/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 251 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), 86 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 98 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:

   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   average(x, y) -> aver(plus(x, y), 0)
   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
   if(true, sum, z) -> aver(sum, s(z))
   if(false, sum, z) -> z

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:

   gt(0', y) -> false
   gt(s(x), 0') -> true
   gt(s(x), s(y)) -> gt(x, y)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   average(x, y) -> aver(plus(x, y), 0')
   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
   if(true, sum, z) -> aver(sum, s(z))
   if(false, sum, z) -> z

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

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

(4)
Obligation:
TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 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

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
gt, plus, double, aver

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

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

(6)
Obligation:
TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 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:
gt, plus, double, aver

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

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

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

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

Induction Step:
gt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
gt(gen_0':s3_0(n5_0), gen_0':s3_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)

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

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

TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 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:
gt, plus, double, aver

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 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


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, 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:
plus, double, aver

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(n258_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n258_0, b)), rt in Omega(1 + n258_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(+(n258_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(plus(gen_0':s3_0(n258_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c259_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:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 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


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n258_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n258_0, b)), rt in Omega(1 + n258_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:
double, aver

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s3_0(n801_0)) -> gen_0':s3_0(*(2, n801_0)), rt in Omega(1 + n801_0)

Induction Base:
double(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s3_0(+(n801_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s3_0(n801_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c802_0))))

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

(16)
Obligation:
TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 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


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n258_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n258_0, b)), rt in Omega(1 + n258_0)
double(gen_0':s3_0(n801_0)) -> gen_0':s3_0(*(2, n801_0)), rt in Omega(1 + n801_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:
aver