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:

eq(0, 0) → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0
length(cons(X, L)) → s(length(L))

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:

eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

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

Heuristically decided to analyse the following defined symbols:
eq, inf, take, length

(6) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
eq, inf, take, length

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

Proved the following rewrite lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →_IH
true

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:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
inf, take, length

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

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

(11) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
take, length

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

Proved the following rewrite lemma:
take(gen_0':s4_0(n288_0), gen_cons:nil5_0(n288_0)) → gen_cons:nil5_0(n288_0), rt ∈ Ω(1 + n288₀)

Induction Base:
take(gen_0':s4_0(0), gen_cons:nil5_0(0)) →_R^Ω(1)
nil

Induction Step:
take(gen_0':s4_0(+(n288_0, 1)), gen_cons:nil5_0(+(n288_0, 1))) →_R^Ω(1)
cons(0', take(gen_0':s4_0(n288_0), gen_cons:nil5_0(n288_0))) →_IH
cons(0', gen_cons:nil5_0(c289_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:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
take(gen_0':s4_0(n288_0), gen_cons:nil5_0(n288_0)) → gen_cons:nil5_0(n288_0), rt ∈ Ω(1 + n288₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
length

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
length(gen_cons:nil5_0(n632_0)) → gen_0':s4_0(n632_0), rt ∈ Ω(1 + n632₀)

Induction Base:
length(gen_cons:nil5_0(0)) →_R^Ω(1)
0'

Induction Step:
length(gen_cons:nil5_0(+(n632_0, 1))) →_R^Ω(1)
s(length(gen_cons:nil5_0(n632_0))) →_IH
s(gen_0':s4_0(c633_0))

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
take(gen_0':s4_0(n288_0), gen_cons:nil5_0(n288_0)) → gen_cons:nil5_0(n288_0), rt ∈ Ω(1 + n288₀)
length(gen_cons:nil5_0(n632_0)) → gen_0':s4_0(n632_0), rt ∈ Ω(1 + n632₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
take(gen_0':s4_0(n288_0), gen_cons:nil5_0(n288_0)) → gen_cons:nil5_0(n288_0), rt ∈ Ω(1 + n288₀)
length(gen_cons:nil5_0(n632_0)) → gen_0':s4_0(n632_0), rt ∈ Ω(1 + n632₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
take(gen_0':s4_0(n288_0), gen_cons:nil5_0(n288_0)) → gen_cons:nil5_0(n288_0), rt ∈ Ω(1 + n288₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf(X) → cons(X, inf(s(X)))
take(0', X) → nil
take(s(X), cons(Y, L)) → cons(Y, take(X, L))
length(nil) → 0'
length(cons(X, L)) → s(length(L))

Types:
eq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
inf :: 0':s → cons:nil
cons :: 0':s → cons:nil → cons:nil
take :: 0':s → cons:nil → cons:nil
nil :: cons:nil
length :: cons:nil → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(28) BOUNDS(n^1, INF)