/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 266 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), 72 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:

   1024 -> 1024_1(0)
   1024_1(x) -> if(lt(x, 10), x)
   if(true, x) -> double(1024_1(s(x)))
   if(false, x) -> s(0)
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   10 -> double(s(double(s(s(0)))))

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:

   1024' -> 1024_1(0')
   1024_1(x) -> if(lt(x, 10'), x)
   if(true, x) -> double(1024_1(s(x)))
   if(false, x) -> s(0')
   lt(0', s(y)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   10' -> double(s(double(s(s(0')))))

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

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

(4)
Obligation:
Innermost TRS:
Rules:
1024' -> 1024_1(0')
1024_1(x) -> if(lt(x, 10'), x)
if(true, x) -> double(1024_1(s(x)))
if(false, x) -> s(0')
lt(0', s(y)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
10' -> double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
10' :: 0':s
true :: true:false
double :: 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
1024_1, lt, double

They will be analysed ascendingly in the following order:
lt < 1024_1
double < 1024_1

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

(6)
Obligation:
Innermost TRS:
Rules:
1024' -> 1024_1(0')
1024_1(x) -> if(lt(x, 10'), x)
if(true, x) -> double(1024_1(s(x)))
if(false, x) -> s(0')
lt(0', s(y)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
10' -> double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
10' :: 0':s
true :: true:false
double :: 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
lt, 1024_1, double

They will be analysed ascendingly in the following order:
lt < 1024_1
double < 1024_1

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

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

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) ->_R^Omega(1)
true

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_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:

Innermost TRS:
Rules:
1024' -> 1024_1(0')
1024_1(x) -> if(lt(x, 10'), x)
if(true, x) -> double(1024_1(s(x)))
if(false, x) -> s(0')
lt(0', s(y)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
10' -> double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
10' :: 0':s
true :: true:false
double :: 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
lt, 1024_1, double

They will be analysed ascendingly in the following order:
lt < 1024_1
double < 1024_1

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
1024' -> 1024_1(0')
1024_1(x) -> if(lt(x, 10'), x)
if(true, x) -> double(1024_1(s(x)))
if(false, x) -> s(0')
lt(0', s(y)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
10' -> double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
10' :: 0':s
true :: true:false
double :: 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, 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:
double, 1024_1

They will be analysed ascendingly in the following order:
double < 1024_1

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s3_0(n293_0)) -> gen_0':s3_0(*(2, n293_0)), rt in Omega(1 + n293_0)

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

Induction Step:
double(gen_0':s3_0(+(n293_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s3_0(n293_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c294_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:
1024' -> 1024_1(0')
1024_1(x) -> if(lt(x, 10'), x)
if(true, x) -> double(1024_1(s(x)))
if(false, x) -> s(0')
lt(0', s(y)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
10' -> double(s(double(s(s(0')))))

Types:
1024' :: 0':s
1024_1 :: 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
10' :: 0':s
true :: true:false
double :: 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
double(gen_0':s3_0(n293_0)) -> gen_0':s3_0(*(2, n293_0)), rt in Omega(1 + n293_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:
1024_1