/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 262 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), 73 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 133 ms]
        (16) typed CpxTrs


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

(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:

   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0, x), x, y)
   ifTimes(true, x, y) -> 0
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0) -> s(s(0))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0, s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0) -> divisible(x, y)
   div(0, y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   a -> b
   a -> c

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

(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:

   lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
   lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
   if(true, x, y, z, u) -> z
   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
   if2(true, x, y, z, u) -> z
   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(x, y) -> ifTimes(ge(0', x), x, y)
   ifTimes(true, x, y) -> 0'
   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
   p(s(x)) -> x
   p(0') -> s(s(0'))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   or(true, y) -> true
   or(false, y) -> y
   divisible(0', s(y)) -> true
   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
   div(x, y, 0') -> divisible(x, y)
   div(0', y, s(z)) -> false
   div(s(x), y, s(z)) -> div(x, y, z)
   a -> b
   a -> c

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

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

(4)
Obligation:
TRS:
Rules:
lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) -> z
if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) -> z
if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) -> 0'
ifTimes(false, x, y) -> plus(y, times(y, p(x)))
p(s(x)) -> x
p(0') -> s(s(0'))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
or(true, y) -> true
or(false, y) -> y
divisible(0', s(y)) -> true
divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
div(x, y, 0') -> divisible(x, y)
div(0', y, s(z)) -> false
div(s(x), y, s(z)) -> div(x, y, z)
a -> b
a -> c

Types:
lcm :: 0':s -> 0':s -> 0':s
lcmIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
times :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
or :: true:false -> true:false -> true:false
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
divisible :: 0':s -> 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
ifTimes :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
lcmIter, times, ge, divisible, plus, div

They will be analysed ascendingly in the following order:
ge < lcmIter
divisible < lcmIter
plus < lcmIter
ge < times
plus < times
divisible = div

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

(6)
Obligation:
TRS:
Rules:
lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) -> z
if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) -> z
if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) -> 0'
ifTimes(false, x, y) -> plus(y, times(y, p(x)))
p(s(x)) -> x
p(0') -> s(s(0'))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
or(true, y) -> true
or(false, y) -> y
divisible(0', s(y)) -> true
divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
div(x, y, 0') -> divisible(x, y)
div(0', y, s(z)) -> false
div(s(x), y, s(z)) -> div(x, y, z)
a -> b
a -> c

Types:
lcm :: 0':s -> 0':s -> 0':s
lcmIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
times :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
or :: true:false -> true:false -> true:false
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
divisible :: 0':s -> 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
ifTimes :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
ge, lcmIter, times, divisible, plus, div

They will be analysed ascendingly in the following order:
ge < lcmIter
divisible < lcmIter
plus < lcmIter
ge < times
plus < times
divisible = div

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> true, rt in Omega(1 + n6_0)

Induction Base:
ge(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
true

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:

TRS:
Rules:
lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) -> z
if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) -> z
if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) -> 0'
ifTimes(false, x, y) -> plus(y, times(y, p(x)))
p(s(x)) -> x
p(0') -> s(s(0'))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
or(true, y) -> true
or(false, y) -> y
divisible(0', s(y)) -> true
divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
div(x, y, 0') -> divisible(x, y)
div(0', y, s(z)) -> false
div(s(x), y, s(z)) -> div(x, y, z)
a -> b
a -> c

Types:
lcm :: 0':s -> 0':s -> 0':s
lcmIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
times :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
or :: true:false -> true:false -> true:false
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
divisible :: 0':s -> 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
ifTimes :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
ge, lcmIter, times, divisible, plus, div

They will be analysed ascendingly in the following order:
ge < lcmIter
divisible < lcmIter
plus < lcmIter
ge < times
plus < times
divisible = div

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) -> z
if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) -> z
if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) -> 0'
ifTimes(false, x, y) -> plus(y, times(y, p(x)))
p(s(x)) -> x
p(0') -> s(s(0'))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
or(true, y) -> true
or(false, y) -> y
divisible(0', s(y)) -> true
divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
div(x, y, 0') -> divisible(x, y)
div(0', y, s(z)) -> false
div(s(x), y, s(z)) -> div(x, y, z)
a -> b
a -> c

Types:
lcm :: 0':s -> 0':s -> 0':s
lcmIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
times :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
or :: true:false -> true:false -> true:false
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
divisible :: 0':s -> 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
ifTimes :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> true, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
plus, lcmIter, times, divisible, div

They will be analysed ascendingly in the following order:
divisible < lcmIter
plus < lcmIter
plus < times
divisible = div

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s4_0(n342_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n342_0, b)), rt in Omega(1 + n342_0)

Induction Base:
plus(gen_0':s4_0(0), gen_0':s4_0(b)) ->_R^Omega(1)
gen_0':s4_0(b)

Induction Step:
plus(gen_0':s4_0(+(n342_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
s(plus(gen_0':s4_0(n342_0), gen_0':s4_0(b))) ->_IH
s(gen_0':s4_0(+(b, c343_0)))

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

(14)
Obligation:
TRS:
Rules:
lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) -> z
if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) -> z
if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) -> 0'
ifTimes(false, x, y) -> plus(y, times(y, p(x)))
p(s(x)) -> x
p(0') -> s(s(0'))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
or(true, y) -> true
or(false, y) -> y
divisible(0', s(y)) -> true
divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
div(x, y, 0') -> divisible(x, y)
div(0', y, s(z)) -> false
div(s(x), y, s(z)) -> div(x, y, z)
a -> b
a -> c

Types:
lcm :: 0':s -> 0':s -> 0':s
lcmIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
times :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
or :: true:false -> true:false -> true:false
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
divisible :: 0':s -> 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
ifTimes :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> true, rt in Omega(1 + n6_0)
plus(gen_0':s4_0(n342_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n342_0, b)), rt in Omega(1 + n342_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
times, lcmIter, divisible, div

They will be analysed ascendingly in the following order:
divisible < lcmIter
divisible = div

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div(gen_0':s4_0(n1203_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1203_0))) -> false, rt in Omega(1 + n1203_0)

Induction Base:
div(gen_0':s4_0(0), gen_0':s4_0(b), gen_0':s4_0(+(1, 0))) ->_R^Omega(1)
false

Induction Step:
div(gen_0':s4_0(+(n1203_0, 1)), gen_0':s4_0(b), gen_0':s4_0(+(1, +(n1203_0, 1)))) ->_R^Omega(1)
div(gen_0':s4_0(n1203_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1203_0))) ->_IH
false

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

(16)
Obligation:
TRS:
Rules:
lcm(x, y) -> lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) -> if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) -> z
if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) -> z
if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(x, y) -> ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) -> 0'
ifTimes(false, x, y) -> plus(y, times(y, p(x)))
p(s(x)) -> x
p(0') -> s(s(0'))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
or(true, y) -> true
or(false, y) -> y
divisible(0', s(y)) -> true
divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
div(x, y, 0') -> divisible(x, y)
div(0', y, s(z)) -> false
div(s(x), y, s(z)) -> div(x, y, z)
a -> b
a -> c

Types:
lcm :: 0':s -> 0':s -> 0':s
lcmIter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
times :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
or :: true:false -> true:false -> true:false
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if2 :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
divisible :: 0':s -> 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
ifTimes :: true:false -> 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> true, rt in Omega(1 + n6_0)
plus(gen_0':s4_0(n342_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n342_0, b)), rt in Omega(1 + n342_0)
div(gen_0':s4_0(n1203_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1203_0))) -> false, rt in Omega(1 + n1203_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
divisible, lcmIter

They will be analysed ascendingly in the following order:
divisible < lcmIter
divisible = div