/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 282 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) 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:

   p(0) -> 0
   p(s(x)) -> x
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(x, s(y)) -> p(minus(x, y))
   div(0, s(y)) -> 0
   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
   log(s(0), s(s(y))) -> 0
   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))

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:

   p(0') -> 0'
   p(s(x)) -> x
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(x, s(y)) -> p(minus(x, y))
   div(0', s(y)) -> 0'
   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
   log(s(0'), s(s(y))) -> 0'
   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
minus(x, s(y)) -> p(minus(x, y))
div(0', s(y)) -> 0'
div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) -> 0'
log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, div, log

They will be analysed ascendingly in the following order:
minus < div
minus < log
div < log

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(6)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
minus(x, s(y)) -> p(minus(x, y))
div(0', s(y)) -> 0'
div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) -> 0'
log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, div, log

They will be analysed ascendingly in the following order:
minus < div
minus < log
div < log

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
gen_0':s2_0(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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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
minus(x, s(y)) -> p(minus(x, y))
div(0', s(y)) -> 0'
div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) -> 0'
log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, div, log

They will be analysed ascendingly in the following order:
minus < div
minus < log
div < log

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
minus(x, s(y)) -> p(minus(x, y))
div(0', s(y)) -> 0'
div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) -> 0'
log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
log :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
div, log

They will be analysed ascendingly in the following order:
div < log