/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^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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), 243 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 1870 ms]
        (16) proven lower bound
        (17) LowerBoundPropagationProof [FINISHED, 0 ms]
        (18) BOUNDS(n^2, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(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^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *'(fac(p(s(x))), s(x))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))

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

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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *'(fac(p(s(x))))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(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:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
fac, p

They will be analysed ascendingly in the following order:
p < fac

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(8)
Obligation:
TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
p, fac

They will be analysed ascendingly in the following order:
p < fac

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_s:0'4_0(+(1, n6_0))) -> gen_s:0'4_0(n6_0), rt in Omega(1 + n6_0)

Induction Base:
p(gen_s:0'4_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_s:0'4_0(+(1, +(n6_0, 1)))) ->_R^Omega(1)
s(p(s(gen_s:0'4_0(n6_0)))) ->_IH
s(gen_s:0'4_0(c7_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:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
p, fac

They will be analysed ascendingly in the following order:
p < fac

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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:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
p(gen_s:0'4_0(+(1, n6_0))) -> gen_s:0'4_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
fac
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_s:0'4_0(+(1, n213_0))) -> *5_0, rt in Omega(n213_0 + n213_0^2)

Induction Base:
fac(gen_s:0'4_0(+(1, 0)))

Induction Step:
fac(gen_s:0'4_0(+(1, +(n213_0, 1)))) ->_R^Omega(1)
*'(fac(p(s(gen_s:0'4_0(+(1, n213_0)))))) ->_L^Omega(2 + n213_0)
*'(fac(gen_s:0'4_0(+(1, n213_0)))) ->_IH
*'(*5_0)

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(16)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
p(gen_s:0'4_0(+(1, n6_0))) -> gen_s:0'4_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
fac
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(17) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(18)
BOUNDS(n^2, INF)