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

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

a__from'(X) → cons'(mark'(X), from'(s'(X)))
a__head'(cons'(X, XS)) → mark'(X)
a__2nd'(cons'(X, XS)) → a__head'(mark'(XS))
a__take'(0', XS) → nil'
a__take'(s'(N), cons'(X, XS)) → cons'(mark'(X), take'(N, XS))
a__sel'(0', cons'(X, XS)) → mark'(X)
a__sel'(s'(N), cons'(X, XS)) → a__sel'(mark'(N), mark'(XS))
mark'(from'(X)) → a__from'(mark'(X))
mark'(head'(X)) → a__head'(mark'(X))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(take'(X1, X2)) → a__take'(mark'(X1), mark'(X2))
mark'(sel'(X1, X2)) → a__sel'(mark'(X1), mark'(X2))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(nil') → nil'
a__from'(X) → from'(X)
a__head'(X) → head'(X)
a__2nd'(X) → 2nd'(X)
a__take'(X1, X2) → take'(X1, X2)
a__sel'(X1, X2) → sel'(X1, X2)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__from'(X) → cons'(mark'(X), from'(s'(X)))
a__head'(cons'(X, XS)) → mark'(X)
a__2nd'(cons'(X, XS)) → a__head'(mark'(XS))
a__take'(0', XS) → nil'
a__take'(s'(N), cons'(X, XS)) → cons'(mark'(X), take'(N, XS))
a__sel'(0', cons'(X, XS)) → mark'(X)
a__sel'(s'(N), cons'(X, XS)) → a__sel'(mark'(N), mark'(XS))
mark'(from'(X)) → a__from'(mark'(X))
mark'(head'(X)) → a__head'(mark'(X))
mark'(2nd'(X)) → a__2nd'(mark'(X))
mark'(take'(X1, X2)) → a__take'(mark'(X1), mark'(X2))
mark'(sel'(X1, X2)) → a__sel'(mark'(X1), mark'(X2))
mark'(cons'(X1, X2)) → cons'(mark'(X1), X2)
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
mark'(nil') → nil'
a__from'(X) → from'(X)
a__head'(X) → head'(X)
a__2nd'(X) → 2nd'(X)
a__take'(X1, X2) → take'(X1, X2)
a__sel'(X1, X2) → sel'(X1, X2)

Types:
a__from' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
cons' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
mark' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
from' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
s' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
a__head' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
a__2nd' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
a__take' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
0' :: s':from':cons':0':nil':take':head':2nd':sel'
nil' :: s':from':cons':0':nil':take':head':2nd':sel'
take' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
a__sel' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
head' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
2nd' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
sel' :: s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel' → s':from':cons':0':nil':take':head':2nd':sel'
_hole_s':from':cons':0':nil':take':head':2nd':sel'1 :: s':from':cons':0':nil':take':head':2nd':sel'
_gen_s':from':cons':0':nil':take':head':2nd':sel'2 :: Nat → s':from':cons':0':nil':take':head':2nd':sel'

Heuristically decided to analyse the following defined symbols:
a__from', mark', a__head', a__2nd', a__sel'

They will be analysed ascendingly in the following order:
a__from' = mark'
a__from' = a__head'
a__from' = a__2nd'
a__from' = a__sel'
mark' = a__head'
mark' = a__2nd'
mark' = a__sel'
a__head' = a__2nd'
a__head' = a__sel'
a__2nd' = a__sel'

Generator Equations:
_gen_s':from':cons':0':nil':take':head':2nd':sel'2(0) ⇔ 0'
_gen_s':from':cons':0':nil':take':head':2nd':sel'2(+(x, 1)) ⇔ cons'(0', _gen_s':from':cons':0':nil':take':head':2nd':sel'2(x))

The following defined symbols remain to be analysed:
mark', a__from', a__head', a__2nd', a__sel'

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

The following defined symbols remain to be analysed:
a__from', a__head', a__2nd', a__sel'

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

The following defined symbols remain to be analysed:
a__head', a__2nd', a__sel'

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

The following defined symbols remain to be analysed:
a__2nd', a__sel'

Could not prove a rewrite lemma for the defined symbol a__2nd'.

The following defined symbols remain to be analysed:
a__sel'

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

No more defined symbols left to analyse.