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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 291 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), 26 ms]
        (14) 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:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   inc(0) -> 0
   inc(s(x)) -> s(inc(x))
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(x)) -> x
   bits(x) -> bitIter(x, 0)
   bitIter(x, y) -> if(zero(x), x, inc(y))
   if(true, x, y) -> p(y)
   if(false, x, y) -> bitIter(half(x), y)

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:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   inc(0') -> 0'
   inc(s(x)) -> s(inc(x))
   zero(0') -> true
   zero(s(x)) -> false
   p(0') -> 0'
   p(s(x)) -> x
   bits(x) -> bitIter(x, 0')
   bitIter(x, y) -> if(zero(x), x, inc(y))
   if(true, x, y) -> p(y)
   if(false, x, y) -> bitIter(half(x), y)

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:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, inc, bitIter

They will be analysed ascendingly in the following order:
half < bitIter
inc < bitIter

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(6)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
half, inc, bitIter

They will be analysed ascendingly in the following order:
half < bitIter
inc < bitIter

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n5_0)))) ->_IH
s(gen_0':s3_0(c6_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:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
half, inc, bitIter

They will be analysed ascendingly in the following order:
half < bitIter
inc < bitIter

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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:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
inc, bitIter

They will be analysed ascendingly in the following order:
inc < bitIter

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s3_0(n311_0)) -> gen_0':s3_0(n311_0), rt in Omega(1 + n311_0)

Induction Base:
inc(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
inc(gen_0':s3_0(+(n311_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s3_0(n311_0))) ->_IH
s(gen_0':s3_0(c312_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)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
inc(gen_0':s3_0(n311_0)) -> gen_0':s3_0(n311_0), rt in Omega(1 + n311_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
bitIter