/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) 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), 260 ms]
(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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(0) -> cons(0, f(s(0)))
   a__f(s(0)) -> a__f(a__p(s(0)))
   a__p(s(X)) -> mark(X)
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0) -> 0
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   a__f(X) -> f(X)
   a__p(X) -> p(X)

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(0') -> cons(0', f(s(0')))
   a__f(s(0')) -> a__f(a__p(s(0')))
   a__p(s(X)) -> mark(X)
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0') -> 0'
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   a__f(X) -> f(X)
   a__p(X) -> p(X)

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

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


The TRS R consists of the following rules:

   a__f(0') -> cons(0')
   a__f(s(0')) -> a__f(a__p(s(0')))
   a__p(s(X)) -> mark(X)
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0') -> 0'
   mark(cons(X1)) -> cons(mark(X1))
   mark(s(X)) -> s(mark(X))
   a__f(X) -> f(X)
   a__p(X) -> p(X)

S is empty.
Rewrite Strategy: FULL
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
TRS:
Rules:
a__f(0') -> cons(0')
a__f(s(0')) -> a__f(a__p(s(0')))
a__p(s(X)) -> mark(X)
mark(f(X)) -> a__f(mark(X))
mark(p(X)) -> a__p(mark(X))
mark(0') -> 0'
mark(cons(X1)) -> cons(mark(X1))
mark(s(X)) -> s(mark(X))
a__f(X) -> f(X)
a__p(X) -> p(X)

Types:
a__f :: 0':cons:s:f:p -> 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p -> 0':cons:s:f:p
s :: 0':cons:s:f:p -> 0':cons:s:f:p
a__p :: 0':cons:s:f:p -> 0':cons:s:f:p
mark :: 0':cons:s:f:p -> 0':cons:s:f:p
f :: 0':cons:s:f:p -> 0':cons:s:f:p
p :: 0':cons:s:f:p -> 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat -> 0':cons:s:f:p

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

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

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(8)
Obligation:
TRS:
Rules:
a__f(0') -> cons(0')
a__f(s(0')) -> a__f(a__p(s(0')))
a__p(s(X)) -> mark(X)
mark(f(X)) -> a__f(mark(X))
mark(p(X)) -> a__p(mark(X))
mark(0') -> 0'
mark(cons(X1)) -> cons(mark(X1))
mark(s(X)) -> s(mark(X))
a__f(X) -> f(X)
a__p(X) -> p(X)

Types:
a__f :: 0':cons:s:f:p -> 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p -> 0':cons:s:f:p
s :: 0':cons:s:f:p -> 0':cons:s:f:p
a__p :: 0':cons:s:f:p -> 0':cons:s:f:p
mark :: 0':cons:s:f:p -> 0':cons:s:f:p
f :: 0':cons:s:f:p -> 0':cons:s:f:p
p :: 0':cons:s:f:p -> 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat -> 0':cons:s:f:p


Generator Equations:
gen_0':cons:s:f:p2_0(0) <=> 0'
gen_0':cons:s:f:p2_0(+(x, 1)) <=> cons(gen_0':cons:s:f:p2_0(x))


The following defined symbols remain to be analysed:
a__p, a__f, mark

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

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':cons:s:f:p2_0(n10_0)) -> gen_0':cons:s:f:p2_0(n10_0), rt in Omega(1 + n10_0)

Induction Base:
mark(gen_0':cons:s:f:p2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':cons:s:f:p2_0(+(n10_0, 1))) ->_R^Omega(1)
cons(mark(gen_0':cons:s:f:p2_0(n10_0))) ->_IH
cons(gen_0':cons:s:f:p2_0(c11_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:

TRS:
Rules:
a__f(0') -> cons(0')
a__f(s(0')) -> a__f(a__p(s(0')))
a__p(s(X)) -> mark(X)
mark(f(X)) -> a__f(mark(X))
mark(p(X)) -> a__p(mark(X))
mark(0') -> 0'
mark(cons(X1)) -> cons(mark(X1))
mark(s(X)) -> s(mark(X))
a__f(X) -> f(X)
a__p(X) -> p(X)

Types:
a__f :: 0':cons:s:f:p -> 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p -> 0':cons:s:f:p
s :: 0':cons:s:f:p -> 0':cons:s:f:p
a__p :: 0':cons:s:f:p -> 0':cons:s:f:p
mark :: 0':cons:s:f:p -> 0':cons:s:f:p
f :: 0':cons:s:f:p -> 0':cons:s:f:p
p :: 0':cons:s:f:p -> 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat -> 0':cons:s:f:p


Generator Equations:
gen_0':cons:s:f:p2_0(0) <=> 0'
gen_0':cons:s:f:p2_0(+(x, 1)) <=> cons(gen_0':cons:s:f:p2_0(x))


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

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

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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:
TRS:
Rules:
a__f(0') -> cons(0')
a__f(s(0')) -> a__f(a__p(s(0')))
a__p(s(X)) -> mark(X)
mark(f(X)) -> a__f(mark(X))
mark(p(X)) -> a__p(mark(X))
mark(0') -> 0'
mark(cons(X1)) -> cons(mark(X1))
mark(s(X)) -> s(mark(X))
a__f(X) -> f(X)
a__p(X) -> p(X)

Types:
a__f :: 0':cons:s:f:p -> 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p -> 0':cons:s:f:p
s :: 0':cons:s:f:p -> 0':cons:s:f:p
a__p :: 0':cons:s:f:p -> 0':cons:s:f:p
mark :: 0':cons:s:f:p -> 0':cons:s:f:p
f :: 0':cons:s:f:p -> 0':cons:s:f:p
p :: 0':cons:s:f:p -> 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat -> 0':cons:s:f:p


Lemmas:
mark(gen_0':cons:s:f:p2_0(n10_0)) -> gen_0':cons:s:f:p2_0(n10_0), rt in Omega(1 + n10_0)


Generator Equations:
gen_0':cons:s:f:p2_0(0) <=> 0'
gen_0':cons:s:f:p2_0(+(x, 1)) <=> cons(gen_0':cons:s:f:p2_0(x))


The following defined symbols remain to be analysed:
a__f, a__p

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