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:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

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:

dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
dbl, dbls, sel, indx, from

They will be analysed ascendingly in the following order:
dbl < dbls
sel < indx

(6) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
dbl, dbls, sel, indx, from

They will be analysed ascendingly in the following order:
dbl < dbls
sel < indx

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

Proved the following rewrite lemma:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

Induction Base:
dbl(gen_0':s3_0(0)) →_R^Ω(1)
0'

Induction Step:
dbl(gen_0':s3_0(+(n6_0, 1))) →_R^Ω(1)
s(s(dbl(gen_0':s3_0(n6_0)))) →_IH
s(s(gen_0':s3_0(*(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:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
dbls, sel, indx, from

They will be analysed ascendingly in the following order:
sel < indx

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)

Induction Base:
dbls(gen_nil:cons4_0(0)) →_R^Ω(1)
nil

Induction Step:
dbls(gen_nil:cons4_0(+(n270_0, 1))) →_R^Ω(1)
cons(dbl(0'), dbls(gen_nil:cons4_0(n270_0))) →_L^Ω(1)
cons(gen_0':s3_0(*(2, 0)), dbls(gen_nil:cons4_0(n270_0))) →_IH
cons(gen_0':s3_0(0), gen_nil:cons4_0(c271_0))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
sel, indx, from

They will be analysed ascendingly in the following order:
sel < indx

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n599₀)

Induction Base:
sel(gen_0':s3_0(0), gen_nil:cons4_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
sel(gen_0':s3_0(+(n599_0, 1)), gen_nil:cons4_0(+(1, +(n599_0, 1)))) →_R^Ω(1)
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) →_IH
gen_0':s3_0(0)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n599₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
indx, from

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
indx(gen_nil:cons4_0(n946_0), gen_nil:cons4_0(1)) → gen_nil:cons4_0(n946_0), rt ∈ Ω(1 + n946₀)

Induction Base:
indx(gen_nil:cons4_0(0), gen_nil:cons4_0(1)) →_R^Ω(1)
nil

Induction Step:
indx(gen_nil:cons4_0(+(n946_0, 1)), gen_nil:cons4_0(1)) →_R^Ω(1)
cons(sel(0', gen_nil:cons4_0(1)), indx(gen_nil:cons4_0(n946_0), gen_nil:cons4_0(1))) →_L^Ω(1)
cons(gen_0':s3_0(0), indx(gen_nil:cons4_0(n946_0), gen_nil:cons4_0(1))) →_IH
cons(gen_0':s3_0(0), gen_nil:cons4_0(c947_0))

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

(17) Complex Obligation (BEST)

(18) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n599₀)
indx(gen_nil:cons4_0(n946_0), gen_nil:cons4_0(1)) → gen_nil:cons4_0(n946_0), rt ∈ Ω(1 + n946₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
from

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n599₀)
indx(gen_nil:cons4_0(n946_0), gen_nil:cons4_0(1)) → gen_nil:cons4_0(n946_0), rt ∈ Ω(1 + n946₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n599₀)
indx(gen_nil:cons4_0(n946_0), gen_nil:cons4_0(1)) → gen_nil:cons4_0(n946_0), rt ∈ Ω(1 + n946₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)
sel(gen_0':s3_0(n599_0), gen_nil:cons4_0(+(1, n599_0))) → gen_0':s3_0(0), rt ∈ Ω(1 + n599₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
dbls(gen_nil:cons4_0(n270_0)) → gen_nil:cons4_0(n270_0), rt ∈ Ω(1 + n270₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

Innermost TRS:
Rules:
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0', cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Types:
dbl :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
dbls :: nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sel :: 0':s → nil:cons → 0':s
indx :: nil:cons → nil:cons → nil:cons
from :: 0':s → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
dbl(gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

(34) BOUNDS(n^1, INF)