen_leaf:cons3_0(n460_0))) →_L^Ω(1)
lessleaves(gen_leaf:cons3_0(+(0, n460_0)), concat(leaf, gen_leaf:cons3_0(n460_0))) →_L^Ω(1)
lessleaves(gen_leaf:cons3_0(n460_0), gen_leaf:cons3_0(+(0, n460_0))) →_IH
false

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Types:
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
lessleaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_leaf:cons1_0 :: leaf:cons
hole_false:true2_0 :: false:true
gen_leaf:cons3_0 :: Nat → leaf:cons

Lemmas:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
lessleaves(gen_leaf:cons3_0(n460_0), gen_leaf:cons3_0(n460_0)) → false, rt ∈ Ω(1 + n460₀)

Generator Equations:
gen_leaf:cons3_0(0) ⇔ leaf
gen_leaf:cons3_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
Rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Types:
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
lessleaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_leaf:cons1_0 :: leaf:cons
hole_false:true2_0 :: false:true
gen_leaf:cons3_0 :: Nat → leaf:cons

Lemmas:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
lessleaves(gen_leaf:cons3_0(n460_0), gen_leaf:cons3_0(n460_0)) → false, rt ∈ Ω(1 + n460₀)

Generator Equations:
gen_leaf:cons3_0(0) ⇔ leaf
gen_leaf:cons3_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons3_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Types:
concat :: leaf:cons → leaf:cons → leaf:cons
leaf :: leaf:cons
cons :: leaf:cons → leaf:cons → leaf:cons
lessleaves :: leaf:cons → leaf:cons → false:true
false :: false:true
true :: false:true
hole_leaf:cons1_0 :: leaf:cons
hole_false:true2_0 :: false:true
gen_leaf:cons3_0 :: Nat → leaf:cons

Lemmas:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_leaf:cons3_0(0) ⇔ leaf
gen_leaf:cons3_0(+(x, 1)) ⇔ cons(leaf, gen_leaf:cons3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) → gen_leaf:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(20) BOUNDS(n^1, INF)