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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
inf/0
cons/0

(4) 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 → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(L)) → s(length(L))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

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

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

(8) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(gen_cons:nil5_0(x))

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

(9) 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).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(gen_cons:nil5_0(x))

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(gen_cons:nil5_0(x))

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
take(gen_0':s4_0(n240_0), gen_cons:nil5_0(n240_0)) → gen_cons:nil5_0(n240_0), rt ∈ Ω(1 + n240₀)

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

Induction Step:
take(gen_0':s4_0(+(n240_0, 1)), gen_cons:nil5_0(+(n240_0, 1))) →_R^Ω(1)
cons(take(gen_0':s4_0(n240_0), gen_cons:nil5_0(n240_0))) →_IH
cons(gen_cons:nil5_0(c241_0))

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(n240_0), gen_cons:nil5_0(n240_0)) → gen_cons:nil5_0(n240_0), rt ∈ Ω(1 + n240₀)

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(gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
length

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
length(gen_cons:nil5_0(n566_0)) → gen_0':s4_0(n566_0), rt ∈ Ω(1 + n566₀)

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

Induction Step:
length(gen_cons:nil5_0(+(n566_0, 1))) →_R^Ω(1)
s(length(gen_cons:nil5_0(n566_0))) →_IH
s(gen_0':s4_0(c567_0))

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

(18) Complex Obligation (BEST)

(19) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(n240_0), gen_cons:nil5_0(n240_0)) → gen_cons:nil5_0(n240_0), rt ∈ Ω(1 + n240₀)
length(gen_cons:nil5_0(n566_0)) → gen_0':s4_0(n566_0), rt ∈ Ω(1 + n566₀)

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(gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(20) 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)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(n240_0), gen_cons:nil5_0(n240_0)) → gen_cons:nil5_0(n240_0), rt ∈ Ω(1 + n240₀)
length(gen_cons:nil5_0(n566_0)) → gen_0':s4_0(n566_0), rt ∈ Ω(1 + n566₀)

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(gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(23) 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)

(24) BOUNDS(n^1, INF)

(25) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(n240_0), gen_cons:nil5_0(n240_0)) → gen_cons:nil5_0(n240_0), rt ∈ Ω(1 + n240₀)

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(gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(26) 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)

(27) BOUNDS(n^1, INF)

(28) Obligation:

Innermost TRS:
Rules:
eq(0', 0') → true
eq(s(X), s(Y)) → eq(X, Y)
eq(X, Y) → false
inf → cons(inf)
take(0', X) → nil
take(s(X), cons(L)) → cons(take(X, L))
length(nil) → 0'
length(cons(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 :: cons:nil
cons :: 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(gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(29) 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)

(30) BOUNDS(n^1, INF)