Runtime Complexity TRS:
The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(X))
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__dbl'(0') → 0'
a__dbl'(s'(X)) → s'(s'(dbl'(X)))
a__dbls'(nil') → nil'
a__dbls'(cons'(X, Y)) → cons'(dbl'(X), dbls'(Y))
a__sel'(0', cons'(X, Y)) → mark'(X)
a__sel'(s'(X), cons'(Y, Z)) → a__sel'(mark'(X), mark'(Z))
a__indx'(nil', X) → nil'
a__indx'(cons'(X, Y), Z) → cons'(sel'(X, Z), indx'(Y, Z))
a__from'(X) → cons'(X, from'(s'(X)))
a__dbl1'(0') → 01'
a__dbl1'(s'(X)) → s1'(s1'(a__dbl1'(mark'(X))))
a__sel1'(0', cons'(X, Y)) → mark'(X)
a__sel1'(s'(X), cons'(Y, Z)) → a__sel1'(mark'(X), mark'(Z))
a__quote'(0') → 01'
a__quote'(s'(X)) → s1'(a__quote'(mark'(X)))
a__quote'(dbl'(X)) → a__dbl1'(mark'(X))
a__quote'(sel'(X, Y)) → a__sel1'(mark'(X), mark'(Y))
mark'(dbl'(X)) → a__dbl'(mark'(X))
mark'(dbls'(X)) → a__dbls'(mark'(X))
mark'(sel'(X1, X2)) → a__sel'(mark'(X1), mark'(X2))
mark'(indx'(X1, X2)) → a__indx'(mark'(X1), X2)
mark'(from'(X)) → a__from'(X)
mark'(dbl1'(X)) → a__dbl1'(mark'(X))
mark'(sel1'(X1, X2)) → a__sel1'(mark'(X1), mark'(X2))
mark'(quote'(X)) → a__quote'(mark'(X))
mark'(0') → 0'
mark'(s'(X)) → s'(X)
mark'(nil') → nil'
mark'(cons'(X1, X2)) → cons'(X1, X2)
mark'(01') → 01'
mark'(s1'(X)) → s1'(mark'(X))
a__dbl'(X) → dbl'(X)
a__dbls'(X) → dbls'(X)
a__sel'(X1, X2) → sel'(X1, X2)
a__indx'(X1, X2) → indx'(X1, X2)
a__from'(X) → from'(X)
a__dbl1'(X) → dbl1'(X)
a__sel1'(X1, X2) → sel1'(X1, X2)
a__quote'(X) → quote'(X)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__dbl'(0') → 0'
a__dbl'(s'(X)) → s'(s'(dbl'(X)))
a__dbls'(nil') → nil'
a__dbls'(cons'(X, Y)) → cons'(dbl'(X), dbls'(Y))
a__sel'(0', cons'(X, Y)) → mark'(X)
a__sel'(s'(X), cons'(Y, Z)) → a__sel'(mark'(X), mark'(Z))
a__indx'(nil', X) → nil'
a__indx'(cons'(X, Y), Z) → cons'(sel'(X, Z), indx'(Y, Z))
a__from'(X) → cons'(X, from'(s'(X)))
a__dbl1'(0') → 01'
a__dbl1'(s'(X)) → s1'(s1'(a__dbl1'(mark'(X))))
a__sel1'(0', cons'(X, Y)) → mark'(X)
a__sel1'(s'(X), cons'(Y, Z)) → a__sel1'(mark'(X), mark'(Z))
a__quote'(0') → 01'
a__quote'(s'(X)) → s1'(a__quote'(mark'(X)))
a__quote'(dbl'(X)) → a__dbl1'(mark'(X))
a__quote'(sel'(X, Y)) → a__sel1'(mark'(X), mark'(Y))
mark'(dbl'(X)) → a__dbl'(mark'(X))
mark'(dbls'(X)) → a__dbls'(mark'(X))
mark'(sel'(X1, X2)) → a__sel'(mark'(X1), mark'(X2))
mark'(indx'(X1, X2)) → a__indx'(mark'(X1), X2)
mark'(from'(X)) → a__from'(X)
mark'(dbl1'(X)) → a__dbl1'(mark'(X))
mark'(sel1'(X1, X2)) → a__sel1'(mark'(X1), mark'(X2))
mark'(quote'(X)) → a__quote'(mark'(X))
mark'(0') → 0'
mark'(s'(X)) → s'(X)
mark'(nil') → nil'
mark'(cons'(X1, X2)) → cons'(X1, X2)
mark'(01') → 01'
mark'(s1'(X)) → s1'(mark'(X))
a__dbl'(X) → dbl'(X)
a__dbls'(X) → dbls'(X)
a__sel'(X1, X2) → sel'(X1, X2)
a__indx'(X1, X2) → indx'(X1, X2)
a__from'(X) → from'(X)
a__dbl1'(X) → dbl1'(X)
a__sel1'(X1, X2) → sel1'(X1, X2)
a__quote'(X) → quote'(X)

Types:
a__dbl' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
0' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
s' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
dbl' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__dbls' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
nil' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
cons' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
dbls' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__sel' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
mark' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__indx' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
sel' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
indx' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__from' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
from' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__dbl1' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
01' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
s1' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__sel1' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
a__quote' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
dbl1' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
sel1' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
quote' :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote' → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
_hole_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'1 :: 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'
_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2 :: Nat → 0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'

Heuristically decided to analyse the following defined symbols:
a__sel', mark', a__dbl1', a__sel1', a__quote'

They will be analysed ascendingly in the following order:
a__sel' = mark'
a__sel' = a__dbl1'
a__sel' = a__sel1'
a__sel' = a__quote'
mark' = a__dbl1'
mark' = a__sel1'
mark' = a__quote'
a__dbl1' = a__sel1'
a__dbl1' = a__quote'
a__sel1' = a__quote'

Generator Equations:
_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(0) ⇔ 0'
_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(x, 1)) ⇔ s'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(x))

The following defined symbols remain to be analysed:
mark', a__sel', a__dbl1', a__sel1', a__quote'

Could not prove a rewrite lemma for the defined symbol mark'.

The following defined symbols remain to be analysed:
a__sel', a__dbl1', a__sel1', a__quote'

Could not prove a rewrite lemma for the defined symbol a__sel'.

The following defined symbols remain to be analysed:
a__dbl1', a__sel1', a__quote'

Proved the following rewrite lemma:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n158224))) → _*3, rt ∈ Ω(_n158224)

Induction Base:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, 0)))

Induction Step:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, +(_$n158225, 1)))) →_R^Ω(1)
s1'(s1'(a__dbl1'(mark'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _$n158225)))))) →_R^Ω(1)
s1'(s1'(a__dbl1'(s'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(_$n158225))))) →_IH
s1'(s1'(_*3))

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

Lemmas:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n158224))) → _*3, rt ∈ Ω(_n158224)

The following defined symbols remain to be analysed:
a__sel1', a__sel', mark', a__quote'

Could not prove a rewrite lemma for the defined symbol a__sel1'.

Lemmas:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n158224))) → _*3, rt ∈ Ω(_n158224)

The following defined symbols remain to be analysed:
a__quote', a__sel', mark'

Proved the following rewrite lemma:
a__quote'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n1188705))) → _*3, rt ∈ Ω(_n1188705)

Induction Base:
a__quote'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, 0)))

Induction Step:
a__quote'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, +(_$n1188706, 1)))) →_R^Ω(1)
s1'(a__quote'(mark'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _$n1188706))))) →_R^Ω(1)
s1'(a__quote'(s'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(_$n1188706)))) →_IH
s1'(_*3)

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

Lemmas:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n158224))) → _*3, rt ∈ Ω(_n158224)
a__quote'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n1188705))) → _*3, rt ∈ Ω(_n1188705)

The following defined symbols remain to be analysed:
mark', a__sel', a__dbl1', a__sel1'

Could not prove a rewrite lemma for the defined symbol mark'.

The following defined symbols remain to be analysed:
a__sel', a__dbl1', a__sel1'

Could not prove a rewrite lemma for the defined symbol a__sel'.

The following defined symbols remain to be analysed:
a__dbl1', a__sel1'

Proved the following rewrite lemma:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n2378618))) → _*3, rt ∈ Ω(_n2378618)

Induction Base:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, 0)))

Induction Step:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, +(_$n2378619, 1)))) →_R^Ω(1)
s1'(s1'(a__dbl1'(mark'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _$n2378619)))))) →_R^Ω(1)
s1'(s1'(a__dbl1'(s'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(_$n2378619))))) →_IH
s1'(s1'(_*3))

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

Lemmas:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n2378618))) → _*3, rt ∈ Ω(_n2378618)
a__quote'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n1188705))) → _*3, rt ∈ Ω(_n1188705)

The following defined symbols remain to be analysed:
a__sel1'

Could not prove a rewrite lemma for the defined symbol a__sel1'.

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
a__dbl1'(_gen_0':s':dbl':nil':cons':dbls':sel':indx':from':01':s1':dbl1':sel1':quote'2(+(1, _n2378618))) → _*3, rt ∈ Ω(_n2378618)