/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 269 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), 88 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
        (16) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   check(0) -> zero
   check(s(0)) -> odd
   check(s(s(0))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0)
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0) -> 0
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   check(0') -> zero
   check(s(0')) -> odd
   check(s(s(0'))) -> even
   check(s(s(s(x)))) -> check(s(x))
   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> timesIter(x, y, 0')
   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
   p(s(x)) -> x
   p(0') -> 0'
   if(zero, x, y, z, u) -> z
   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
check(0') -> zero
check(s(0')) -> odd
check(s(s(0'))) -> even
check(s(s(s(x)))) -> check(s(x))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0')
timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
p(s(x)) -> x
p(0') -> 0'
if(zero, x, y, z, u) -> z
if(odd, x, y, z, u) -> timesIter(p(x), y, u)
if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s -> zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s -> 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
timesIter :: 0':s -> 0':s -> 0':s -> 0':s
if :: zero:odd:even -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
check, half, plus, timesIter

They will be analysed ascendingly in the following order:
check < timesIter
half < timesIter
plus < timesIter

----------------------------------------

(6)
Obligation:
Innermost TRS:
Rules:
check(0') -> zero
check(s(0')) -> odd
check(s(s(0'))) -> even
check(s(s(s(x)))) -> check(s(x))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0')
timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
p(s(x)) -> x
p(0') -> 0'
if(zero, x, y, z, u) -> z
if(odd, x, y, z, u) -> timesIter(p(x), y, u)
if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s -> zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s -> 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
timesIter :: 0':s -> 0':s -> 0':s -> 0':s
if :: zero:odd:even -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
hole_zero:odd:even1_0 :: zero:odd:even
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:
check, half, plus, timesIter

They will be analysed ascendingly in the following order:
check < timesIter
half < timesIter
plus < timesIter

----------------------------------------

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

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

Induction Step:
check(gen_0':s3_0(+(1, *(2, +(n5_0, 1))))) ->_R^Omega(1)
check(s(gen_0':s3_0(*(2, n5_0)))) ->_IH
odd

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
check(0') -> zero
check(s(0')) -> odd
check(s(s(0'))) -> even
check(s(s(s(x)))) -> check(s(x))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0')
timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
p(s(x)) -> x
p(0') -> 0'
if(zero, x, y, z, u) -> z
if(odd, x, y, z, u) -> timesIter(p(x), y, u)
if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s -> zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s -> 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
timesIter :: 0':s -> 0':s -> 0':s -> 0':s
if :: zero:odd:even -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
hole_zero:odd:even1_0 :: zero:odd:even
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:
check, half, plus, timesIter

They will be analysed ascendingly in the following order:
check < timesIter
half < timesIter
plus < timesIter

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
check(0') -> zero
check(s(0')) -> odd
check(s(s(0'))) -> even
check(s(s(s(x)))) -> check(s(x))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0')
timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
p(s(x)) -> x
p(0') -> 0'
if(zero, x, y, z, u) -> z
if(odd, x, y, z, u) -> timesIter(p(x), y, u)
if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s -> zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s -> 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
timesIter :: 0':s -> 0':s -> 0':s -> 0':s
if :: zero:odd:even -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) -> odd, 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, plus, timesIter

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

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n285_0))) -> gen_0':s3_0(n285_0), rt in Omega(1 + n285_0)

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

Induction Step:
half(gen_0':s3_0(*(2, +(n285_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n285_0)))) ->_IH
s(gen_0':s3_0(c286_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
check(0') -> zero
check(s(0')) -> odd
check(s(s(0'))) -> even
check(s(s(s(x)))) -> check(s(x))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0')
timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
p(s(x)) -> x
p(0') -> 0'
if(zero, x, y, z, u) -> z
if(odd, x, y, z, u) -> timesIter(p(x), y, u)
if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s -> zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s -> 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
timesIter :: 0':s -> 0':s -> 0':s -> 0':s
if :: zero:odd:even -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) -> odd, rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n285_0))) -> gen_0':s3_0(n285_0), rt in Omega(1 + n285_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:
plus, timesIter

They will be analysed ascendingly in the following order:
plus < timesIter

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(n655_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n655_0, b)), rt in Omega(1 + n655_0)

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

Induction Step:
plus(gen_0':s3_0(+(n655_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(plus(gen_0':s3_0(n655_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c656_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
check(0') -> zero
check(s(0')) -> odd
check(s(s(0'))) -> even
check(s(s(s(x)))) -> check(s(x))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> timesIter(x, y, 0')
timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
p(s(x)) -> x
p(0') -> 0'
if(zero, x, y, z, u) -> z
if(odd, x, y, z, u) -> timesIter(p(x), y, u)
if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))

Types:
check :: 0':s -> zero:odd:even
0' :: 0':s
zero :: zero:odd:even
s :: 0':s -> 0':s
odd :: zero:odd:even
even :: zero:odd:even
half :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
timesIter :: 0':s -> 0':s -> 0':s -> 0':s
if :: zero:odd:even -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
hole_zero:odd:even1_0 :: zero:odd:even
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
check(gen_0':s3_0(+(1, *(2, n5_0)))) -> odd, rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n285_0))) -> gen_0':s3_0(n285_0), rt in Omega(1 + n285_0)
plus(gen_0':s3_0(n655_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n655_0, b)), rt in Omega(1 + n655_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:
timesIter