/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 321 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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: INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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: INNERMOST
----------------------------------------

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/1

----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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: INNERMOST
----------------------------------------

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
Innermost 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

----------------------------------------

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__app, mark, a__from

They will be analysed ascendingly in the following order:
a__app = mark
a__app = a__from
mark = a__from

----------------------------------------

(8)
Obligation:
Innermost 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


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

----------------------------------------

(9) RewriteLemmaProof (LOWER BOUND(ID))
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 in Omega(1 + n4_0)

Induction Base:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(0)) ->_R^Omega(1)
nil

Induction Step:
mark(gen_nil:cons:app:from:zWadr:prefix:s2_0(+(n4_0, 1))) ->_R^Omega(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 in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(10)
Complex Obligation (BEST)

----------------------------------------

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost 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


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

----------------------------------------

(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(13)
BOUNDS(n^1, INF)

----------------------------------------

(14)
Obligation:
Innermost 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


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 in Omega(1 + n4_0)


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