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:

f(X) → cons(X, f(g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, 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:

f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
f, g, sel

They will be analysed ascendingly in the following order:
g < f

(6) Obligation:

Innermost TRS:
Rules:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
g, f, sel

They will be analysed ascendingly in the following order:
g < f

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

Proved the following rewrite lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Induction Base:
g(gen_0':s4_0(0)) →_R^Ω(1)
s(0')

Induction Step:
g(gen_0':s4_0(+(n6_0, 1))) →_R^Ω(1)
s(s(g(gen_0':s4_0(n6_0)))) →_IH
s(s(gen_0':s4_0(+(1, *(2, c7_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:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
f, sel

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
sel

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

Proved the following rewrite lemma:
sel(gen_0':s4_0(n474_0), gen_cons3_0(+(1, n474_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n474₀)

Induction Base:
sel(gen_0':s4_0(0), gen_cons3_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
sel(gen_0':s4_0(+(n474_0, 1)), gen_cons3_0(+(1, +(n474_0, 1)))) →_R^Ω(1)
sel(gen_0':s4_0(n474_0), gen_cons3_0(+(1, n474_0))) →_IH
gen_0':s4_0(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:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)
sel(gen_0':s4_0(n474_0), gen_cons3_0(+(1, n474_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n474₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)
sel(gen_0':s4_0(n474_0), gen_cons3_0(+(1, n474_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n474₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
f(X) → cons(X, f(g(X)))
g(0') → s(0')
g(s(X)) → s(s(g(X)))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)

Types:
f :: 0':s → cons
cons :: 0':s → cons → cons
g :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
sel :: 0':s → cons → 0':s
hole_cons1_0 :: cons
hole_0':s2_0 :: 0':s
gen_cons3_0 :: Nat → cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_cons3_0(0) ⇔ hole_cons1_0
gen_cons3_0(+(x, 1)) ⇔ cons(0', gen_cons3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_0':s4_0(n6_0)) → gen_0':s4_0(+(1, *(2, n6_0))), rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)