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:

lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

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:

lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

Types:
lookup :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil → Cons:Nil → run:intlookup
intlookup :: Cons:Nil → Cons:Nil → run:intlookup
hole_Cons:Nil1_0 :: Cons:Nil
hole_run:intlookup2_0 :: run:intlookup
gen_Cons:Nil3_0 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
lookup, intlookup

They will be analysed ascendingly in the following order:
lookup < intlookup

(6) Obligation:

Innermost TRS:
Rules:
lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

Types:
lookup :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil → Cons:Nil → run:intlookup
intlookup :: Cons:Nil → Cons:Nil → run:intlookup
hole_Cons:Nil1_0 :: Cons:Nil
hole_run:intlookup2_0 :: run:intlookup
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
lookup, intlookup

They will be analysed ascendingly in the following order:
lookup < intlookup

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

Proved the following rewrite lemma:
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) → gen_Cons:Nil3_0(0), rt ∈ Ω(1 + n5₀)

Induction Base:
lookup(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(+(1, 0))) →_R^Ω(1)
Nil

Induction Step:
lookup(gen_Cons:Nil3_0(+(n5_0, 1)), gen_Cons:Nil3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) →_IH
gen_Cons:Nil3_0(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:
lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

Types:
lookup :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil → Cons:Nil → run:intlookup
intlookup :: Cons:Nil → Cons:Nil → run:intlookup
hole_Cons:Nil1_0 :: Cons:Nil
hole_run:intlookup2_0 :: run:intlookup
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) → gen_Cons:Nil3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil3_0(x))

The following defined symbols remain to be analysed:
intlookup

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

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

(11) Obligation:

Innermost TRS:
Rules:
lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

Types:
lookup :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil → Cons:Nil → run:intlookup
intlookup :: Cons:Nil → Cons:Nil → run:intlookup
hole_Cons:Nil1_0 :: Cons:Nil
hole_run:intlookup2_0 :: run:intlookup
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) → gen_Cons:Nil3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) → gen_Cons:Nil3_0(0), rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
lookup(Cons(x', xs'), Cons(x, xs)) → lookup(xs', xs)
lookup(Nil, Cons(x, xs)) → x
run(e, p) → intlookup(e, p)
intlookup(e, p) → intlookup(lookup(e, p), p)

Types:
lookup :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
run :: Cons:Nil → Cons:Nil → run:intlookup
intlookup :: Cons:Nil → Cons:Nil → run:intlookup
hole_Cons:Nil1_0 :: Cons:Nil
hole_run:intlookup2_0 :: run:intlookup
gen_Cons:Nil3_0 :: Nat → Cons:Nil

Lemmas:
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) → gen_Cons:Nil3_0(0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Cons:Nil3_0(0) ⇔ Nil
gen_Cons:Nil3_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lookup(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(+(1, n5_0))) → gen_Cons:Nil3_0(0), rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)