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)))
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(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
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)

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)))
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'(0') → 0'
mark'(s'(X)) → s'(X)
mark'(nil') → nil'
mark'(cons'(X1, X2)) → cons'(X1, X2)
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)

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)))
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'(0') → 0'
mark'(s'(X)) → s'(X)
mark'(nil') → nil'
mark'(cons'(X1, X2)) → cons'(X1, X2)
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)

Types:
a__dbl' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
0' :: 0':s':dbl':nil':cons':dbls':sel':indx':from'
s' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
dbl' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
a__dbls' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
nil' :: 0':s':dbl':nil':cons':dbls':sel':indx':from'
cons' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
dbls' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
a__sel' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
mark' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
a__indx' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
sel' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
indx' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
a__from' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
from' :: 0':s':dbl':nil':cons':dbls':sel':indx':from' → 0':s':dbl':nil':cons':dbls':sel':indx':from'
_hole_0':s':dbl':nil':cons':dbls':sel':indx':from'1 :: 0':s':dbl':nil':cons':dbls':sel':indx':from'
_gen_0':s':dbl':nil':cons':dbls':sel':indx':from'2 :: Nat → 0':s':dbl':nil':cons':dbls':sel':indx':from'

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

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

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

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

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

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

The following defined symbols remain to be analysed:
a__sel'

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

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

No more defined symbols left to analyse.