(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 [ ].
(2) BOUNDS(2^n, INF)
(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
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
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__quotThey 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) →
nila__zWquot(
nil,
XS) →
nila__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) →
nila__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) →
nila__zWquot(
nil,
XS) →
nila__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) →
nila__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) →
nila__zWquot(
nil,
XS) →
nila__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) →
nila__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) →
nila__zWquot(
nil,
XS) →
nila__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) →
nila__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) →
nila__zWquot(
nil,
XS) →
nila__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) →
nila__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) →
nila__zWquot(
nil,
XS) →
nila__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) →
nila__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.