/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 254 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), 83 ms]
        (14) 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:

   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   log(0) -> logError
   log(s(x)) -> loop(s(x), s(0), 0)
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
   if(true, x, y, z) -> z
   if(false, x, y, z) -> loop(x, double(y), 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:

   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   log(0') -> logError
   log(s(x)) -> loop(s(x), s(0'), 0')
   loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
   if(true, x, y, z) -> z
   if(false, x, y, z) -> loop(x, double(y), s(z))

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

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

(4)
Obligation:
Innermost TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
log(0') -> logError
log(s(x)) -> loop(s(x), s(0'), 0')
loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) -> z
if(false, x, y, z) -> loop(x, double(y), s(z))

Types:
le :: s:0':logError -> s:0':logError -> false:true
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat -> s:0':logError

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, double, loop

They will be analysed ascendingly in the following order:
le < loop
double < loop

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(6)
Obligation:
Innermost TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
log(0') -> logError
log(s(x)) -> loop(s(x), s(0'), 0')
loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) -> z
if(false, x, y, z) -> loop(x, double(y), s(z))

Types:
le :: s:0':logError -> s:0':logError -> false:true
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat -> s:0':logError


Generator Equations:
gen_s:0':logError3_0(0) <=> 0'
gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x))


The following defined symbols remain to be analysed:
le, double, loop

They will be analysed ascendingly in the following order:
le < loop
double < loop

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
le(gen_s:0':logError3_0(+(1, 0)), gen_s:0':logError3_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_s:0':logError3_0(+(1, +(n5_0, 1))), gen_s:0':logError3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) ->_IH
false

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

(8)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
log(0') -> logError
log(s(x)) -> loop(s(x), s(0'), 0')
loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) -> z
if(false, x, y, z) -> loop(x, double(y), s(z))

Types:
le :: s:0':logError -> s:0':logError -> false:true
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat -> s:0':logError


Generator Equations:
gen_s:0':logError3_0(0) <=> 0'
gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x))


The following defined symbols remain to be analysed:
le, double, loop

They will be analysed ascendingly in the following order:
le < loop
double < loop

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
log(0') -> logError
log(s(x)) -> loop(s(x), s(0'), 0')
loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) -> z
if(false, x, y, z) -> loop(x, double(y), s(z))

Types:
le :: s:0':logError -> s:0':logError -> false:true
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat -> s:0':logError


Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0':logError3_0(0) <=> 0'
gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x))


The following defined symbols remain to be analysed:
double, loop

They will be analysed ascendingly in the following order:
double < loop

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_s:0':logError3_0(n294_0)) -> gen_s:0':logError3_0(*(2, n294_0)), rt in Omega(1 + n294_0)

Induction Base:
double(gen_s:0':logError3_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_s:0':logError3_0(+(n294_0, 1))) ->_R^Omega(1)
s(s(double(gen_s:0':logError3_0(n294_0)))) ->_IH
s(s(gen_s:0':logError3_0(*(2, c295_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:
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
log(0') -> logError
log(s(x)) -> loop(s(x), s(0'), 0')
loop(x, s(y), z) -> if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) -> z
if(false, x, y, z) -> loop(x, double(y), s(z))

Types:
le :: s:0':logError -> s:0':logError -> false:true
s :: s:0':logError -> s:0':logError
0' :: s:0':logError
false :: false:true
true :: false:true
double :: s:0':logError -> s:0':logError
log :: s:0':logError -> s:0':logError
logError :: s:0':logError
loop :: s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
if :: false:true -> s:0':logError -> s:0':logError -> s:0':logError -> s:0':logError
hole_false:true1_0 :: false:true
hole_s:0':logError2_0 :: s:0':logError
gen_s:0':logError3_0 :: Nat -> s:0':logError


Lemmas:
le(gen_s:0':logError3_0(+(1, n5_0)), gen_s:0':logError3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
double(gen_s:0':logError3_0(n294_0)) -> gen_s:0':logError3_0(*(2, n294_0)), rt in Omega(1 + n294_0)


Generator Equations:
gen_s:0':logError3_0(0) <=> 0'
gen_s:0':logError3_0(+(x, 1)) <=> s(gen_s:0':logError3_0(x))


The following defined symbols remain to be analysed:
loop