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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 12.0 s]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs


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(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__first(0, X) -> nil
   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(X)) -> s(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__first(X1, X2) -> first(X1, X2)
   a__from(X) -> from(X)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(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__first(0', X) -> nil
   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(0') -> 0'
   mark(nil) -> nil
   mark(s(X)) -> s(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__first(X1, X2) -> first(X1, X2)
   a__from(X) -> from(X)

S is empty.
Rewrite Strategy: INNERMOST
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(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/1

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(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__first(0', X) -> nil
   a__first(s(X), cons(Y)) -> cons(mark(Y))
   a__from(X) -> cons(mark(X))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(0') -> 0'
   mark(nil) -> nil
   mark(s(X)) -> s(mark(X))
   mark(cons(X1)) -> cons(mark(X1))
   a__first(X1, X2) -> first(X1, X2)
   a__from(X) -> from(X)

S is empty.
Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Innermost TRS:
Rules:
a__first(0', X) -> nil
a__first(s(X), cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(0') -> 0'
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1)) -> cons(mark(X1))
a__first(X1, X2) -> first(X1, X2)
a__from(X) -> from(X)

Types:
a__first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
0' :: 0':nil:s:cons:first:from
nil :: 0':nil:s:cons:first:from
s :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
cons :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
mark :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
a__from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
hole_0':nil:s:cons:first:from1_0 :: 0':nil:s:cons:first:from
gen_0':nil:s:cons:first:from2_0 :: Nat -> 0':nil:s:cons:first:from

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

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

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(8)
Obligation:
Innermost TRS:
Rules:
a__first(0', X) -> nil
a__first(s(X), cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(0') -> 0'
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1)) -> cons(mark(X1))
a__first(X1, X2) -> first(X1, X2)
a__from(X) -> from(X)

Types:
a__first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
0' :: 0':nil:s:cons:first:from
nil :: 0':nil:s:cons:first:from
s :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
cons :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
mark :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
a__from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
hole_0':nil:s:cons:first:from1_0 :: 0':nil:s:cons:first:from
gen_0':nil:s:cons:first:from2_0 :: Nat -> 0':nil:s:cons:first:from


Generator Equations:
gen_0':nil:s:cons:first:from2_0(0) <=> 0'
gen_0':nil:s:cons:first:from2_0(+(x, 1)) <=> s(gen_0':nil:s:cons:first:from2_0(x))


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

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

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':nil:s:cons:first:from2_0(n9771_0)) -> gen_0':nil:s:cons:first:from2_0(n9771_0), rt in Omega(1 + n9771_0)

Induction Base:
mark(gen_0':nil:s:cons:first:from2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':nil:s:cons:first:from2_0(+(n9771_0, 1))) ->_R^Omega(1)
s(mark(gen_0':nil:s:cons:first:from2_0(n9771_0))) ->_IH
s(gen_0':nil:s:cons:first:from2_0(c9772_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(10)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
a__first(0', X) -> nil
a__first(s(X), cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(0') -> 0'
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1)) -> cons(mark(X1))
a__first(X1, X2) -> first(X1, X2)
a__from(X) -> from(X)

Types:
a__first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
0' :: 0':nil:s:cons:first:from
nil :: 0':nil:s:cons:first:from
s :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
cons :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
mark :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
a__from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
hole_0':nil:s:cons:first:from1_0 :: 0':nil:s:cons:first:from
gen_0':nil:s:cons:first:from2_0 :: Nat -> 0':nil:s:cons:first:from


Generator Equations:
gen_0':nil:s:cons:first:from2_0(0) <=> 0'
gen_0':nil:s:cons:first:from2_0(+(x, 1)) <=> s(gen_0':nil:s:cons:first:from2_0(x))


The following defined symbols remain to be analysed:
mark

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

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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
Innermost TRS:
Rules:
a__first(0', X) -> nil
a__first(s(X), cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(0') -> 0'
mark(nil) -> nil
mark(s(X)) -> s(mark(X))
mark(cons(X1)) -> cons(mark(X1))
a__first(X1, X2) -> first(X1, X2)
a__from(X) -> from(X)

Types:
a__first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
0' :: 0':nil:s:cons:first:from
nil :: 0':nil:s:cons:first:from
s :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
cons :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
mark :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
a__from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
first :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
from :: 0':nil:s:cons:first:from -> 0':nil:s:cons:first:from
hole_0':nil:s:cons:first:from1_0 :: 0':nil:s:cons:first:from
gen_0':nil:s:cons:first:from2_0 :: Nat -> 0':nil:s:cons:first:from


Lemmas:
mark(gen_0':nil:s:cons:first:from2_0(n9771_0)) -> gen_0':nil:s:cons:first:from2_0(n9771_0), rt in Omega(1 + n9771_0)


Generator Equations:
gen_0':nil:s:cons:first:from2_0(0) <=> 0'
gen_0':nil:s:cons:first:from2_0(+(x, 1)) <=> s(gen_0':nil:s:cons:first:from2_0(x))


The following defined symbols remain to be analysed:
a__from

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