/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 151 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) SlicingProof [LOWER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 1436 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
        (18) BOUNDS(1, INF)


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST
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(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: INNERMOST
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(5) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0
mapconsapp/0

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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   subsets(Cons(xs)) -> subsets[Ite][True][Let](Cons(xs), subsets(xs))
   subsets(Nil) -> Cons(Nil)
   mapconsapp(Cons(xs), rest) -> Cons(mapconsapp(xs, rest))
   mapconsapp(Nil, rest) -> rest
   notEmpty(Cons(xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

The (relative) TRS S consists of the following rules:

   subsets[Ite][True][Let](Cons(xs), subs) -> mapconsapp(subs, subs)

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
subsets(Cons(xs)) -> subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) -> Cons(Nil)
mapconsapp(Cons(xs), rest) -> Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) -> rest
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) -> mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
subsets, mapconsapp
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(10)
Obligation:
Innermost TRS:
Rules:
subsets(Cons(xs)) -> subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) -> Cons(Nil)
mapconsapp(Cons(xs), rest) -> Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) -> rest
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) -> mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
subsets, mapconsapp
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
subsets(gen_Cons:Nil3_0(+(1, 0)))

Induction Step:
subsets(gen_Cons:Nil3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
subsets[Ite][True][Let](Cons(gen_Cons:Nil3_0(+(1, n5_0))), subsets(gen_Cons:Nil3_0(+(1, n5_0)))) ->_IH
subsets[Ite][True][Let](Cons(gen_Cons:Nil3_0(+(1, n5_0))), *4_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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(12)
Complex Obligation (BEST)

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(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
subsets(Cons(xs)) -> subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) -> Cons(Nil)
mapconsapp(Cons(xs), rest) -> Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) -> rest
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) -> mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
subsets, mapconsapp
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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
subsets(Cons(xs)) -> subsets[Ite][True][Let](Cons(xs), subsets(xs))
subsets(Nil) -> Cons(Nil)
mapconsapp(Cons(xs), rest) -> Cons(mapconsapp(xs, rest))
mapconsapp(Nil, rest) -> rest
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> subsets(xs)
subsets[Ite][True][Let](Cons(xs), subs) -> mapconsapp(subs, subs)

Types:
subsets :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
subsets(gen_Cons:Nil3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
mapconsapp
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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mapconsapp(gen_Cons:Nil3_0(n3634_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n3634_0, b)), rt in Omega(1 + n3634_0)

Induction Base:
mapconsapp(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) ->_R^Omega(1)
gen_Cons:Nil3_0(b)

Induction Step:
mapconsapp(gen_Cons:Nil3_0(+(n3634_0, 1)), gen_Cons:Nil3_0(b)) ->_R^Omega(1)
Cons(mapconsapp(gen_Cons:Nil3_0(n3634_0), gen_Cons:Nil3_0(b))) ->_IH
Cons(gen_Cons:Nil3_0(+(b, c3635_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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(18)
BOUNDS(1, INF)