(0) Obligation:

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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(from(X)) →+ cons(mark(mark(X)), from(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X / from(X)].
The result substitution is [ ].

The rewrite sequence
mark(from(X)) →+ cons(mark(mark(X)), from(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / from(X)].
The result substitution is [ ].

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative 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)

S is empty.
Rewrite Strategy: FULL

Infered types.

(6) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

(7) OrderProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

Generator Equations:
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(0) ⇔ 0'
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(+(x, 1)) ⇔ cons(0', gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(x))

The following defined symbols remain to be analysed:
mark, a__from, 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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

Generator Equations:
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(0) ⇔ 0'
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(+(x, 1)) ⇔ cons(0', gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(x))

The following defined symbols remain to be analysed:
a__from, 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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

Generator Equations:
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(0) ⇔ 0'
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(+(x, 1)) ⇔ cons(0', gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(x))

The following defined symbols remain to be analysed:
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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

Generator Equations:
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(0) ⇔ 0'
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(+(x, 1)) ⇔ cons(0', gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(x))

The following defined symbols remain to be analysed:
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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

Generator Equations:
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(0) ⇔ 0'
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(+(x, 1)) ⇔ cons(0', gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(x))

The following defined symbols remain to be analysed:
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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
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:quot1_0 :: s:from:cons:0':nil:zWquot:sel:minus:quot
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0 :: Nat → s:from:cons:0':nil:zWquot:sel:minus:quot

Generator Equations:
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(0) ⇔ 0'
gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(+(x, 1)) ⇔ cons(0', gen_s:from:cons:0':nil:zWquot:sel:minus:quot2_0(x))

No more defined symbols left to analyse.