(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__app(nil, YS) → mark(YS)
a__app(cons(X, XS), YS) → cons(mark(X), app(XS, YS))
a__from(X) → cons(mark(X), from(s(X)))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X, XS), cons(Y, YS)) → cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a__prefix(L) → cons(nil, zWadr(L, prefix(L)))
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(X)
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(app(from(X36060_0), X2)) →+ a__app(cons(mark(mark(X36060_0)), from(s(mark(X36060_0)))), mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X36060_0 / app(from(X36060_0), X2)].
The result substitution is [ ].
The rewrite sequence
mark(app(from(X36060_0), X2)) →+ a__app(cons(mark(mark(X36060_0)), from(s(mark(X36060_0)))), mark(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X36060_0 / app(from(X36060_0), X2)].
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__app(nil, YS) → mark(YS)
a__app(cons(X, XS), YS) → cons(mark(X), app(XS, YS))
a__from(X) → cons(mark(X), from(s(X)))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X, XS), cons(Y, YS)) → cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
a__prefix(L) → cons(nil, zWadr(L, prefix(L)))
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(X)
S is empty.
Rewrite Strategy: FULL
(5) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
cons/1
(6) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
a__app(nil, YS) → mark(YS)
a__app(cons(X), YS) → cons(mark(X))
a__from(X) → cons(mark(X))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X), cons(Y)) → cons(a__app(mark(Y), cons(mark(X))))
a__prefix(L) → cons(nil)
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(X)
S is empty.
Rewrite Strategy: FULL
(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(8) Obligation:
TRS:
Rules:
a__app(nil, YS) → mark(YS)
a__app(cons(X), YS) → cons(mark(X))
a__from(X) → cons(mark(X))
a__zWadr(nil, YS) → nil
a__zWadr(XS, nil) → nil
a__zWadr(cons(X), cons(Y)) → cons(a__app(mark(Y), cons(mark(X))))
a__prefix(L) → cons(nil)
mark(app(X1, X2)) → a__app(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(zWadr(X1, X2)) → a__zWadr(mark(X1), mark(X2))
mark(prefix(X)) → a__prefix(mark(X))
mark(nil) → nil
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__app(X1, X2) → app(X1, X2)
a__from(X) → from(X)
a__zWadr(X1, X2) → zWadr(X1, X2)
a__prefix(X) → prefix(X)
Types:
a__app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
nil :: nil:cons:app:from:zWadr:prefix:s
mark :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
cons :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
s :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
hole_nil:cons:app:from:zWadr:prefix:s1_0 :: nil:cons:app:from:zWadr:prefix:s
gen_nil:cons:app:from:zWadr:prefix:s2_0 :: Nat → nil:cons:app:from:zWadr:prefix:s
(9) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
a__app,
mark,
a__fromThey will be analysed ascendingly in the following order:
a__app = mark
a__app = a__from
mark = a__from
(10) Obligation:
TRS:
Rules:
a__app(
nil,
YS) →
mark(
YS)
a__app(
cons(
X),
YS) →
cons(
mark(
X))
a__from(
X) →
cons(
mark(
X))
a__zWadr(
nil,
YS) →
nila__zWadr(
XS,
nil) →
nila__zWadr(
cons(
X),
cons(
Y)) →
cons(
a__app(
mark(
Y),
cons(
mark(
X))))
a__prefix(
L) →
cons(
nil)
mark(
app(
X1,
X2)) →
a__app(
mark(
X1),
mark(
X2))
mark(
from(
X)) →
a__from(
mark(
X))
mark(
zWadr(
X1,
X2)) →
a__zWadr(
mark(
X1),
mark(
X2))
mark(
prefix(
X)) →
a__prefix(
mark(
X))
mark(
nil) →
nilmark(
cons(
X1)) →
cons(
mark(
X1))
mark(
s(
X)) →
s(
mark(
X))
a__app(
X1,
X2) →
app(
X1,
X2)
a__from(
X) →
from(
X)
a__zWadr(
X1,
X2) →
zWadr(
X1,
X2)
a__prefix(
X) →
prefix(
X)
Types:
a__app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
nil :: nil:cons:app:from:zWadr:prefix:s
mark :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
cons :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
s :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
hole_nil:cons:app:from:zWadr:prefix:s1_0 :: nil:cons:app:from:zWadr:prefix:s
gen_nil:cons:app:from:zWadr:prefix:s2_0 :: Nat → nil:cons:app:from:zWadr:prefix:s
Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s2_0(0) ⇔ nil
gen_nil:cons:app:from:zWadr:prefix:s2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:from:zWadr:prefix:s2_0(x))
The following defined symbols remain to be analysed:
mark, a__app, a__from
They will be analysed ascendingly in the following order:
a__app = mark
a__app = a__from
mark = a__from
(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
mark(
gen_nil:cons:app:from:zWadr:prefix:s2_0(
n4_0)) →
gen_nil:cons:app:from:zWadr:prefix:s2_0(
n4_0), rt ∈ Ω(1 + n4
0)
Induction Base:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(0)) →RΩ(1)
nil
Induction Step:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(+(n4_0, 1))) →RΩ(1)
cons(mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0))) →IH
cons(gen_nil:cons:app:from:zWadr:prefix:s2_0(c5_0))
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(12) Complex Obligation (BEST)
(13) Obligation:
TRS:
Rules:
a__app(
nil,
YS) →
mark(
YS)
a__app(
cons(
X),
YS) →
cons(
mark(
X))
a__from(
X) →
cons(
mark(
X))
a__zWadr(
nil,
YS) →
nila__zWadr(
XS,
nil) →
nila__zWadr(
cons(
X),
cons(
Y)) →
cons(
a__app(
mark(
Y),
cons(
mark(
X))))
a__prefix(
L) →
cons(
nil)
mark(
app(
X1,
X2)) →
a__app(
mark(
X1),
mark(
X2))
mark(
from(
X)) →
a__from(
mark(
X))
mark(
zWadr(
X1,
X2)) →
a__zWadr(
mark(
X1),
mark(
X2))
mark(
prefix(
X)) →
a__prefix(
mark(
X))
mark(
nil) →
nilmark(
cons(
X1)) →
cons(
mark(
X1))
mark(
s(
X)) →
s(
mark(
X))
a__app(
X1,
X2) →
app(
X1,
X2)
a__from(
X) →
from(
X)
a__zWadr(
X1,
X2) →
zWadr(
X1,
X2)
a__prefix(
X) →
prefix(
X)
Types:
a__app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
nil :: nil:cons:app:from:zWadr:prefix:s
mark :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
cons :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
s :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
hole_nil:cons:app:from:zWadr:prefix:s1_0 :: nil:cons:app:from:zWadr:prefix:s
gen_nil:cons:app:from:zWadr:prefix:s2_0 :: Nat → nil:cons:app:from:zWadr:prefix:s
Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0)) → gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s2_0(0) ⇔ nil
gen_nil:cons:app:from:zWadr:prefix:s2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:from:zWadr:prefix:s2_0(x))
The following defined symbols remain to be analysed:
a__app, a__from
They will be analysed ascendingly in the following order:
a__app = mark
a__app = a__from
mark = a__from
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__app.
(15) Obligation:
TRS:
Rules:
a__app(
nil,
YS) →
mark(
YS)
a__app(
cons(
X),
YS) →
cons(
mark(
X))
a__from(
X) →
cons(
mark(
X))
a__zWadr(
nil,
YS) →
nila__zWadr(
XS,
nil) →
nila__zWadr(
cons(
X),
cons(
Y)) →
cons(
a__app(
mark(
Y),
cons(
mark(
X))))
a__prefix(
L) →
cons(
nil)
mark(
app(
X1,
X2)) →
a__app(
mark(
X1),
mark(
X2))
mark(
from(
X)) →
a__from(
mark(
X))
mark(
zWadr(
X1,
X2)) →
a__zWadr(
mark(
X1),
mark(
X2))
mark(
prefix(
X)) →
a__prefix(
mark(
X))
mark(
nil) →
nilmark(
cons(
X1)) →
cons(
mark(
X1))
mark(
s(
X)) →
s(
mark(
X))
a__app(
X1,
X2) →
app(
X1,
X2)
a__from(
X) →
from(
X)
a__zWadr(
X1,
X2) →
zWadr(
X1,
X2)
a__prefix(
X) →
prefix(
X)
Types:
a__app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
nil :: nil:cons:app:from:zWadr:prefix:s
mark :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
cons :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
s :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
hole_nil:cons:app:from:zWadr:prefix:s1_0 :: nil:cons:app:from:zWadr:prefix:s
gen_nil:cons:app:from:zWadr:prefix:s2_0 :: Nat → nil:cons:app:from:zWadr:prefix:s
Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0)) → gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s2_0(0) ⇔ nil
gen_nil:cons:app:from:zWadr:prefix:s2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:from:zWadr:prefix:s2_0(x))
The following defined symbols remain to be analysed:
a__from
They will be analysed ascendingly in the following order:
a__app = mark
a__app = a__from
mark = a__from
(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__from.
(17) Obligation:
TRS:
Rules:
a__app(
nil,
YS) →
mark(
YS)
a__app(
cons(
X),
YS) →
cons(
mark(
X))
a__from(
X) →
cons(
mark(
X))
a__zWadr(
nil,
YS) →
nila__zWadr(
XS,
nil) →
nila__zWadr(
cons(
X),
cons(
Y)) →
cons(
a__app(
mark(
Y),
cons(
mark(
X))))
a__prefix(
L) →
cons(
nil)
mark(
app(
X1,
X2)) →
a__app(
mark(
X1),
mark(
X2))
mark(
from(
X)) →
a__from(
mark(
X))
mark(
zWadr(
X1,
X2)) →
a__zWadr(
mark(
X1),
mark(
X2))
mark(
prefix(
X)) →
a__prefix(
mark(
X))
mark(
nil) →
nilmark(
cons(
X1)) →
cons(
mark(
X1))
mark(
s(
X)) →
s(
mark(
X))
a__app(
X1,
X2) →
app(
X1,
X2)
a__from(
X) →
from(
X)
a__zWadr(
X1,
X2) →
zWadr(
X1,
X2)
a__prefix(
X) →
prefix(
X)
Types:
a__app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
nil :: nil:cons:app:from:zWadr:prefix:s
mark :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
cons :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
s :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
hole_nil:cons:app:from:zWadr:prefix:s1_0 :: nil:cons:app:from:zWadr:prefix:s
gen_nil:cons:app:from:zWadr:prefix:s2_0 :: Nat → nil:cons:app:from:zWadr:prefix:s
Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0)) → gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s2_0(0) ⇔ nil
gen_nil:cons:app:from:zWadr:prefix:s2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:from:zWadr:prefix:s2_0(x))
No more defined symbols left to analyse.
(18) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0)) → gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0), rt ∈ Ω(1 + n40)
(19) BOUNDS(n^1, INF)
(20) Obligation:
TRS:
Rules:
a__app(
nil,
YS) →
mark(
YS)
a__app(
cons(
X),
YS) →
cons(
mark(
X))
a__from(
X) →
cons(
mark(
X))
a__zWadr(
nil,
YS) →
nila__zWadr(
XS,
nil) →
nila__zWadr(
cons(
X),
cons(
Y)) →
cons(
a__app(
mark(
Y),
cons(
mark(
X))))
a__prefix(
L) →
cons(
nil)
mark(
app(
X1,
X2)) →
a__app(
mark(
X1),
mark(
X2))
mark(
from(
X)) →
a__from(
mark(
X))
mark(
zWadr(
X1,
X2)) →
a__zWadr(
mark(
X1),
mark(
X2))
mark(
prefix(
X)) →
a__prefix(
mark(
X))
mark(
nil) →
nilmark(
cons(
X1)) →
cons(
mark(
X1))
mark(
s(
X)) →
s(
mark(
X))
a__app(
X1,
X2) →
app(
X1,
X2)
a__from(
X) →
from(
X)
a__zWadr(
X1,
X2) →
zWadr(
X1,
X2)
a__prefix(
X) →
prefix(
X)
Types:
a__app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
nil :: nil:cons:app:from:zWadr:prefix:s
mark :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
cons :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
a__prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
app :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
from :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
zWadr :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
prefix :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
s :: nil:cons:app:from:zWadr:prefix:s → nil:cons:app:from:zWadr:prefix:s
hole_nil:cons:app:from:zWadr:prefix:s1_0 :: nil:cons:app:from:zWadr:prefix:s
gen_nil:cons:app:from:zWadr:prefix:s2_0 :: Nat → nil:cons:app:from:zWadr:prefix:s
Lemmas:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0)) → gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0), rt ∈ Ω(1 + n40)
Generator Equations:
gen_nil:cons:app:from:zWadr:prefix:s2_0(0) ⇔ nil
gen_nil:cons:app:from:zWadr:prefix:s2_0(+(x, 1)) ⇔ cons(gen_nil:cons:app:from:zWadr:prefix:s2_0(x))
No more defined symbols left to analyse.
(21) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0)) → gen_nil:cons:app:from:zWadr:prefix:s2_0(n4_0), rt ∈ Ω(1 + n40)
(22) BOUNDS(n^1, INF)