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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 206 ms]
(8) proven lower bound
(9) LowerBoundPropagationProof [FINISHED, 0 ms]
(10) BOUNDS(n^1, INF)


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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:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(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:

   nonZero(0') -> false
   nonZero(s(x)) -> true
   p(0') -> 0'
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0')
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(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:
nonZero(0') -> false
nonZero(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
rand
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(6)
Obligation:
TRS:
Rules:
nonZero(0') -> false
nonZero(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
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:
rand
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(b), rt in Omega(1 + n5_0)

Induction Base:
rand(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
if(nonZero(gen_0':s3_0(0)), gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
if(false, gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
gen_0':s3_0(b)

Induction Step:
rand(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
if(nonZero(gen_0':s3_0(+(n5_0, 1))), gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
if(true, gen_0':s3_0(+(1, n5_0)), gen_0':s3_0(b)) ->_R^Omega(1)
rand(p(gen_0':s3_0(+(1, n5_0))), id_inc(gen_0':s3_0(b))) ->_R^Omega(1)
rand(gen_0':s3_0(n5_0), id_inc(gen_0':s3_0(b))) ->_R^Omega(1)
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) ->_IH
gen_0':s3_0(b)

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

TRS:
Rules:
nonZero(0') -> false
nonZero(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x
id_inc(x) -> x
id_inc(x) -> s(x)
random(x) -> rand(x, 0')
rand(x, y) -> if(nonZero(x), x, y)
if(false, x, y) -> y
if(true, x, y) -> rand(p(x), id_inc(y))

Types:
nonZero :: 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
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:
rand
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(9) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(10)
BOUNDS(n^1, INF)