WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 455 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:

   minus(minus(x)) -> x
   minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
   minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
   f(minus(x)) -> minus(minus(minus(f(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^1, INF).


The TRS R consists of the following rules:

   minus(minus(x)) -> x
   minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
   minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
   f(minus(x)) -> minus(minus(minus(f(x))))

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:
minus(minus(x)) -> x
minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Types:
minus :: +':*' -> +':*'
+' :: +':*' -> +':*' -> +':*'
*' :: +':*' -> +':*' -> +':*'
f :: +':*' -> +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat -> +':*'

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

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

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(6)
Obligation:
TRS:
Rules:
minus(minus(x)) -> x
minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Types:
minus :: +':*' -> +':*'
+' :: +':*' -> +':*' -> +':*'
*' :: +':*' -> +':*' -> +':*'
f :: +':*' -> +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat -> +':*'


Generator Equations:
gen_+':*'2_0(0) <=> hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) <=> +'(hole_+':*'1_0, gen_+':*'2_0(x))


The following defined symbols remain to be analysed:
minus, f

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_+':*'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
minus(gen_+':*'2_0(0))

Induction Step:
minus(gen_+':*'2_0(+(n4_0, 1))) ->_R^Omega(1)
*'(minus(minus(minus(hole_+':*'1_0))), minus(minus(minus(gen_+':*'2_0(n4_0))))) ->_R^Omega(1)
*'(minus(hole_+':*'1_0), minus(minus(minus(gen_+':*'2_0(n4_0))))) ->_IH
*'(minus(hole_+':*'1_0), minus(minus(*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:
minus(minus(x)) -> x
minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Types:
minus :: +':*' -> +':*'
+' :: +':*' -> +':*' -> +':*'
*' :: +':*' -> +':*' -> +':*'
f :: +':*' -> +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat -> +':*'


Generator Equations:
gen_+':*'2_0(0) <=> hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) <=> +'(hole_+':*'1_0, gen_+':*'2_0(x))


The following defined symbols remain to be analysed:
minus, f

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

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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:
minus(minus(x)) -> x
minus(+'(x, y)) -> *'(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*'(x, y)) -> +'(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Types:
minus :: +':*' -> +':*'
+' :: +':*' -> +':*' -> +':*'
*' :: +':*' -> +':*' -> +':*'
f :: +':*' -> +':*'
hole_+':*'1_0 :: +':*'
gen_+':*'2_0 :: Nat -> +':*'


Lemmas:
minus(gen_+':*'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_+':*'2_0(0) <=> hole_+':*'1_0
gen_+':*'2_0(+(x, 1)) <=> +'(hole_+':*'1_0, gen_+':*'2_0(x))


The following defined symbols remain to be analysed:
f