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:

walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
walk#1(Node(x5, x3)) →⁺ comp_f_g(walk#1(x5), walk#1(x3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x5 / Node(x5, x3)].
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:

walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Types:
walk#1 :: Leaf:Node → cons_x:comp_f_g
Leaf :: a → Leaf:Node
cons_x :: a → cons_x:comp_f_g
Node :: Leaf:Node → Leaf:Node → Leaf:Node
comp_f_g :: cons_x:comp_f_g → cons_x:comp_f_g → cons_x:comp_f_g
comp_f_g#1 :: cons_x:comp_f_g → cons_x:comp_f_g → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
main :: Leaf:Node → Cons:Nil
Nil :: Cons:Nil
hole_cons_x:comp_f_g1_0 :: cons_x:comp_f_g
hole_Leaf:Node2_0 :: Leaf:Node
hole_a3_0 :: a
hole_Cons:Nil4_0 :: Cons:Nil
gen_cons_x:comp_f_g5_0 :: Nat → cons_x:comp_f_g
gen_Leaf:Node6_0 :: Nat → Leaf:Node
gen_Cons:Nil7_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
walk#1, comp_f_g#1

(8) Obligation:

Innermost TRS:
Rules:
walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Types:
walk#1 :: Leaf:Node → cons_x:comp_f_g
Leaf :: a → Leaf:Node
cons_x :: a → cons_x:comp_f_g
Node :: Leaf:Node → Leaf:Node → Leaf:Node
comp_f_g :: cons_x:comp_f_g → cons_x:comp_f_g → cons_x:comp_f_g
comp_f_g#1 :: cons_x:comp_f_g → cons_x:comp_f_g → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
main :: Leaf:Node → Cons:Nil
Nil :: Cons:Nil
hole_cons_x:comp_f_g1_0 :: cons_x:comp_f_g
hole_Leaf:Node2_0 :: Leaf:Node
hole_a3_0 :: a
hole_Cons:Nil4_0 :: Cons:Nil
gen_cons_x:comp_f_g5_0 :: Nat → cons_x:comp_f_g
gen_Leaf:Node6_0 :: Nat → Leaf:Node
gen_Cons:Nil7_0 :: Nat → Cons:Nil

Generator Equations:
gen_cons_x:comp_f_g5_0(0) ⇔ cons_x(hole_a3_0)
gen_cons_x:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(x))
gen_Leaf:Node6_0(0) ⇔ Leaf(hole_a3_0)
gen_Leaf:Node6_0(+(x, 1)) ⇔ Node(Leaf(hole_a3_0), gen_Leaf:Node6_0(x))
gen_Cons:Nil7_0(0) ⇔ Nil
gen_Cons:Nil7_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil7_0(x))

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

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

Proved the following rewrite lemma:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)

Induction Base:
walk#1(gen_Leaf:Node6_0(0)) →_R^Ω(1)
cons_x(hole_a3_0)

Induction Step:
walk#1(gen_Leaf:Node6_0(+(n9_0, 1))) →_R^Ω(1)
comp_f_g(walk#1(Leaf(hole_a3_0)), walk#1(gen_Leaf:Node6_0(n9_0))) →_R^Ω(1)
comp_f_g(cons_x(hole_a3_0), walk#1(gen_Leaf:Node6_0(n9_0))) →_IH
comp_f_g(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(c10_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:
walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Types:
walk#1 :: Leaf:Node → cons_x:comp_f_g
Leaf :: a → Leaf:Node
cons_x :: a → cons_x:comp_f_g
Node :: Leaf:Node → Leaf:Node → Leaf:Node
comp_f_g :: cons_x:comp_f_g → cons_x:comp_f_g → cons_x:comp_f_g
comp_f_g#1 :: cons_x:comp_f_g → cons_x:comp_f_g → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
main :: Leaf:Node → Cons:Nil
Nil :: Cons:Nil
hole_cons_x:comp_f_g1_0 :: cons_x:comp_f_g
hole_Leaf:Node2_0 :: Leaf:Node
hole_a3_0 :: a
hole_Cons:Nil4_0 :: Cons:Nil
gen_cons_x:comp_f_g5_0 :: Nat → cons_x:comp_f_g
gen_Leaf:Node6_0 :: Nat → Leaf:Node
gen_Cons:Nil7_0 :: Nat → Cons:Nil

Lemmas:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons_x:comp_f_g5_0(0) ⇔ cons_x(hole_a3_0)
gen_cons_x:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(x))
gen_Leaf:Node6_0(0) ⇔ Leaf(hole_a3_0)
gen_Leaf:Node6_0(+(x, 1)) ⇔ Node(Leaf(hole_a3_0), gen_Leaf:Node6_0(x))
gen_Cons:Nil7_0(0) ⇔ Nil
gen_Cons:Nil7_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil7_0(x))

The following defined symbols remain to be analysed:
comp_f_g#1

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
comp_f_g#1(gen_cons_x:comp_f_g5_0(0), gen_cons_x:comp_f_g5_0(n165_0), gen_Cons:Nil7_0(c)) → gen_Cons:Nil7_0(+(+(2, n165_0), c)), rt ∈ Ω(1 + n165₀)

Induction Base:
comp_f_g#1(gen_cons_x:comp_f_g5_0(0), gen_cons_x:comp_f_g5_0(0), gen_Cons:Nil7_0(c)) →_R^Ω(1)
Cons(hole_a3_0, Cons(hole_a3_0, gen_Cons:Nil7_0(c)))

Induction Step:
comp_f_g#1(gen_cons_x:comp_f_g5_0(0), gen_cons_x:comp_f_g5_0(+(n165_0, 1)), gen_Cons:Nil7_0(c)) →_R^Ω(1)
Cons(hole_a3_0, comp_f_g#1(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(n165_0), gen_Cons:Nil7_0(c))) →_IH
Cons(hole_a3_0, gen_Cons:Nil7_0(+(+(2, c), c166_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Types:
walk#1 :: Leaf:Node → cons_x:comp_f_g
Leaf :: a → Leaf:Node
cons_x :: a → cons_x:comp_f_g
Node :: Leaf:Node → Leaf:Node → Leaf:Node
comp_f_g :: cons_x:comp_f_g → cons_x:comp_f_g → cons_x:comp_f_g
comp_f_g#1 :: cons_x:comp_f_g → cons_x:comp_f_g → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
main :: Leaf:Node → Cons:Nil
Nil :: Cons:Nil
hole_cons_x:comp_f_g1_0 :: cons_x:comp_f_g
hole_Leaf:Node2_0 :: Leaf:Node
hole_a3_0 :: a
hole_Cons:Nil4_0 :: Cons:Nil
gen_cons_x:comp_f_g5_0 :: Nat → cons_x:comp_f_g
gen_Leaf:Node6_0 :: Nat → Leaf:Node
gen_Cons:Nil7_0 :: Nat → Cons:Nil

Lemmas:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)
comp_f_g#1(gen_cons_x:comp_f_g5_0(0), gen_cons_x:comp_f_g5_0(n165_0), gen_Cons:Nil7_0(c)) → gen_Cons:Nil7_0(+(+(2, n165_0), c)), rt ∈ Ω(1 + n165₀)

Generator Equations:
gen_cons_x:comp_f_g5_0(0) ⇔ cons_x(hole_a3_0)
gen_cons_x:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(x))
gen_Leaf:Node6_0(0) ⇔ Leaf(hole_a3_0)
gen_Leaf:Node6_0(+(x, 1)) ⇔ Node(Leaf(hole_a3_0), gen_Leaf:Node6_0(x))
gen_Cons:Nil7_0(0) ⇔ Nil
gen_Cons:Nil7_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil7_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Types:
walk#1 :: Leaf:Node → cons_x:comp_f_g
Leaf :: a → Leaf:Node
cons_x :: a → cons_x:comp_f_g
Node :: Leaf:Node → Leaf:Node → Leaf:Node
comp_f_g :: cons_x:comp_f_g → cons_x:comp_f_g → cons_x:comp_f_g
comp_f_g#1 :: cons_x:comp_f_g → cons_x:comp_f_g → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
main :: Leaf:Node → Cons:Nil
Nil :: Cons:Nil
hole_cons_x:comp_f_g1_0 :: cons_x:comp_f_g
hole_Leaf:Node2_0 :: Leaf:Node
hole_a3_0 :: a
hole_Cons:Nil4_0 :: Cons:Nil
gen_cons_x:comp_f_g5_0 :: Nat → cons_x:comp_f_g
gen_Leaf:Node6_0 :: Nat → Leaf:Node
gen_Cons:Nil7_0 :: Nat → Cons:Nil

Lemmas:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)
comp_f_g#1(gen_cons_x:comp_f_g5_0(0), gen_cons_x:comp_f_g5_0(n165_0), gen_Cons:Nil7_0(c)) → gen_Cons:Nil7_0(+(+(2, n165_0), c)), rt ∈ Ω(1 + n165₀)

Generator Equations:
gen_cons_x:comp_f_g5_0(0) ⇔ cons_x(hole_a3_0)
gen_cons_x:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(x))
gen_Leaf:Node6_0(0) ⇔ Leaf(hole_a3_0)
gen_Leaf:Node6_0(+(x, 1)) ⇔ Node(Leaf(hole_a3_0), gen_Leaf:Node6_0(x))
gen_Cons:Nil7_0(0) ⇔ Nil
gen_Cons:Nil7_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil7_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
walk#1(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), Nil)

Types:
walk#1 :: Leaf:Node → cons_x:comp_f_g
Leaf :: a → Leaf:Node
cons_x :: a → cons_x:comp_f_g
Node :: Leaf:Node → Leaf:Node → Leaf:Node
comp_f_g :: cons_x:comp_f_g → cons_x:comp_f_g → cons_x:comp_f_g
comp_f_g#1 :: cons_x:comp_f_g → cons_x:comp_f_g → Cons:Nil → Cons:Nil
Cons :: a → Cons:Nil → Cons:Nil
main :: Leaf:Node → Cons:Nil
Nil :: Cons:Nil
hole_cons_x:comp_f_g1_0 :: cons_x:comp_f_g
hole_Leaf:Node2_0 :: Leaf:Node
hole_a3_0 :: a
hole_Cons:Nil4_0 :: Cons:Nil
gen_cons_x:comp_f_g5_0 :: Nat → cons_x:comp_f_g
gen_Leaf:Node6_0 :: Nat → Leaf:Node
gen_Cons:Nil7_0 :: Nat → Cons:Nil

Lemmas:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)

Generator Equations:
gen_cons_x:comp_f_g5_0(0) ⇔ cons_x(hole_a3_0)
gen_cons_x:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(cons_x(hole_a3_0), gen_cons_x:comp_f_g5_0(x))
gen_Leaf:Node6_0(0) ⇔ Leaf(hole_a3_0)
gen_Leaf:Node6_0(+(x, 1)) ⇔ Node(Leaf(hole_a3_0), gen_Leaf:Node6_0(x))
gen_Cons:Nil7_0(0) ⇔ Nil
gen_Cons:Nil7_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Cons:Nil7_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Leaf:Node6_0(n9_0)) → gen_cons_x:comp_f_g5_0(n9_0), rt ∈ Ω(1 + n9₀)

(22) BOUNDS(n^1, INF)