/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 561 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 280 ms]
        (14) BOUNDS(1, INF)


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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:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

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:

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))
   and(x, or(y, z)) -> or(and(x, y), and(x, z))

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:
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))
and(x, or(y, z)) -> or(and(x, y), and(x, z))

Types:
not :: or -> or
and :: or -> or -> or
or :: or -> or -> or
hole_or1_0 :: or
gen_or2_0 :: Nat -> or

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
not, and

They will be analysed ascendingly in the following order:
and < not

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(6)
Obligation:
TRS:
Rules:
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))
and(x, or(y, z)) -> or(and(x, y), and(x, z))

Types:
not :: or -> or
and :: or -> or -> or
or :: or -> or -> or
hole_or1_0 :: or
gen_or2_0 :: Nat -> or


Generator Equations:
gen_or2_0(0) <=> hole_or1_0
gen_or2_0(+(x, 1)) <=> or(hole_or1_0, gen_or2_0(x))


The following defined symbols remain to be analysed:
and, not

They will be analysed ascendingly in the following order:
and < not

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
and(gen_or2_0(a), gen_or2_0(+(1, 0)))

Induction Step:
and(gen_or2_0(a), gen_or2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
or(and(gen_or2_0(a), hole_or1_0), and(gen_or2_0(a), gen_or2_0(+(1, n4_0)))) ->_IH
or(and(gen_or2_0(a), hole_or1_0), *3_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:
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))
and(x, or(y, z)) -> or(and(x, y), and(x, z))

Types:
not :: or -> or
and :: or -> or -> or
or :: or -> or -> or
hole_or1_0 :: or
gen_or2_0 :: Nat -> or


Generator Equations:
gen_or2_0(0) <=> hole_or1_0
gen_or2_0(+(x, 1)) <=> or(hole_or1_0, gen_or2_0(x))


The following defined symbols remain to be analysed:
and, not

They will be analysed ascendingly in the following order:
and < not

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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:
not(and(x, y)) -> or(not(x), not(y))
not(or(x, y)) -> and(not(x), not(y))
and(x, or(y, z)) -> or(and(x, y), and(x, z))

Types:
not :: or -> or
and :: or -> or -> or
or :: or -> or -> or
hole_or1_0 :: or
gen_or2_0 :: Nat -> or


Lemmas:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_or2_0(0) <=> hole_or1_0
gen_or2_0(+(x, 1)) <=> or(hole_or1_0, gen_or2_0(x))


The following defined symbols remain to be analysed:
not
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
not(gen_or2_0(+(1, n3893_0))) -> *3_0, rt in Omega(n3893_0)

Induction Base:
not(gen_or2_0(+(1, 0)))

Induction Step:
not(gen_or2_0(+(1, +(n3893_0, 1)))) ->_R^Omega(1)
and(not(hole_or1_0), not(gen_or2_0(+(1, n3893_0)))) ->_IH
and(not(hole_or1_0), *3_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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(14)
BOUNDS(1, INF)