/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 254 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:

   half(x) -> if(ge(x, s(s(0))), x)
   if(false, x) -> 0
   if(true, x) -> s(half(p(p(x))))
   p(0) -> 0
   p(s(x)) -> x
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   log(0) -> 0
   log(s(x)) -> s(log(half(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.
----------------------------------------

(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(x) -> if(ge(x, s(s(0'))), x)
   if(false, x) -> 0'
   if(true, x) -> s(half(p(p(x))))
   p(0') -> 0'
   p(s(x)) -> x
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   log(0') -> 0'
   log(s(x)) -> s(log(half(s(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:
half(x) -> if(ge(x, s(s(0'))), x)
if(false, x) -> 0'
if(true, x) -> s(half(p(p(x))))
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
log(0') -> 0'
log(s(x)) -> s(log(half(s(x))))

Types:
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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, ge, log

They will be analysed ascendingly in the following order:
ge < half
half < log

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(6)
Obligation:
TRS:
Rules:
half(x) -> if(ge(x, s(s(0'))), x)
if(false, x) -> 0'
if(true, x) -> s(half(p(p(x))))
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
log(0') -> 0'
log(s(x)) -> s(log(half(s(x))))

Types:
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
ge, half, log

They will be analysed ascendingly in the following order:
ge < half
half < log

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

Induction Base:
ge(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
ge(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

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(x) -> if(ge(x, s(s(0'))), x)
if(false, x) -> 0'
if(true, x) -> s(half(p(p(x))))
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
log(0') -> 0'
log(s(x)) -> s(log(half(s(x))))

Types:
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
ge, half, log

They will be analysed ascendingly in the following order:
ge < half
half < log

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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(x) -> if(ge(x, s(s(0'))), x)
if(false, x) -> 0'
if(true, x) -> s(half(p(p(x))))
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
log(0') -> 0'
log(s(x)) -> s(log(half(s(x))))

Types:
half :: 0':s -> 0':s
if :: false:true -> 0':s -> 0':s
ge :: 0':s -> 0':s -> false:true
s :: 0':s -> 0':s
0' :: 0':s
false :: false:true
true :: false:true
p :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
ge(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, 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:
half, log

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