/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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) 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), 198 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), 574 ms]
        (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:

   rev(nil) -> nil
   rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
   rev1(x, nil) -> x
   rev1(x, ++(y, z)) -> rev1(y, z)
   rev2(x, nil) -> nil
   rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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:

   rev(nil) -> nil
   rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
   rev1(x, nil) -> x
   rev1(x, ++(y, z)) -> rev1(y, z)
   rev2(x, nil) -> nil
   rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(4)
Obligation:
Innermost TRS:
Rules:
rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Types:
rev :: nil:++ -> nil:++
nil :: nil:++
++ :: rev1 -> nil:++ -> nil:++
rev1 :: rev1 -> nil:++ -> rev1
rev2 :: rev1 -> nil:++ -> nil:++
hole_nil:++1_0 :: nil:++
hole_rev12_0 :: rev1
gen_nil:++3_0 :: Nat -> nil:++

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
rev, rev1, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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(6)
Obligation:
Innermost TRS:
Rules:
rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Types:
rev :: nil:++ -> nil:++
nil :: nil:++
++ :: rev1 -> nil:++ -> nil:++
rev1 :: rev1 -> nil:++ -> rev1
rev2 :: rev1 -> nil:++ -> nil:++
hole_nil:++1_0 :: nil:++
hole_rev12_0 :: rev1
gen_nil:++3_0 :: Nat -> nil:++


Generator Equations:
gen_nil:++3_0(0) <=> nil
gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))


The following defined symbols remain to be analysed:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) -> hole_rev12_0, rt in Omega(1 + n5_0)

Induction Base:
rev1(hole_rev12_0, gen_nil:++3_0(0)) ->_R^Omega(1)
hole_rev12_0

Induction Step:
rev1(hole_rev12_0, gen_nil:++3_0(+(n5_0, 1))) ->_R^Omega(1)
rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) ->_IH
hole_rev12_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:

Innermost TRS:
Rules:
rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Types:
rev :: nil:++ -> nil:++
nil :: nil:++
++ :: rev1 -> nil:++ -> nil:++
rev1 :: rev1 -> nil:++ -> rev1
rev2 :: rev1 -> nil:++ -> nil:++
hole_nil:++1_0 :: nil:++
hole_rev12_0 :: rev1
gen_nil:++3_0 :: Nat -> nil:++


Generator Equations:
gen_nil:++3_0(0) <=> nil
gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))


The following defined symbols remain to be analysed:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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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:
Innermost TRS:
Rules:
rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Types:
rev :: nil:++ -> nil:++
nil :: nil:++
++ :: rev1 -> nil:++ -> nil:++
rev1 :: rev1 -> nil:++ -> rev1
rev2 :: rev1 -> nil:++ -> nil:++
hole_nil:++1_0 :: nil:++
hole_rev12_0 :: rev1
gen_nil:++3_0 :: Nat -> nil:++


Lemmas:
rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) -> hole_rev12_0, rt in Omega(1 + n5_0)


Generator Equations:
gen_nil:++3_0(0) <=> nil
gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))


The following defined symbols remain to be analysed:
rev2, rev

They will be analysed ascendingly in the following order:
rev = rev2

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev2(hole_rev12_0, gen_nil:++3_0(+(1, n128_0))) -> *4_0, rt in Omega(n128_0)

Induction Base:
rev2(hole_rev12_0, gen_nil:++3_0(+(1, 0)))

Induction Step:
rev2(hole_rev12_0, gen_nil:++3_0(+(1, +(n128_0, 1)))) ->_R^Omega(1)
rev(++(hole_rev12_0, rev(rev2(hole_rev12_0, gen_nil:++3_0(+(1, n128_0)))))) ->_IH
rev(++(hole_rev12_0, rev(*4_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
rev(nil) -> nil
rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
rev1(x, nil) -> x
rev1(x, ++(y, z)) -> rev1(y, z)
rev2(x, nil) -> nil
rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

Types:
rev :: nil:++ -> nil:++
nil :: nil:++
++ :: rev1 -> nil:++ -> nil:++
rev1 :: rev1 -> nil:++ -> rev1
rev2 :: rev1 -> nil:++ -> nil:++
hole_nil:++1_0 :: nil:++
hole_rev12_0 :: rev1
gen_nil:++3_0 :: Nat -> nil:++


Lemmas:
rev1(hole_rev12_0, gen_nil:++3_0(n5_0)) -> hole_rev12_0, rt in Omega(1 + n5_0)
rev2(hole_rev12_0, gen_nil:++3_0(+(1, n128_0))) -> *4_0, rt in Omega(n128_0)


Generator Equations:
gen_nil:++3_0(0) <=> nil
gen_nil:++3_0(+(x, 1)) <=> ++(hole_rev12_0, gen_nil:++3_0(x))


The following defined symbols remain to be analysed:
rev

They will be analysed ascendingly in the following order:
rev = rev2