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

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__minus(X, 0) → 0
a__minus(s(X), s(Y)) → a__minus(mark(X), mark(Y))
a__quot(0, s(Y)) → 0
a__quot(s(X), s(Y)) → s(a__quot(a__minus(mark(X), mark(Y)), s(mark(Y))))
a__zWquot(XS, nil) → nil
a__zWquot(nil, XS) → nil
a__zWquot(cons(X, XS), cons(Y, YS)) → cons(a__quot(mark(X), mark(Y)), zWquot(XS, YS))
mark(from(X)) → a__from(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(minus(X1, X2)) → a__minus(mark(X1), mark(X2))
mark(quot(X1, X2)) → a__quot(mark(X1), mark(X2))
mark(zWquot(X1, X2)) → a__zWquot(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__sel(X1, X2) → sel(X1, X2)
a__minus(X1, X2) → minus(X1, X2)
a__quot(X1, X2) → quot(X1, X2)
a__zWquot(X1, X2) → zWquot(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__sel'(0', cons'(X, XS)) → mark'(X)
a__sel'(s'(N), cons'(X, XS)) → a__sel'(mark'(N), mark'(XS))
a__minus'(X, 0') → 0'
a__minus'(s'(X), s'(Y)) → a__minus'(mark'(X), mark'(Y))
a__quot'(0', s'(Y)) → 0'
a__quot'(s'(X), s'(Y)) → s'(a__quot'(a__minus'(mark'(X), mark'(Y)), s'(mark'(Y))))
a__zWquot'(XS, nil') → nil'
a__zWquot'(nil', XS) → nil'
a__zWquot'(cons'(X, XS), cons'(Y, YS)) → cons'(a__quot'(mark'(X), mark'(Y)), zWquot'(XS, YS))
mark'(from'(X)) → a__from'(mark'(X))
mark'(sel'(X1, X2)) → a__sel'(mark'(X1), mark'(X2))
mark'(minus'(X1, X2)) → a__minus'(mark'(X1), mark'(X2))
mark'(quot'(X1, X2)) → a__quot'(mark'(X1), mark'(X2))
mark'(zWquot'(X1, X2)) → a__zWquot'(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__sel'(X1, X2) → sel'(X1, X2)
a__minus'(X1, X2) → minus'(X1, X2)
a__quot'(X1, X2) → quot'(X1, X2)
a__zWquot'(X1, X2) → zWquot'(X1, X2)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__from'(X) → cons'(mark'(X), from'(s'(X)))
a__sel'(0', cons'(X, XS)) → mark'(X)
a__sel'(s'(N), cons'(X, XS)) → a__sel'(mark'(N), mark'(XS))
a__minus'(X, 0') → 0'
a__minus'(s'(X), s'(Y)) → a__minus'(mark'(X), mark'(Y))
a__quot'(0', s'(Y)) → 0'
a__quot'(s'(X), s'(Y)) → s'(a__quot'(a__minus'(mark'(X), mark'(Y)), s'(mark'(Y))))
a__zWquot'(XS, nil') → nil'
a__zWquot'(nil', XS) → nil'
a__zWquot'(cons'(X, XS), cons'(Y, YS)) → cons'(a__quot'(mark'(X), mark'(Y)), zWquot'(XS, YS))
mark'(from'(X)) → a__from'(mark'(X))
mark'(sel'(X1, X2)) → a__sel'(mark'(X1), mark'(X2))
mark'(minus'(X1, X2)) → a__minus'(mark'(X1), mark'(X2))
mark'(quot'(X1, X2)) → a__quot'(mark'(X1), mark'(X2))
mark'(zWquot'(X1, X2)) → a__zWquot'(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__sel'(X1, X2) → sel'(X1, X2)
a__minus'(X1, X2) → minus'(X1, X2)
a__quot'(X1, X2) → quot'(X1, X2)
a__zWquot'(X1, X2) → zWquot'(X1, X2)

Types:
a__from' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
cons' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
mark' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
from' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
s' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
a__sel' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
0' :: s':from':cons':0':nil':zWquot':sel':minus':quot'
a__minus' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
a__quot' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
a__zWquot' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
nil' :: s':from':cons':0':nil':zWquot':sel':minus':quot'
zWquot' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
sel' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
minus' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
quot' :: s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot' → s':from':cons':0':nil':zWquot':sel':minus':quot'
_hole_s':from':cons':0':nil':zWquot':sel':minus':quot'1 :: s':from':cons':0':nil':zWquot':sel':minus':quot'
_gen_s':from':cons':0':nil':zWquot':sel':minus':quot'2 :: Nat → s':from':cons':0':nil':zWquot':sel':minus':quot'

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

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

Generator Equations:
_gen_s':from':cons':0':nil':zWquot':sel':minus':quot'2(0) ⇔ 0'
_gen_s':from':cons':0':nil':zWquot':sel':minus':quot'2(+(x, 1)) ⇔ cons'(0', _gen_s':from':cons':0':nil':zWquot':sel':minus':quot'2(x))

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

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

The following defined symbols remain to be analysed:
a__from', a__sel', a__minus', a__quot'

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

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

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

The following defined symbols remain to be analysed:
a__minus', a__quot'

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

The following defined symbols remain to be analysed:
a__quot'

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

No more defined symbols left to analyse.