/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 247 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), 65 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 53 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:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))

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:

   division(x, y) -> div(x, y, 0')
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0') -> false
   lt(0', s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0') -> s(0')
   inc(s(x)) -> s(inc(x))

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

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

(4)
Obligation:
TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
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:
div, lt, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

(6)
Obligation:
TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
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, div, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

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

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

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
lt(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:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
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, div, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
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(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:
inc, div, minus

They will be analysed ascendingly in the following order:
inc < div
minus < div

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s3_0(n257_0)) -> gen_0':s3_0(+(1, n257_0)), rt in Omega(1 + n257_0)

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

Induction Step:
inc(gen_0':s3_0(+(n257_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s3_0(n257_0))) ->_IH
s(gen_0':s3_0(+(1, c258_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:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
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(n5_0)) -> false, rt in Omega(1 + n5_0)
inc(gen_0':s3_0(n257_0)) -> gen_0':s3_0(+(1, n257_0)), rt in Omega(1 + n257_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:
minus, div

They will be analysed ascendingly in the following order:
minus < div

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n467_0), gen_0':s3_0(n467_0)) -> gen_0':s3_0(0), rt in Omega(1 + n467_0)

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

Induction Step:
minus(gen_0':s3_0(+(n467_0, 1)), gen_0':s3_0(+(n467_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n467_0), gen_0':s3_0(n467_0)) ->_IH
gen_0':s3_0(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:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
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(n5_0)) -> false, rt in Omega(1 + n5_0)
inc(gen_0':s3_0(n257_0)) -> gen_0':s3_0(+(1, n257_0)), rt in Omega(1 + n257_0)
minus(gen_0':s3_0(n467_0), gen_0':s3_0(n467_0)) -> gen_0':s3_0(0), rt in Omega(1 + n467_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:
div